Chapter Name

The differential calculus (Secs. 4.5-1 to 4.5-7) describes relations between small changes of suitable variables. The integral calculus (Secs.4.6-1 to 4.6-19, 4.9-3, and 4.9-4) yields measures for over-all or average properties of a set of objects and furnishes techniques for adding many small changes. Sections 4.8-1 to 4.9-2 outline the properties of infinite series, and Secs. 4.10-1 to 4.11-8 deal with the representation of functions by power series, Fourier series, and Fourier integrals.

The definitions and theorems presented in this chapter apply to complex functions and variables as well as to real functions and variables unless a restriction to real quantities is specifically indicated. Analytic functions of a complex variable are discussed in Chap. 7.

4.2. FUNCTIONS

4.2-1. Functions and Variables (see also Sec. 12.1-4). (a) Given a rule of correspondence which associates a real or complex number

with each given real or complex number x of a set S_x, y is called a (numerical) function y = y(x) = ƒ(x) of the argument x. Equation (1) specifies a value (or values) y = Y = ƒ(X) of the variable y corresponding to each suitable value x = X of the variable x. If the relation (1) is primarily intended to describe the dependence of y on x, x is called the independent variable, and y is called the dependent variable.

The term “variable x” essentially refers to a set of values X, and Eq. (1) symbolizes a set of correspondences relating values X of x and values Y = ƒ(X) of y. In order to conform with the notation employed in most textbooks, the symbol x will be used to denote both the variable x and a value of the variable x whenever this notation does not result in ambiguities.

If one interprets x and y as plane cartesian coordinates (Sec. 2.1-2), a real function y = ƒ(x) of a real variable x is often represented by a curve (graph of y vs. x, see also Sec. 2.1-9).

(b) A function

of n variables x₁, x₂, . . . , x_n similarly associates values of a (dependent) variable y with ordered sets of values of the (independent) variables x₁, x₂. . . , x_n.

A function may be defined by a table of function values, or by rules for computing such a table by means of known operations (constructive definitions). A function may also be defined implicitly (Sec. 4.5-7), or in terms of defining properties described by functional, differential, or integral equations, extreme-value properties (Sec. 11.5-2), behavior for certain values of the argument, etc. Each nonconstructive definition requires an existence proof demonstrating, by example or construction, a function with the specified properties.

(c) In most applications, the variables x, y or x₁, x₂, . . . , x_n, y label physical objects or quantities, so that suitable relations (1) or (2) describe physical relationships (EXAMPLE: y = x₁x₂ if x₁, x₂, and y respectively label the voltage, current, and power in a simple electric circuit).

(d) The set S_x of values of x (or of sets of values of x₁, x₂, . . . , x_n) for which the relationship (1) or (2) is defined is the domain of definition of the function f(x) or ƒ(x₁, x₂, . . . , x_n). The corresponding set S_y of values of y is the range of the function.

(e) A sequence of real or complex numbers s₀, s₁, s₂, . . . represents a function s_n = s_n(n) defined on the set of nonnegative integers n.

4.2-2. Functions with Special Properties (see also Secs. 1.4-3, 7.3-3, 7.6-5, and 7.6-7). (a) A function is single-valued * wherever a single function value corresponds to the value of the argument. A function is multiple-valued wherever two or more function values correspond to the value of the argument. The function y(x) has an inverse function x(y) if y = y(x) implies x = x(y) for all x in S_x.

(b) A function ƒ(x) of a real or complex variable x is even if and only if ƒ( — x) ≡ ƒ (x); odd if and only if ƒ( — x) ≡ — ƒ(x); periodic with the period T if and only if ƒ(t + T) ≡ ƒ(t).

Every function ƒ(x) defined for the values of x in question can be expressed as the sum of the even function 1/2[ƒ(x) + ƒ(— x) and the odd function 1/2[ƒ(x) — ƒ(— x]. A periodic function ƒ(t) with period T is antiperiodic if and only if ƒ(t+ T/2) = — ƒ(t). Every periodic function ƒ(t)can be expressed as the sum of the antiperiodic function 1/2[ƒ(t) — ƒ(t + T/2)] and the function 1/2[ƒ(t)+ ƒ(t + T/2)], which is periodic with period T/2.

(c) y = ƒ(x) is an algebraic function of x if and only if x and y satisfy a relation of the form F(x, y) = 0, where F(x,y) is a polynomial in x and y(Sec. 1.4-3). In particular, y = f(x) is a rational (rational algebraic) function of x if ƒ(x) is a polynomial (integral rational function) or a quotient of two polynomials (fractional rational function). y is a linear function of x if y = ax + b.

4.3. POINT SETS, INTERVALS, AND REGIONS

4.3-1. Introduction. When discussing the properties of a function ƒ(x) of a real variable x, one is often required to specify a set of values of x such that ƒ(x) is defined and satisfies given conditions. Note that either functions or sets may be described in this manner. It is customary to refer to the values * of a real variable x (or to objects labeled by values of x) as points(x)of a line, and to sets of such real numbers as linear point sets.

Properties of a function f(x₁, x₂,. . . , x_n) of n real variables x₁, x₂, . . . , x_n are similarly related to sets of “points” (x₁, x₂, . . . , x_n) in an n-dimensional “space” which comprises all points (x₁, x₂,. . . , x_n) under consideration (Sec. 14.1-2).

The use of geometrical language is prompted by the Cantor-Dedekind axiom of continuity, which postulates the existence of a one-to-one reciprocal correspondence between the real numbers and the points of a straight line. This “coordinate axiom” (Sec. 2.1-2) is compatible with the properties of real numbers as well as with the postulates defining Euclidean and other geometries.

Sections 4.3-2 to 4.3-6 deal mainly with those properties of point sets which apply directly to the theory of functions of real variables; Secs. 7.2-2 to 7.2-4 deal with regions of the complex number plane. In a more general context, any set (class) of objects (in particular, objects labeled with values of a real variable or variables) may be referred to as a set of points. The properties of such sets are further discussed in Secs. 12.5-1 to 12.5-4.

4.3-2. Properties of Sets. (a) Algebra of Sets (Classes). An object (point) P contained in a set (class) S is an element of S (P ∊ S). A set S₁ is a subset of another set S₂ (S₁ is contained in S₂, S₁ ⊂ S₂) if

and only if each element of S₁ is an element of S₂. S₁ and S₂ are equal (S₁ = S₂) if and only if both contain the same elements, i.e., if and only if S₁ ⊂ S₂ and S₂ ⊂ S₁. The empty set 0 is, by definition, a subset of every set S. A proper subset (proper part) of S is a nonempty subset of S not equal to S. The union (join, logical sum) S₁ ∪ S₂ (or S₁ + S₂) is the set of all elements contained in either S₁ or S₂, or both. The intersection (meet, logical product) S₁ ∩ S₂ (or S₁S₂) of S₁ and S₂ is the set of all elements contained in both S₁ and S₂. The complement of a set S with respect to a set I containing S is the set of all elements of I not contained in S. The subsets of any set (class) I constitute a Boolean algebra (Sec. 12.8-1) under the operations of logical addition and multiplication.

(b) Cardinal Numbers and Countability. Two sets S₁ and S₂ have the same cardinal number if and only if there exists a reciprocal one-to-one correspondence between their respective elements. S is an infinite set if it has the same cardinal number as one of its proper subsets; otherwise, S is a finite set.

A finite or infinite set S is countable (enumerable, denumerable) if and only if it is possible to establish a reciprocal one-to-one correspondence between its elements and those of a set of real integers. Every finite set is countable; the cardinal number of a finite set is identical with the number of its elements. Every subset of a countable set is countable. The union of a countable set of countable sets is a countable set.

Cardinal numbers corresponding to infinite sets are called transfinite numbers. The cardinal number of every countable infinite set is the same as the cardinal number of the set of the positive real integers and is denoted by The set of all real numbers (or the set of points of a continuous line, Sec. 4.3-1) is not countable; the corresponding cardinal number is denoted by

4.3-3. Bounds. (a) A real number M is an upper bound or a lower bound of a set S_y of real numbers y if and only if, respectively, y ≤ M or y ≥ M for all y in S_y. A set of real or complex numbers is bounded (has an absolute bound) if the set of their absolute values has an upper bound; otherwise the set is unbounded.

Every (nonempty) set S_y of real numbers y having an upper bound has a least upper bound(l.u.b.) Sup y, and every (nonempty) set of real numbers y having a lower bound has a greatest lower bound (g.l.b.) inf y. If S_y is finite, sup y is necessarily equal to the maximum value max y actually assumed by a number y in S_y, and inf y is equal to the minimum min y.

EXAMPLE: The set of all real numbers less than 1 has the least upper bound 1, but no maximum.

(b) A real or complex function y = ƒ(x) or y = ƒ(x₁,x₂, . . . , x_n)is bounded on a set S of “points” (x) or (x₁, x₂, . . . , x_n) if and only if the corresponding set S_y of function values y is bounded. Similarly, a real function y = ƒ(x) or y = ƒ(x₁, x₂, . . . ,x_n) has an upper bound, lower bound, least upper bound, greatest lower bound, (absolute) maximum, and/or (absolute) minimum on a set S of “points” (x) or (x₁, x₂, . . . , x_n) if this is true of the corresponding set S_y of function values y.

(c) A real or complex function ƒ(x,y) or ƒ(x₁, x₂, . . . , x_n; y)is uniformly bounded on a set S of “points” (x) or (x₁, x₂, . . . , x_n) if and only if ƒ as a function of y has an upper bound independent of x or x₁, x₂, . . . , x_n on S. Uniform upper bounds and uniform lower bounds are similarly defined.

4.3-4. Intervals (see also Secs. 4.3-3 and 4.3-5). Given a real variable x, the set of all values of x (points) such that

1.a < x < b is the bounded open interval (a, b).

2.a < x is the unbounded open interval (a, ∞).

3.x < a is the unbounded open interval (— ∞, a).

4.a ≤ x ≤ b is the bounded closed interval [a, b].

Sets of points (x) such that a ≤ x < b, a < x ≤ b, a ≤ x, x ≤ a may be referred to as semiclosed intervals. Every interval I₁ contained in another interval I₂ is a subinterval of I₂.

4.3-5. Definition of Neighborhoods. (a) Given any finite real number a, an (open) δ-neighborhood of the point (x = a) in the space of real numbers is any open interval (a — δ, a + δ) containing x = a; or, the set of all points (x) such that |x — a| < δ, for some positive real number δ. A neighborhood of x = a is any set containing a δ-neighborhood of x = a.

(b) Every set containing all points (x) such that x > M for some real number M is a neighborhood of plus infinity (+ ∞) in the space of real numbers; every set containing all points (x) such that x < N for some real number N is a neighborhood of minus infinity (— ∞) in the space of real numbers.

(c) In a space whose “points” are (described as) ordered sets (x₁x₂, …, x_n) of real numbers, one may define an (open) δ-neighborhood of the point (a₁, a₂, …, a_n), where a₁, a₂, … , a_n are finite, as the set of all points (x₁, x₂, …, x_n)such that |x₁ — a₁| < δ, |x₂ — a₂| < δ, …, and |x_n — a_n| < δ for some positive real number δ. A neighborhood of the point (a₁, a₂,…, a_n) is any set containing a δ-neighborhood of this point.

NOTE: Definitions of neighborhoods, like those given above, are not necessarily self-evident; they amount to postulates defining the “topological” properties of the space in question (see also Sec. 12.5-1). In particular, neighborhoods involving unbounded values of the variables [as in (b) above] may be defined in a variety of ways (thus, + ∞ and — ∞ may or may not be regarded as the same point, see also Sec. 7.2-2). In applied mathematics, the choice of such definitions will depend on the nature of the objects “represented” by the points (x) or (x₁, x₂ …, x_n). The definition of neighborhoods is closely related to the definition of open sets (Sec. 4.3-6a), regions (Sec. 4.3-6b), and limits of functions on the space in question (Sec. 4.4-1; see also Sec. 12.5-3).

4.3-6. Open and Closed Sets and Regions. All the following definitions imply a specific definition of neighborhoods in a space containing the sets and regions in question (topology, see also Secs. 12.5-1 to 12.5-4).

(a) A point P is a limit point (cluster point, accumulation point) of the point set S if and only if every neighborhood of P contains points of S other than P itself. A limit point P is an interior point of S if and only if S is a neighborhood of P; otherwise, P is a boundary point of S. A point P is an isolated point of S if and only if P has a neighborhood in which P is the only point belonging to S.

A point set S is

An open set if and only if it contains only interior points.

A closed set if and only if it contains all its limit points; a finite set is closed.

A discrete (isolated) set if and only if it contains only isolated points; every discrete set is countable.

(See also Sec. 12.5-4),

(b) In the Euclidean plane (Sec. 2.1-1) or space, a simply connected open region D is an open set of points such that every closed curve in D can be continuously contracted into a point without leaving D. If a region is not simply connected, it is said to be multiply connected. A point P is on the boundary of the region D if every neighborhood of P contains points in D and points not in D; the boundary of a simply connected open region is a simple surface or simple closed surface. Connected regions in D are subregions of D. An open region and its boundary or boundaries constitute a closed region. A region of a Euclidean plane or space is bounded if and only if all its points can be described by bounded cartesian coordinates (see also Sec. 3.1-2).

4.4. LIMITS, CONTINUOUS FUNCTIONS, AND RELATED TOPICS

4.4-1. Limits of Functions and Sequences (see also Secs. 4.8-1 and 12.5-3; see Table 4.7-1 for examples). (a) A single-valued function ƒ(x) has (approaches, converges to, tends to)

A (necessarily finite and unique) limit as x approaches a finite value x = a [ƒ(x) → L as x → a] if and only if for each positive real number ∊ there exists a real number δ > 0 such that 0 < |x — a| < δ implies that ƒ (x) is defined and |f(x) — L| < ∊.

A (necessarily finite and unique) limit as x increases indefinitely [increases without bound, tends to infinity; ƒ(x) → L as x → ∞ ] if and only if for each positive real number ∊ there exists a real number N such that x > N implies that f(x) is defined and |f(x) — L| < ∊.

(b) A sequence of numbers (Sec. 4.2-le) s₀, s₁, s₂, … [ ≡ s(n)] converges to a (necessarily finite and unique) limit if and only if for each positive real number ∊ there exists a real integer N such that n > N implies |s_n — S| < ∊. Criteria for the convergence of sequences are presented in Sec. 4.9-1 in connection with the convergence of infinite series.

(c) A real function ƒ(x) increases indefinitely (increases without bound, tends to infinity)

As x approaches a finite value x = a [ƒ(x) → ∞ as x → a; some authors write ] if and only if for each positive real number M there exists a real number δ > 0 such that 0 < |x — a| < δ implies that ƒ (x) is defined, and f(x) > M.

As x increases indefinitely [ƒ(x) → ∞ as x → ∞ ; some authors write ∞] if and only if for each real number M there exists a real number N such that x > N implies that ƒ(x) is defined and ƒ(x) > M.

These definitions apply, in particular, to indefinitely increasing sequences. A real variable x or ƒ(x) decreases indefinitely [x→ — ∞ or ƒ(x) → — ∞] if, respectively, — x → ∞, or —ƒ(x) → ∞. Section 4.4-1 specifies the mathematical meaning of infinity in the context of the real-number system (see also Secs. 4.3-5 and 7.2-2).

4.4-2. Operations with Limits (see also Secs. 4.4-6c and 4.8-4). If the limits in question exist,

a may be finite or infinite; these rules apply to limits of sequences (Sec. 4.4-1b) and also to multiple limits (Sec. 4.4-5).

4.4-3. Asymptotic Relations between Two Functions (see also Sec. 4.8-6). Given two real or complex functions ƒ(x), g(x) of a real or complex variable x, one writes

ƒ(x = 0[g(x)]{f(x) is 0[g(x)], f(x)is of the order of g(x)} as x →a if and only if there exists a neighborhood of x = a such that |f(x/g(x| is bounded.

f(x) ~g(x) [f(x) is asymptotically proportional to g(x)]as x →a if and only if exists and differs from zero.

f(x) g(x) [f(x) is asymptotically equal to f(x)]as x → a if and only if this implies that the percentage difference between f(x) and g(x) converges to zero as x →a.

f(x) = o[g(x)] as x→a if and only if This may often be read “f(x) becomes negligible compared with g(x) as x →a.”

In each of the above definitions, a may be finite or infinite. Functions of order 1, 2, . . . and functions of exponential order are functions of the order of x, x², . . . , and e^x as x → ∞;functions of order — 1, — 2, . . . are functions of the order of x^—l, x^—2, . . . as x →0.

Asymptotic relations often yield estimates or approximations of f(x) in terms of g(x) in a neighborhood of x = a; note that f{x) ~g(x) implies that the fractional error decreases in absolute value as x→ a. One writes

4.4-4. Uniform Convergence.(a) A single-valued function f(x₁,x₂) converges uniformly on a set S of values of x₂

To the (necessarily finite and unique) function if and only if for each positive real number ∊ there exists a real number δ> 0 such that 0 < |x₁ — a| < δ implies that f(x₁,x₂) is defined and |f(x₁, x₂) — L(x₂)| < ∊ for all x₂ in S (δis independent of x₂).

To the (necessarily finite and unique) function if and only if for each positive real number ∊ there exists a real number N such that x₁ > N implies that f(x₁, x₂) is defined and |f (x₁, x₂₁) — L(x₂)| < ∊ for all x₂ in S.

(b) A sequence of functions s_o(x),S₁(x), s₂(x), …converges uniformly on a set S of values of x to the (necessarily finite and unique) function

if and only if for each positive real number ∊ there exists a real integer N such that n > N implies |s_n(x) — s(x)|< ∊for all x in S (see also Secs. 4.6-2c and 4.8-2).

4.4-5. Multiple Limits and Iterated Limits. (a) Given the definition of a neighborhood formulated in Sec. 4.3-5c, a single-valued function f{x₁, x₂) has the(necessarily finite and unique)limit if and only if for each positive real number ∊ there exists a neighborhood D of the point (a₁, a₂)such that f(x₁, x₂) is defined and |f(x1₁, x₂ |— L| < ∊ for all points (x₁, x₂)in D, except possibly at (a₁ a₂ ).a₁ and/or a₂ may be finite or infinite.

In particular, a double sequence s₀₀, s₁₀, … converges to a limit if and only if for each positive real number ∊ there exists a pair of real numbers M,N such that ∞ > m > M, ∞ > n > N implies |s_mn — s |< ∊. Limits of functions of more than two variables are defined in an analogous manner.

(b) If there exists a positive real number δ such that

and at least one of these limiting processes converges uniformly in the interval specified, then the three limits exist and are equal. Analogous theorems apply if a₁ and/or a₂ are infinite.

4.4-6. Continuous Functions. (a) A single-valued function f(x) defined throughout a neighborhood of x = a is continuous at x = a [at the point (x = a)] if and only if exists and equals f(a), i.e., if and only if for every positive real number ∊ there exists a real number δ > 0 such that |x — a| < δ implies |f(x) — f(a)| < ∊.

Similarly, a single-valued function f(x₁, x₂, . . . , x_n) defined throughout a neighborhood of the point (a₁, a₂ … , a_n) is continuous at (a₁, a₂, . . . , a_n) if and only if

A single-valued function f(x₁, x₂, . . . , a;_n) is continuous in x₁ at the point (a₁, a₂, . . . , a_n) if and only if f(x₁, a₂, . . . , a_n) is continuous for x₁= a₁. A function continuous in x₁, x₂ . . . , and x_n separately at (a₁, a₂, . . . , a_n) is not necessarily continuous at (a₁, a₂, . . . , a_n).

(b) A function is continuous on a set of points (e.g., an interval or region) if and only if it is continuous at each point of the set. A real function continuous on a bounded closed interval [a, b] is bounded on [a, b] and assumes every value between and including its g.l.b. and its l.u.b. (Sec. 4.3-3) at least once on [a, b]. An analogous theorem holds for a real function of two or more variables continuous on a bounded singly connected closed region.

f{x) is uniformly continuous on S if and only if for each positive real number∊ there exists a real number δ such that |x — X| < δ implies |f(x) — f(X)| < ∊ for all X in S. A function continuous on a bounded closed interval [a, b] is uniformly continuous on [a, b].

(c) If two functions f and g are continuous at a given point x or (x₁, x₂, . . . , x_n), the same is true for f + g and fg. Given and a function F(y₁, y₂, … , y_n) continuous for y₁ = A₁ y₂ = A₂, . . . , y_n = A_n,

(see also Sec. 4.4-2). In particular, if each y_i(x) is continuous for x = a, the same is true for F[y₁ (x), y₂(x), . . . , y_n(x)].

The limit s(x) of a uniformly convergent sequence of functions S₀ (x), S₁ (x), . . . all continuous on a set S of values of x is continuous on S.

4.4-7. One-sided (Unilateral) Limits. Unilateral Continuity.(a) A function f(x) of a real variable x has the (necessarily finite and unique) right-hand limit at x = a if and only if for each positive real number ∊ there exists a real number δ > 0 such that 0 < x — a < δ implies that f(x) is defined, and |f(x) — L₊| < ∊. f(x) has the left-hand limit≡f(a — 0) = L_— at x = a if and only if for each positive real number ∊ there exists a real number δ > 0 such that 0 < a — x < δ implies that f(x) is defined, and |f(x) — L_—| < ∊. If exists, then

Conversely, implies the existence of

(b) f(x) is right continuous or left continuous at x = a if f(a + 0) = f(a) or f(a — 0) = f(a), respectively. A discontinuity of the first kind of a real function f(x) is a point x = a such that f(a + 0) and f(a — 0) exist; the greatest difference between two of the numbers f(a), f(a + 0), f(a — 0) is the saltus of f(x) at such a discontinuity. The discontinuities of the first kind of f(x) constitute a discrete (and thus countable) set (Sec. 4.3-6a).

(c) f(x) is piecewise continuous on an interval I if and only if f(x) is continuous throughout I except for a finite number of discontinuities of the first kind.

f(x₁, x₂, . . . , x_n) is piecewise continuous on a region V of n-dimensional space if and only if f(x₁ x₂, . . . , x_n )is continuous throughout V, except possibly on a set of regular hypersurfaces (regular curves for n = 2, regular surfaces for n = 3, Secs. 3.1-13 and 3.1-14) which divide V into a finite number of subregions such that f(x ₁, x₂, … , x_n )has a finite and unique unilateral limit on approaching any boundary point of a subregion from its interior.

4.4-8. Monotonic Functions and Functions of Bounded Variation.(a) A real function f(x) of a real variable x is strongly monotonic in (a, b) if f(x )increases as x increases in (a, b )(increasing or positive monotonic function), or if f(x) decreases as x increases in (a, b) (decreasing or negative monotonic function). f(x is weakly monotonic in (a, b )if f(x )does not decrease in (a,b )(nondecreasing function), or if f(x)does not increase in (a, b)(nonincreasing function). Analogous definitions apply to monotonic sequences (Sec. 4.2-ld).

(b) A real function f(x)of a real variable x is of bounded variation in the interval (a,b ) if and only if there exists a real number M such that for all partitions a = x₀ < X₁ < x₂ < ...< x_m = b of the interval (a,b ). f(x)is of bounded variation in (a,b ) if and only if it can be expressed in the formf(x) ≡ f₁(x)— f₂(x), where f₁(x) and f₂(x) are bounded and nondecreasing in (a,b )(alternative definition). If f(x) and g(x) are of bounded variation in (a, b), the same is true for f(x) + g(x) and f(x)g(x). f(x) is of bounded variation in every finite open interval where f(x) is bounded and has a finite number of relative maxima and minima (Sec. 11.2-1) and discontinuities (Dirichlet conditions).

A function of bounded variation in (a,b ) is bounded in (a,b ), and its only discontinuities are discontinuities of the first kind (Sec. 4.4-7).

In physical applications, the condition that f(x) is of bounded variation in every bounded interval expresses the fact that f(x) is bounded, and that components of very high frequency cannot contribute significantly to its total intensity (Sec. 18.10-1).

4.5.DIFFERENTIAL CALCULUS

4.5-1. Derivatives and Differentiation. (a) Let y= f(x) be a real, single-valued function of the real variable x throughout a neighborhood of the point (x ). The (first, first-order) derivative or (first-order) differential coefficient of f(x)with respect to x at the point (x )is the limit

The function dy/dx = ƒ′(x)is a measure of the rate of change of y with respect to x at each point (x)where the limit (1) exists. On a graph of y = f(x) (Sec. 4.2-la), ƒ′(x)corresponds to the slope of the tangent (Sec. 17.1-1).

The corresponding unilateral limits (Sec. 4.4-7a) are the left-hand derivative ƒ′ _- (x)and the right-hand derivative ƒ′ ₊ (x)of f(x)at the point (x).

(b) The second, third, . . . , n^th derivatives (second-order, third-order, . . . , n^th-order differential coefficients) of y = f(x) with respect to (x)at the point (x) are respectively defined as

if the limits in question exist.

Table 4.5-1. Derivatives of Frequently Used Functions (see also Chap. 21 for derivatives of special functions)

(c) The operation of determining ƒ′(x)for a given function f(x) is called differentiation of f(x) with respect to x. f(x)is differentiable for any value or set of values of (x)such that f(x)exists; f(x)is continuously differentiable (smooth) wherever ƒ′(x)exists and is continuous. f(x)is piecewise continuously differentiable on an interval I if and only if f(x)and ƒ′(x)exist and are piecewise continuous (Sec. 4.4-7c) on I. A function is continuous wherever it is differentiable.

Derivatives of a number of frequently used functions are tabulated in Table 4.5-1. Additional derivatives may be obtained through use of the differentiation rules of Sec. 4.5-3.

4.5-2. Partial Derivatives. (a) Let y = f(x₁ x₂, . . . , x_n) be a real single-valued function of the real variables x₁ x₂ . . . , x_n in a neighborhood of the point (x₁, x₂, . . . , x_n). The (first-order) partial derivative of f(x₁, x₂, . . . , x_n) with respect to (x₁ at the point x₂, . . . , x_n)is the limit

The function is a measure of the rate of change of y with respect to(x₁) for fixed values of the remaining independent variables at each point (x₁ x₂, . . . , x_nwhere the limit (3) exists. The partial derivatives dy/dx₂, dy/dx₃. . . , dy/dx_n are defined in an analogous manner. Each partial derivative dy/dx_k may be found by differentiation of f(x₁ x₂, . . . , x_n) with respect to x_k while the remaining n — 1 independent variables are regarded as constant parameters [partial differentiation of f (x₁ x₂, . . . , x_n)with respect to x_k].

(b) Higher-order partial derivatives of y = f(x₁ x₂, . . . , x_n) are defined by

if the limits in question exist; in each case, the number of differentiations involved is the order of the partial derivative. Note that

if (1) exists throughout a neighborhood of the point (x1, x2 . . . , x_n) and is continuous at (x₁ x₂, . . . , x_n), and (2) exists at (x₁ x₂, . . . , x_n).

(c) y = f(x₁, x₂,. . . , x_n) is differentiable with respect to x_k at any point (x₁, x₂,. . . , x_n) where dy/dx_kexists. y = f(x₁, x₂,. . . , x_n) is differentiable wherever dy/dx₁, dy/dx₂,. . . , and dy/dx_n exist. y = f(x₁, x₂,. . . , x_n) is continuously differentiable wherever all these partial derivatives exist and are continuous. y = f(x₁, x₂,. . . , x_n)is piecewise continuously differentiable on a region V if and only if f(x₁, x₂. . . , x_n), dy/dx₁, dy/dx₂,. . . , and dy/dx_n exist and are piecewise continuous on V (Sec. 4.4-7c).

4.5-3. Differentials. (a) Given an arbitrary small change (increment) dx of the independent variable x (differential of the independent variable x), the corresponding term in the expansion

(Sec. 4.10-4) of a differentiable function y = f(x)is called the (first-order) differential of the dependent variable y at the point (x). Similarly, the (first-order) differential of a differentiable function

of n variables x₁, x₂,...,x_n is

The operation of obtaining dy ≡ df(x₁, x₂,...,x_n) is sometimes called total differentiation, and df is referred to as a total differential made up of the ‘partial changes’ (∂f/∂xk) dx_k.

(b) The differential of each independent variable is regarded as a constant, so that d²x ≡ d(dx) ≡ d^zx = d(d³x) ≡ d(d²x) ≡ · · · ≡ 0. The differential of a dependent variable is a function of the independent variable or variables. The second-order, third-order, . . . differentials of suitably differentiable functions are obtained by successive differentiations of the first-order differential, e.g.,

Note also

(c) Given a problem involving n independent variables x₁, x₂,. . . , x_n any function of the order of dx₁^r₁ dx₂^r₂ . . . dx_n^r_n as dx₁ → 0, dx₂ → 0, . . . , dx_n → 0 (Sec. 4.4-3) is an infinitesimal of order r₁ + r₂ + … + r_n. In particular, the r^th-order differential d^rf of a suitably differentiable function is an infinitesimal of order r.

4.5-4. Differentiation Rules. (a) Table 4.5-2 summarizes the most important differentiation rules. The formulas of Table 4.5-2a and b apply to partial differentiation if d/dx_k is substituted for d/dx in each case. Thus, if u_i = u_i(x₁, x₂,. . . , x_n)(i = 1, 2, . . . ,m),

Multiplication of each formula of Table 4.5-2a and 6 by dx or dx^r yields an analogous rule for total differentiation (see also Sec. 4.5-3); thus

Refer to Table 4.6-1 for the differentiation of integrals, and to Sec. 4.8-4c for the differentiation of infinite series.

4.5-5. Homogeneous Functions. f(x₁, x₂,. . . , x_n)is homogeneous of degree r in its arguments x₁, x₂,. . . , x_n if and only if f{αx₁, αx₂, . . . , αx_n) ≡ α^rf(x₁, x₂, . . . , x_n)(see also Sec. 1.4-3a). If f(x₁, x₂, . . . , x_n) is continuously differentiable and homogeneous of degree r, then

4.5-6. Jacobians and Functional Dependence (see also Secs. 4.6-12, 6.2-3b, and 16.1-2). A set of transformation equations (see also Sec. 14.1-3)

define a reciprocal one-to-one correspondence between sets (x₁, x₂, . . . , x_n) and (y₁, y₂, . . . , y_n)throughout a neighborhood of a “point” (x₁, x₂,

Table 4.5.2 Differentiation Rules (Sec.4.5-4; existence of continuous derivatives is assumed in each case)

(a) Basic Rules

(b) Sums, Products, and Quotients. Logarithmic Differentiation

NOTE: To differentiate functions of the form it may be convenient to find the logarithmic derivative first.

(d) Implicit Functions (see also Sec. 4.5-7). If y=y(x) is given implicity in terms of a suitably differentiable relation F(x, y) =0, where F_y ≠ 0,

(e) Function Given in Terms of a Parameter t. Given x = x(t,

. . . , x_n) where the functions (13) are single-valued and continuously differentiable, and where the Jacobian or functional determinant

is different from zero.

The Jacobian (14) vanishes whenever the differentials

are linearly dependent (Sec. 1.9-3). If the functions (13) are continuously differentiable throughout a neighborhood of a point (x₁, x₂, . . . , x_n) where the rank of the matrix [dy_i/dx_kk] (Sec. 13.2-7) is everywhere m < n, then (x₁, x₂, . . . , x_n) has a neighborhood V where the y_i are related by a continuously differentiable relation

such that the dΦ/dy_i do not all vanish simultaneously. In this case the functions (13) are said to be functionally dependent over V (see also Sec. 4.5-7c).

4.5-7. Implicit Functions. (a) If a function y = y(x) is given implicitly in terms of a suitably differentiable relation F(x, y ) = 0, then

(b) If m functions y₁ = y₁(x₁, x₂ ,. . . , x_n), y₂ = y₂(x₁, x₂, . . . , x_n), …, y_m = y_m (x₁, x₂, . . . , x_n) of n independent variables x₁, x₂, . . . , x_n are given implicitly in terms of m continuously differentiable relations

(4.5-17)

where the F_i are single-valued functions, then

1.The differentials dy_j and dx_k are related by m linear equations dF_i = 0.

2.For each value of k = 1, 2, . . . , n, the m derivatives dy_j/dx_k may be obtained by Cramer's rule (Sec. 1.9-2) from the m linear equations

(4.5-18)

provided that

for the values of x₁, x₂, . . . , x_n in question. Differentiation of the Eqs. (18) yields relations involving higher derivatives of the y_j.

In particular, two continuously differentiable relations F(x,y,z)= 0, G(x, y,z)=0 imply

whenever the determinant in the denominator is different from zero. The computation of implicit derivatives is of particular importance in thermodynamics (Ref. 4.6).

(c) An Existence Theorem for Implicit Functions (see also Sec. 4.2-1). Given a “point” P ≡ (x₁, x₂, . . . , x_n) (y₁, y₂, . . . , y_m) such that Eqs. (17) and (19) hold, m relations (17) define the y, as single-valued continuous functions within a neighborhood of the “point” (x₁, x₂, . . . , x_n) if all F_i and ∂F_i/∂y_j exist and are continuous throughout a neighborhood D of P. If, in addition, all ∂F_i/∂x_k exist and are continuous throughout D, then the derivatives dy_j/dx_k exist and are continuous within a neighborhood of (x₁, x₂, . . . , x_n). If the Jacobian (19) vanishes, then the y_j are not uniquely determined.

4.6. INTEGRALS AND INTEGRATION

4.6-1. Definite Integrals (Riemann Integrals). (a) A real function f(x) bounded on the bounded closed interval [a, b]is integrable over (a,b) in the sense of Riemann if and only if the sum tends to a unique finite limit I for every sequence of partitions a = x₀ < ξ x₁ < ξ₂ < x₂ ... < ξ_m < x_m = b asmax |x_i — x_i-1| → 0. In this case

is the definite integral of f(x) over (a, b) in the sense of Riemann (Riemann integral). f(x) is called the integrand; a and b are the limits of integration. Table 4.6-1 summarizes important properties of definite integrals.

represents the area bounded by the curve y = f(x) (Sec. 4.2-la) and the x axis between the lines x = a and x = b; areas below the x axis are represented by negative numbers.

(b) A function f(x) bounded on the bounded closed interval [a,b] is integrable over (a, b) in the sense of Riemann if and only if f(x) is continuous almost everywhere on (a, b) Sec. 4.6-14b). This is true, in particular, (1) if f(x) is continuous on [a, b] (2) if f(x)is bounded on (a, b) and continuous on (a, b) except possibly for a discrete set of discontinuities; (3) if f(x) is bounded and monotonic on (a, b); (4) if f(x) is of bounded variation in (a, b)(see also Sec. 4.4-8). If f(x) is integrable over (on) (a, b), f(x) is necessarily integrable over every subinterval of (a, b).

4.6-2. Improper Integrals. (a) Given a function f(x)bounded and integrable on every bounded subinterval of (a,b )the concept of a definite integral(Sec. 4.6-1) can be extended to apply even if

Table 4.6-1. Properties of Integrals

(a) Elementary Properties. If the integrals exist,

(b) Integration by Parts. If u(x) and v(x) are differentiable for a ≤ x ≤ b, and if the integrals exist,

(c) Change of Variable (Integration by Substitution). If u = u(x) and its inverse function x = x(u) are single-valued and continously differentiable for a ≤ x ≤ b, and if the integral exists,

(d) Differentiation with Respect to a Parameter. If f(x, λ), u(λ), and v(λ) are continuously differentiable with respect to λ,

provided that the integrals exist and, in the case of improper integrals, converge uniformly in a neighborhood of the point (λ).

The second case can often be reduced to the first by a suitable change of variables. Note also

(e) Inequalities (see also sec. 4.6-19). If the integrals exist,

If|f(x)| ≤ M on the bounded interval (a, b), the existence of implies the existence of , and

1.f(x)is unbounded in a neighborhood of a finite limit of integration x = a or x = b(see also Sec. 4.6-2b).

2.The interval (a,b )is unbounded.

Thus, if f(x) is bounded and integrable on every finite interval (a, X) for a < X < b, one defines

and, in particular, for b = ∞

(b) The integration rules of Table 4.6-1 apply to suitably convergent improper integrals. Given a bounded or unbounded interval (a,b) or [a, b] containing a discrete set of points x = c₁, x = c₂,. . . such that f(x)is unbounded in a neighborhood of x = c_i (i = 1, 2, . . . ), may be defined as an improper integral equal to a sum of improper integrals (2) ; e.g.,

if the limits exist.

Even if the integrals (4) and (5) do not exist, their respective Cauchy principal values

may exist; if either integral exists it is necessarily equal to its principal value.

(c) An improper integral converges uniformly on a set S of values of y if and only if the corresponding limit (Secs. 4.6-2a and b) converges uniformly on S (Sec. 4.4-4). If f(x, y) is a continuous function, then is a continuous function of y in every open interval where the integral converges uniformly (Continuity Theorem).

(d) Criteria for convergence and uniform convergence of improper integrals are listed in Secs. 4.9-3 and 4.9-4.

4.6-3. Arithmetic Means. The arithmetic means (averages) of f(x)over the respective intervals (a, b), (0, ∞), and (— ∞, ∞) are defined as

if these quantities exist.

4.6-4. Indefinite Integrals. A given single-valued function f(x) has an indefinite integral F(x) in [a, b] if and only if there exists a function F(x) such that F′(x) = f(x) in [a, b]. In this case F(x) is uniquely defined in [a, b] except for an arbitrary additive constant C (constant of integration; see also Sec. 9.1-2); one writes

Note that is uniquely defined for a ≤ x ≤ b.

Note also

if the indefinite integrals exist (see also Tables 4.5-2 and 4.6-1).

4.6-5. The Fundamental Theorem of the Integral Calculus. If f(x) is single-valued, bounded, and integrable on [a, b], and there exists a function F(x) such that F′(x) = f(x) for a ≤ x ≤ b, then

(4.6-11)

In particular, if f(x) is continuous in [a, b ],

and Eq. (11) applies.

More generally, whenever exists, the function exists and is continuous and of bounded variation (Sec.4.4-8b) in (a, b), and Eq. (12) holds for almost all x in [a, b] (Sec. 4.6-14b).

NOTE: The fundamental theorem of the integral calculus enables one (1) to evaluate definite integrals by reversing the process of differentiation, and (2) to solve differential equations by numerical evaluation of definite integrals (see also Sec. 4.6-6).

4.6-6. Integration Methods. (a) Integration is the operation yielding a (definite or indefinite) integral of a given integrand f(x). Definite integrals may be calculated directly as limits of sums (numerical integration, Secs. 20.6-2 and 20.6-3) or by the calculus of residues (Sec. 7.7-3); more frequently, one attempts to find an indefinite integral which may be inserted into Eq. (11). To obtain an indefinite integral, one must reduce the given integrand f(x) to a sum of known derivatives with the aid of the “integration rules” listed in Table 4.6-l a, b, c.

The remainder of this section deals with integration methods applicable to special types of integrands. Comprehensive tables of definite and indefinite integrals are presented in Appendix E.

(b) Integration of Polynomials.

(c) Integration of Rational Functions. The methods outlined in Secs. 1.7-2 and 1.7-4 will reduce every rational integrand to the sum of a polynomial (Sec. 4.6-6b) and a set of partial fractions (1.7-5) and/or (1.7-6). The partial-fraction terms are integrated successively with the aid of the following formulas:

(d) Integrands Which Can Be Reduced to Rational Functions by a Change of Variables(Table 4.6-lc).

If the integrand f(x) is a rational function of sin x and cos x, introduce u = tan (x/2), so that

If the integrand f(x) is a rational function of sinh x and cosh x, introduce u = tanh (x/2), so that

NOTE: If f(x) is a rational function of sin² x, cos², x sin x cos x, and tan x (or of the corresponding hyperbolic functions), one simplifies the calculation by first introducing v = x/2, so that u =tan v (or u = tanh v).

3.If the integrand f(x) is a rational function of x and either or , reduce the problem to case 1 or 2 by the respective substitutions x = cos v or x= cosh v.

4.If the integrand f(x) is a rational function of x and , introduce , so that

5.If the integrand f(x) is a rational function of x and reduce the problem to case 3 (b² — 4ac < 0) or to case 4 (b² — 4ac > 0) through the substitution

6.If the integrand f(x) is a rational function of x and introduce u as a new variable.

7.If the integrand f(x) is a rational function of x, , and , introduce as a new variable.

Many other substitution methods apply in special cases. Note that the integrals may not be real for all values of x.

(e) Integrands of the form xⁿe^ax, xⁿ log_e x, xⁿ sin x, xⁿ cos x (n ≠ — 1); sin^mx cosⁿx (n + m ≠ 0); e^ax sinⁿx, e^ax cosⁿx yield to repeated integration by parts (Table 4.6-1b).

(f) Many integrals cannot be expressed as finite sums involving only algebraic, exponential, and trigonometric functions and their inverses. One may then expand the integrand as an infinite series (Secs. 4.10-4, 4.11-4, 15.2-6), or one resorts to numerical integration. Refer to Chap. 21 for examples of “new” functions of x defined as integrals of the form .

4.6-7. Elliptic Integrals. (a) If f(x) is a rational function of x and is called an elliptic integral; one may except the trivial case that the equation

has multiple roots. Every elliptic integral can be reduced to a weighted sum of elementary functions and normal elliptic integrals (Sec. 21.6-5); values of the latter are available in tabular form.

4.6-8. Multiple Integrals (see also Secs. 4.6-11 to 4.6-13). (a) Let f(x,y) be piecewise continuous (Sec. 4.4-7) on a bounded closed region D which is uniquely defined by a ≤ x ≤ b, g₁(x) ≤y ≤ g₂(x) as well as by α ≤ y ≤ β, γ₁(y) ≤ x ≤ γ₂(y), where g₁(x), g₂(x), γ₂(y), γ₂(y) are piecewise continuous functions. Then Fubini's theorem states

Analogous theorems hold for triple, quadruple, etc., integrals.

The second expression for the integral (19) may be written without the brackets if the meaning of the integration limits is evident.

EXAMPLE: If D is the region bounded by the circle x² + y² = 1, and f(x, y) = c = constant,

(b) If exists and equals for every b > a, then

provided that converges uniformly for α < y < β. Similar theorems hold for other improper multiple integrals.

4.6-9. Arc Length of a Rectifiable Curve (see also Secs. 6.2-3, 6.4-3, and 17.2-1). A continuous curve segment is rectifiable if and only if every subsegment C₁in the finite portion of the plane or space has a unique finite arc length s(C₁); s(C₁) is the limit of the length of an inscribed polygonal curve as the length of the largest straight-line segment approaches zero.

For a continuous arc C₁ in the Euclidean plane or space, described in terms of rectangular cartesian coordinates by

(4.6-21)

(Secs. 2.1-9b and 3.1-13), the length of an infinitesimal regular arc corresponding to the interval (t, t + dt) is the element of arc length

(4.6-22)

The sign of ds is assigned arbitrarily, usually so that ds/dt ≥ 0.

The arc length s(C₁) of a given curve segment C₁ corresponding to a finite interval (t_o, t)is

The arc length s(C₁) is a geometrical object independent of the coordinate system and the particular parameter t used to describe the curve; refer to Secs. 6.2-3 and 6.4-3 for formulas expressing ds (and hence ds/dt) in terms of curvilinear coordinates.

s(C₁) exists if the functions (21) are of bounded variation on [t₀, t]; in this case ds is defined almost everywhere (Sec. 4.6-14b) on C₁. Every regular curve is rectifiable.

4.6-10. Line Integrals (see also Secs. 5.4-5, 6.2-3, and 6.4-3 for vector notation and use of curvilinear coordinates). Given a rectifiable are C described by Eq. (21) for a ≤ t ≤ b, the line integral over a bounded function f(x, y, z) is defined by

with

if the limit exists (see also Sec. 4.6-1). The line integral (24a) can be computed (or directly denned) as an integral over t:

where the element of arc length ds is given by Eq. (22). Omit terms involving z in Eq. (24) for line integrals in the xy plane. Improper line integrals are defined in the manner of Sec. 4.6-2.

4.6-11. Areas and Volumes (see also Secs. 5.4-6, 5.4-7, 6.2-3, 6.4-3, and 17.3-3 for vector notation and use of curvilinear coordinates). (a) The area A of the region S bounded by a simple closed curve (Sec. 3.1-13) in the Euclidean plane is the least limit of the area of an inscribed closed polygon as the largest polygon side approaches zero. This limit exists, in particular, whenever the boundary has a finite arc length.

The surface area A of a curved surface region S in Euclidean space is the smallest possible limit of the area of an inscribed polyhedral surface as the maximum distance between adjacent vertices decreases. The volume U enclosed by a simple closed surface (Sec. 3.1-14) is the limit of the volume of the inscribed polyhedrons.

(b) Areas and volumes can be computed (or even directly defined) as double and triple integrals over suitable coordinates:

Areas and volumes are independent of the particular coordinate system used. In terms of curvilinear coordinates x¹, x², x³ (Sec. 6.2-1)*

In terms of curvilinear surface coordinates u, von a plane or curved surface (Sec. 3.1-14)

The function a(u, v) is specified in Sec. 17.3-3c.

(c) (c).Simple Special Cases. The plane area A bounded by the lines x = aand x = b and two nonintersecting curves y = g₁(x), y = g₂(x) is [A > 0 if g₁(x) < g₂(x); see also Sec. 4.6-1]. The plane area bounded by a suitable closed curve C described by x = x(t), y = y(t) or r = r(6)(Sec. 2.1-8) is

where the symbol signifies counterclockwise integration around the closed curve.

The surface area A and the internal volume U of a surface of revolution (Sec. 3.1-15) generated by rotation of the curve segment y = f(x) > 0 (a ≤ x ≤ b ) about the x axis are given by

4.6-12. Surface Integrals and Volume Integrals (see also Secs. 5.4-6, 5.4-7, 6.2-3, and 6.4-3 for vector notation and use of curvilinear coordinates). Surface integrals and volume integrals can be defined as limits of sums in the manner of Secs. 4.6-1 and 4.6-10; or they can be introduced directly as double and triple integrals with the aid of the elements of area and volume defined in Sec. 4.6-11. The surface integral of a piecewise continuous function f(u,v) over a suitable bounded surface region S with surface coordinates u, v is

This includes integrals over plane surface regions [and, in particular, also ] as special cases.

The volume integral of a piecewise continuous function f(x,y,z) f[x(x^l, x², x³), y(x^l, x², x³), z(x^l, x², x³)] over a bounded region V is

Improper surface and volume integrals (unbounded functions and regions) are defined in the manner of Sec. 4.6-2. Refer to Secs. 5.6-1 and 5.6-2 for formulas relating line integrals, surface integrals, and volume integrals. The following special cases relate line integrals and surface integrals (C is the boundary of S):

where

4.6-13. Change of Variables in Volume Integrals and Surface Integrals. If one introduces new coordinates x¹, x², x³ by a continuously differentiable coordinate transformation

(see also Sec. 6.2-1), each volume integral (32) can be rewritten in terms of the new coordinates:

Surface integrals are transformed in an analogous manner (see also Sec. 17.3-3).

4.6-14. Lebesgue Measure. Measurable Functions. (a) Measure of a Point Set. The exterior Lebesgue measure m_e[S] of any bounded linear point set S (Sec. 4.3-3a) is the g.l.b. of the combined length of a set of intervals covering S.The interior Lebesgue measure mⁱ[S] of S is the difference between the length b — a of any bounded interval (a,b) containing S and the exterior measure of the complement of S with respect to (a,b)(Sec. 4.3-2a). S is a measurable set with the Lebesgue measure m[S] if and only if m_e[S] = m[S]= m[S] (constructive definition of the Lebesgue measure). An unbounded linear point set S is measurable if and only if (—X, X) ∩ S is measurable for all X > 0. In this case one defines may or may not be finite.

More generally, a measure M[S] defined on a suitable class (completely additive Boolean algebra, Sec. .12.8-2) of point sets S is a set function with the properties

and, for every (finite or infinite) countable set of nonintersecting point sets S₁, s₂, . .

The Lebesgue measure m[S] of a linear point set has the additional property

for every bounded interval (a,b); the Lebesgue measure is thus a generalization of length (descriptive definition of the Lebesgue measure; see also Secs. 12.1-1, 12.8-2, and 18.2-2).

The Lebesgue measures of point sets in suitable spaces of two, three, . . . dimensions are defined (either constructively or descriptively) by analogous generalizations of area and volume.

(b) Every bounded open set (Sec. 4.3-6a) is measurable. More generally, a (linear) Borel set is obtained by a finite or infinite sequence of unions, intersections, and/or complementations of intervals and of the resulting combinations; the class of all Borel sets is a completely additive Boolean algebra of measurable sets. Every measurable set is the union of a Borel set and a set of Lebesgue measure zero. Every countable (discrete) set is measurable and has the Lebesgue measure zero. Analogous theorems apply in the multidimensional case.

A property which holds for every point of a given interval, region, or point set with the possible exception of a set of Lebesgue measure zero is said to hold almost everywhere on (at almost all points of) the given interval, region, or set.

(c) (c)Measurable Functions. A function f(x) defined on (a, b)is measurable on (a, b) if and only if the set of points (x) in (a,b)such that f(x)≤ c is measurable for every real value of c.

In this definition, the condition f(x) ≤ c can be replaced by any one of the conditions f(x) < c, f(x)≥ c, f(x) ≥ c.

Every function f(x) continuous on (a,b) is measurable on (a,b). If f₁(x),f₂(x), are measurable on(a,b), the same is true for f₁(x) + f₂(x), αf₁(x), and f₁(x),f₂(x), and also for if the limit exists on (a,b).

Analogous definitions and theorems apply to measurable functions f(x₁, x₂,. . . , x_n) defined on a space of “points” (x₁, x₂,. . . , x_n) permitting the definition of a Lebesgue measure.

4.6-15. Lebesgue Integrals (see also Secs. 4.6-1 and 4.6-2). (a) The Lebesgue Integral of a Bounded Function. Given a real function y = f(x) measurable and bounded on the bounded interval (a, b), subdivide the range of variation of f(x) on (a, b),

and let S_i be the set of points (x) in (a, b) such that y_i-1 < f(x) ≤ y_i. The sum tends to a unique finite limit I for every sequence of partitions (38) as max |y_i — y_i-1| → the quantity

is the definite integral of f(x) over (a, b) in the sense of Lebesgue (Lebesgue integral).

(b) The Lebesgue Integral of an Unbounded Function. If f(x) is measurable and unbounded on the bounded interval (a,b), the Lebesgue Integral is defined by

f(x)is integrable in the sense of Lebesgue (summable) over (a, b) if and only if exists.

(c) Lebesgue Integrals over Unbounded Intervals. If exists for all X > a, one defines the Lebesgue integral by

(d) Lebesgue Integral over a Point Set. Multiple Lebesgue Integrals. The definitions of Secs. 4.6-15a and b apply without change to the Lebesgue integral over any measurable set S of points (x). Analogous definitions apply to multiple Lebesgue integrals over regions or measurable sets of “points” (x₁, x₂. . . , x_n)

(e) Existence and Properties of Lebesgue Integrals. Lebesgue Integrals vs. Riemann Integrals. Every bounded measurable function is summable over any bounded measurable set. A function summable over (on) a measurable set S is summable over every measurable subset of S.

The definitions of Secs. 4.6-15a, b, c, and d imply that the Lebesgue integral exists if and only if the Lebesgue integral exists. Whenever any proper or improper, single or multiple Riemann integral exists in the sense of absolute convergence (Sec.4.6-2a),* the corresponding Lebesgue integral exists and is equal to the Riemann integral.

The theorems of Table 4.6-1 and Secs. 4.6-5 and 4.7-lc, d apply to Lebesgue integrals.

For every countable set of nonintersecting measurable point sets S₁, S₂, . . .

provided that the integrals exist. Every Lebesgue integral over a set of (Lebesgue) measure zero is equal to zero.

NOTE: Lebesgue integration applies to a more general class of functions than Riemann integration and simplifies the statement of many theorems. Many theorems stated in terms of Lebesgue integrals apply directly to absolutely convergent improper Riemann integrals (see also Secs. 4.6-16 and 4.8-4b).

4.6-16. Convergence Theorems (Continuity Theorems, see also Sec. 12.5-lc). (a) If the sequence of functions s₀(x), s₁(x), s₂(x), . . . each bounded and integrable in the sense of Riemann on the bounded interval (a, b) converges uniformly to s(x) on [a, b], then exists and equals .

(b) The following more powerful theorem is phrased specifically in terms of Lebesgue integrals: Let s₀(x), s₁(x), s₂(x), . . . , and a positive comparison function A(x) be summable over a measurable set S such that |s_n(x)| ≤ A(x) for all n and for almost all x in S (Sec. 4.6-14b). Then for almost all x in S implies that exists and equals (Lebesgue's Convergence Theorem; see also Sec. 4.8-4b). Note that uniform convergence is not required.

4.6-17. Stieltjes Integrals. (a) Riemann-Stieltjes Integrals. The Riemann-Stieltjes integral of f(x) with respect to g(x) over the bounded interval [a, b] is defined as

for an arbitrary sequence of partitions

* Note that is not summable over the interval (0, 1) even though the improper Riemann integral

exists.

The limit (43) exists whenever g(x) is of bounded variation (Sec. 4.4-8b), and f(x) is continuous on [a, b].

Improper Riemann-Stieltjes integrals may be defined in the manner of Sec. 4.6-2.

(b) Lebesgue-Stieltjes Integrals. Every function g(x) nondecreasing and right continuous (Sec.4.4-7b) on the bounded interval (a, b) defines a measure (Lebesgue-Stieltjes measure) M[S]for every Borel set (Sec. 4.6-14b) in (a, b) by the relations (35), (36), and

where the expression in the square brackets specifies a set of values of x. Note

Starting with the Lebesgue-Stieltjes measures of bounded intervals, one introduces the Lebesgue-Stieltjes measure M[S] of any measurable set as the common value of an inner and outer measure in the manner of Sec. 4.6-14a (see also Ref. 8.16).

Given a function y = f(x) bounded and measurable on (a, b), the Lebesgue-Stieltjes integral of f(x) with respect to g(x) over (a, b) is then defined as

for an arbitrary subdivision (38) of the range of f(x); S_i is, again, the set of values of x such that y_i-1 < f(x) < y_i.

The Lebesgue-Stieltjes integral of a bounded or unbounded function f(x) over any measurable set may now be defined in the manner of Secs. 4.6-15b, c, and d, with the understanding that g(x) is finite on every bounded set under consideration. In the multidimensional case, g(x) is replaced by a function g(x₁ x₂, … , x_n) non-decreasing with respect to each argument. One may, further, define the Lebesgue-Stieltjes integral with respect to any function g(x) of bounded variation by applying Eq. (48) to the sum of two monotonic functions (Sec. 4.4-8b). Each Lebesgue-Stieltjes integral equals the corresponding Riemann-Stieltjes integral if the latter exists in the sense of absolute convergence.

(c) Properties of Stieltjes Integrals (see also Table 4.6-1). Given a bounded or unbounded interval (a, b) such that the integrals in question exist,

If g(x) is a nondecreasing function on (a, b), then

If g(x) is nondecreasing and f(x) ≤ F(x) on (a, b), then

Stieltjes integrals often have an “intuitive” meaning (line integrals, surface integrals, volume integrals; integrals over distributions of mass, charge, and probability, Sec. 18.3-6). Note that Stieltjes integrals include ordinary integrals and sums as special cases:

whenever g(x) is continuously differentiable on (a, b), and

4.6-18. Convolutions. The Stieltjes convolution of two functions f(x) and g(x) over the interval (a, b) is defined as the function

(4.6-55)

for all values of t such that the two integrals exist and are equal. The classical convolution of f(x) and g(x) over (a, b) is similarly defined as

(4.6-56)

In the literature, either (55) or (56) is often simply referred to as the convolution of f(x) and g(x) over (a, b), with either function denoted by or f * g; the precise meaning is usually evident from the context (see also Secs. 4.11-5, 8.3-1, 8.3-3, and 18.5-7). For -a = b = ∞, Eqs. (55) and (56) hold whenever the integrals exist.

The convolution of two finite or infinite sequences a(0), a(l), a(2), … and b(0), b(1), b(2), … is the sequence

If Eq. (55) or (56) holds,

4.6-19. Minkowski's and Hölder's Inequalities. (a) Given a bounded or unbounded interval (a, b) such that the integrals on the right exist,

These inequalities hold, in particular, for ordinary Riemann and Lebesgue integrals and, more generally, for multidimensional integrals. Equation (60) reduces to the Cauchy-Schwarz inequality (15.2-3) for p = p /(p - 1) = 2.

(b) Equations (59) and (60) imply analogous inequalities for sums and for convergent infinite series. Whenever the sums on the right exist,

Equation (62) reduces to the Cauchy-Schwarz inequality for p = p /(p - 1) = 2 (see also Sec. 1.3-2).

4.7.MEAN-VALUE THEOREMS. VALUES OF INDETERMINATE
FORMS. WEIERSTRASS'S APPROXIMATION THEOREMS

4.7-1. Mean-value Theorems (see also Sec. 4.10-4). The following theorems are useful for estimating values and limits of real functions, derivatives, and integrals.

(a) If f(x) is continuous on [a, b] and continuously differentiable on (a, b) there exists a real number X in(a, b) such that

X is often written as X= a + ϑ(b - a) (0 < ϑ < 1). For f(a) = f(b) = 0 the theorem is known as Rolle's theorem (see also Sec. 1.6-6e).

If f(x₁, x₂, … , x_n) is continuous for a_i ≤ x_i ≤ b_i and continuously differentiable for a_i < x_i < b_i (i = 1, 2, … , n), there exists a set of real numbers X₁ X₂, … , X_n such that X_i= a_i + ϑ(b_i - a_i) (0 < ϑ < 1, i = 1, 2, … , n) and

(b) If (1) u(x) and v(x) are continuous on [a, b], (2) v(b) ≠ v(a), and (3) u′ (x) and v′ (x) exist and do not vanish simultaneously on (a, b), then there exists a real number X in (a, b) such that

(d) If f(x) and g(x) are continuous on [a, b], and g(x) ≠ 0 on [a, b], there exists a value X of x in (a, b) such that

If f(x) and g(x) are continuous on [a, b] and g(x) is a nondecreasing or nonincreasing function on (a, b), then there exists a value X of x in (a, b) such that

If, in addition, g(x) > 0 on (a, b), there exists a value X of x in (a, b) such that

4.7-2. Values of Indeterminate Forms (see Table 4.7-1 for examples). (a) Functions f(x) of the form u(x) / v(x), u(x)v(x), [u(x)]^v(x), and u(x) - v(x) are not defined for x = a if f(a) takes the form 0/0, ∞ / ∞, 0. ∞, 0⁰, ∞ ⁰, 1^∞, or ∞ - ∞ ; but may exist. In such cases it is often desirable to define .

(b) Treatment of 0 / 0 and ∞ / ∞. Let u{a) = v(a) = 0. If there exists a neighborhood of x = a such that (1) v(x) ≠ 0, except for x = a, and (2) u′ (x) and v′ (x) exist and do not vanish simultaneously, then

whenever the limit on the right exists (L'Hôpital's Rule).

Let If there exists a neighborhood of x = a such that x ≠ a implies (1) u(x) ≠ 0, v(x) ≠ 0, and (2) u′(x) and v′(x) exist and do not vanish simultaneously, then Eq. (8) holds whenever the limit on the right exists.

If u′(x) / v′(x) is itself an indeterminate form, the above method may be applied to u′(x) / v′(x)in turn, so that

If necessary, this process may be continued.

(c) Treatment of 0.∞, 0⁰, ∞⁰, 1^∞ and ∞ – ∞. u(x)v(x), [u(x)]^v(x), and u(x) – v(x) can often be reduced to the form φ(x) / ψ(x) with the aid of one of the following relations:

so that the methods of Sec. 4.7-2b become applicable.

(d) It is often helpful to write and to isolate terms of the order of Δx by algebraic manipulation or by a Taylor-series expansion (Sec. 4.10-4).

(e) The methods of Secs. 4.7-2a, b, c, and d are readily modified to apply to the one-sided limits and (Sec. 4.4-7). To find , use .

Table 4.7-1. Some Frequently Used Limits (Values of Indeterminate Forms, Sec. 4.7-2)

4.7-3. Weierstrass's Approximation Theorems (see also Secs. 4.10-4, 4.11-7, and 12.5-4b). Let f(x) be a real function continuous on the bounded closed interval [a, b]. Then for every given positive real number є (maximum error of approximation) there exists

1. A real polynomial such that | f(x) - P(x) | < є for all x in [a, b]

2. A real trigonometric polynomial (α_k cos kwx + β_k sin kwx) such that | f(x) - T(x) | < є for all x in [a, b]

Analogous theorems hold for functions of two or more variables. The Weierstrass approximation theorems permit one to derive various properties of continuous functions from corresponding properties of polynomials or trigonometric polynomials.

4.8. INFINITE SERIES, INFINITE PRODUCTS,
AND CONTINUED FRACTIONS

4.8-1. Infinite Series. Convergence (see also Sec. 4.4-1). An infinite series (infinite sum) a₀ + a₁ + a₂ + … of real or complex numbers (terms) a_o, a₁, a₂, … converges if and only if the sequence s₀, s₁, s₂, … of the partial sums has a (necessarily finite and unique) limit s, i.e., if and only if the sequence of the remainders R_n+1 = s – s_n converges to zero. In this case s is called the sum of the infinite series, and it is permissible to write

A (necessarily convergent) infinite series a₀ + a₁ + a₂ + … is absolutely convergent if and only if the series |a₀| + |a₁| + |a₂| + … converges. An infinite series which does not converge is divergent (diverges, see also Sec. 4.8-6). Sections 4.9-1 and 4.9-2 list a number of tests for the convergence of a given infinite series.

4.8-2. Series of Functions. Uniform Convergence (see also Sec. 4.4-4). An infinite series of functions a₀(x) + a₁(x) + a₂(x)+ … converges to a function (sum) s(x) for every value of x such that

The series converges uniformly to s(x) on a set S of values of x if and only if the sequence of partial sums converges uniformly to s(x) on S. Section 4.9-2 lists a number of tests for uniform convergence.

4.8-3. Operations with Convergent Series (see also Table E-l).

(a) Addition and Multiplication by Constants. If and are convergent series of real or complex terms, and a is a real or complex number, then

In each case the convergence or absolute convergence of the series on the left implies the same for the series on the right.

(b) Rearrangement of Terms. An unconditionally convergent series is a convergent series which converges to the same limit after every rearrangement of its terms; this is true if and only if the series is absolutely convergent. Every subseries (obtained by omission of terms) of such a series is absolutely convergent.

The terms of every conditionally convergent (i.e., not absolutely convergent) infinite series of real terms can be rearranged (1) so that the sequence of partial sums converges to any given limit, (2) so that the sequence of partial sums increases indefinitely, (3) so that the sequence of partial sums decreases indefinitely, and (4) so that the sequence of partial sums oscillates between any two given limits (Riemann's Theorem).

whenever the three limits exist (Pringsheim's Theorem on Summation by Columns and Rows). In particular, converges to a unique limit for any order of summation if the double series is absolutely convergent, i.e., if converges (see also Sec. 4.8-3b).

(d) Product of Two Infinite Series. If and are absolutely convergent series of real or complex terms, then the double series converges absolutely to the limit . More generally,

whenever the three infinite series converge (see also Appendix E).

4.8-4. Operations with Infinite Series of Functions. (a) Addition and Multiplication by Bounded Functions. If and converge uniformly (Sec. 4.8-2) on a set S of values of x, the same is true for and for where φ(x) is any function bounded for all x in S.

(b) Limits, Continuity, and Integration (see also Sec. 4.4-5).

Let converge uniformly on a bounded open interval (a, b) containing x = x₀ and x = x₁. Then

(

In each of these relations the given conditions imply the convergence of the series on the right.

The following theorem is phrased specifically in terms of Lebesgue integrals and may be applied to suitable absolutely convergent improper Riemann integrals (see also Sec. 4.6-15). If the infinite series of functions summable over a (bounded or unbounded) measurable set (or interval) S converges for almost all x in S, then

(4.8-5)

provided that there exists a real number A [or a function A(x) summable over S, Sec. 4.6-16] such that for all n and for almost all x in S. Note that uniform convergence is not required.

(c) Differentiation. Let converge for at least one value of x in (a, b). Then, if a′₀(x), a′₁(x), a′₂(x), … exist and converges uniformly on (a, b)

whenever the derivative on the left exists. Under the given conditions this is necessarily true if each a′_k(x) is continuous on (a, b).

4.8-5. Improvement of Convergence and Summation of Series (see also Secs. 7.7-4 and 20.4-3a). (a) Euler's Transformation. If the infinite series on the left converges,

(4.8-7)

where the differences Δ^ka₀ are defined as in Sec. 20.4-la. The series on the right often converges more rapidly than the series on the left.

(b) Kummer's Transformation. Given two convergent series and

If S is known, Kummer's transformation may be useful in numerical calculations (Ref. 20.11; see Sec. E-5 for useful S series).

(c) Poisson's Summation Formula. If the infinite series converges uniformly to a function of bounded variation in the interval 0 ≤ t < 2π, then

provided that the integrals on the right exist.

(d) The Euler-MacLaurin Summation Formula. The following relation often yields closed expressions, convergent approximation series, or semiconvergent approximation series (Sec. 4.8-6a) for finite sums; the formula is also used for numerical integration (Sec. 20.6-2b). If f^(2m+2) (x) exists and is continuous for 0 ≤x ≤n,

where the B_i are the Bernoulli numbers defined in Sec.21.5-2. Note also the following special summation formulas:

(see also Secs. 1.2-6 and 1.2-7).

(e) Abel's Lemma. The following lemma is sometimes useful for estimating partial sums. Given two sequences α₀, α₁, α₂, … and a₀,a₁, a₂, … of real numbers such that α₀ ≥ α₁ ≥ α₂ ≥ … ≥ α_n ≥ 0 and for all n.for all n. Then for all n.

4.8-6. Divergent Infinite Series. (a) Semiconvergence. A divergent infinite series a₀ + a₁ + a₂ + … may yield useful approximations to a quantity s if the absolute error decreases to a sufficiently small minimum for some value n₀ of n before increasing again (semiconvergent series).

(b) Asymptotic Series. A divergent infinite series is an asymptotic series describing the behavior of f(x) as if and only if there exists a positive real number N such that n > Nimplies

A divergent infinite series is an asymptotic series describing the behavior of f(x) as if and only if there exists a positive real number N such that n > N implies

Asymptotic series yield semiconvergent approximations to f(x), with n₀ → ∞ as x → x₀ or x → ∞ (see also Secs. 4.4-3 and 8.4-9).

(c) Summation by Arithmetic Means. A convergent or divergent series a₀ + a₁ + a₂+ · · · is summable by arithmetic means [summable by Césaro's means of the first order, summable C₁, summable (C, 1)] to the C₁-sum S₁ if and only if the sequence σ₀, σ₁ σ₂, … of the arithmetic means converges to the limit . Every convergent series is summable by arithmetic means, and C₁ = s. An infinite series of positive real terms is summable by arithmetic means if and only if it converges. See also Sec. 4.11-7.

4.8-7. Infinite Products. (a) An infinite product

of real or complex factors 1 + a_k≠ 0 converges to a limit (value of the infinite product) if and only if

This is true if and only if the infinite series converges to one of the values of log_e p (see also Sec. 21.2-10). If the infinite product is said to diverge to zero.

(b) A (necessarily convergent) infinite product converges absolutely if and only if converges; this is true if and only if the infinite series converges absolutely. An infinite product converges unconditionally (i.e., its value is independent of the order of factors) if and only if it is absolutely convergent (see also Sec. 4.8-3b). See Sec. E-8 for examples of infinite products.

(c) converges uniformly on a set S of values of x such that 1 + a_k(x) ≠ 0 for all k if and only if the sequence of functions converges uniformly to a function p(x) different from zero on S. This is, in particular, true if the infinite series converges uniformly on S.

4.8-8. Continued Fractions. A sequence of the form

is called a sequence of continued fractions; the n^th denominator is b_n-1 + a_n/b_n. Continued-fraction expansions of certain functions converge more rapidly than equivalent power-series expansions and are useful in some applications (electrical-network design; see also Sec. 20.5-7 and Ref. 4.7).

4.9. TESTS FOR THE CONVERGENCE AND UNIFORM CONVERGENCE OF INFINITE SERIES AND IMPROPER INTEGRALS

4.9-1. Tests for Convergence of Infinite Series (Sec. 4.8-1). (a) A Necessary and Sufficient Condition for Convergence. A sequence of real or complex numbers S₀, S₁ S₂, … (e.g., the partial sums of an infinite series, Sec, 4.8-1) converges if and only if for every positive real number є there exists a real integer N such that m > N, n > N implies |s_n - s_m| < є (Cauchy's Test for the convergence of sequences or series).

(b) Tests for Series of Real Positive Terms (useful also as tests for absolute convergence of real or complex series, Secs. 4.8-1 and 4.8-3). An infinite series a₀ + a₁ + a₂ + · · · of real positive terms converges if there exists a real number N such that n > N implies one or more of the following conditions:

is a convergent comparison series of real positive terms (comparison tests for convergence; see Secs. E-5 to E-7 for comparison series).

2.At least one of the quantities has an upper bound A < 1.

The fourth of these four tests is stronger than the third (Raabe's Test), which is, in turn, stronger than the first two (Cauchy's Ratio and Root Tests).

3.a_n ≤ f(n), where f(x) is a real positive decreasing function whose (improper) integral exists (Cauchy's Integral Test for convergence, see also Sec. 4.9-3b).

The infinite series a₀ + a₁ + a₂ + · · · diverges if there exists a real number N such that n > N implies one or more of the following conditions:

is a divergent comparison series of real positive terms (comparison tests for divergence).

2.At least one of the quantities (1) has a lower bound A ≥ 1.

3.a_n ≥ f(n), where f(x) is a real positive decreasing function whose integral diverges (Cauchy's Integral Test for divergence).

NOTE: The series which converges for real λ if and only if λ > 1, is useful as a comparison series.

If successive terms are alternatingly positive and negative (alternating series), decrease in absolute value, and

If the sequence s₀, s₁, s₂, … of the partial sums is bounded and monotonic

(d) Given a decreasing sequence of real positive numbers α₀, α₁, α₂, … , the infinite series α₀a₀ + α₁a₁ + α₂a₂ + · · · converges

1.If the series a₀ + a₁+ a₂ + · · · converges (Abel's Test, see also Sec. 4.8-5e)

2.If is bounded for all n (Dirichlet's Test)

4.9-2. Tests for Uniform Convergence of Infinite Series (Sec. 4.8-2). (a) A Necessary and Sufficient Condition for Uniform Convergence. A sequence s₀(x), s₁(x), s₂(x), … of real or complex functions (e.g., partial sums of an infinite series of functions, Sec. 4.8-2) converges uniformly on a set S of values of x if and only if for every positive real number є there exists a real number N independent of x such that m > N, n > N implies |s_n(x) -s_m(x)| < є for all x in S (Cauchy's Test for uniform convergence of sequences or series).

(b) An infinite series a₀(x) + a₁(x) + a₂(x) + · · · of real or complex functions converges uniformly and absolutely on every set S of values of x such that |a_n(x)| ≤ M_n for all n, where M₀ + M₁+ M₂ + · · · is a convergent comparison series of real positive terms (Weierstrass's Test). The convergence of the comparison series may be tested in the manner of Sec. 4.9-1b.

(c) Given a decreasing sequence of real positive terms α₀, α₁, α₂, … , the infinite series α₀a₀(x) + α₁a₁(x) + α₂a₂(x) + · · · converges uniformly on a set S of values of x

1. If the infinite series a₀(x) + a₁(x) + a₂(x) + · · · converges uniformly on S (Abel's Test, see also Sec. 4.9-ld)

2. If and there exists a real number for all n and all x in S (Dirichlet's Test)

4.9-3. Tests for Convergence of Improper Integrals (see also Sec. 4.6-2). Sections 4.9-3 and 4.9-4 list convergence criteria for improper integrals of the form . Other improper integrals can be reduced to these forms (Sec. 4.6-2). It is assumed that f(x) is bounded and integrable on every bounded interval (a, X) which does not contain the upper limit of integration.

(a) Necessary and Sufficient Conditions for Convergence (Cauchy's Test). The improper integral converges if and only if for every positive real number є there exists a real number M > a such that X₂ > X₁ > M implies .

Similarly, converges if and only if for every positive real number є there exists a positive real number δ < b – a such that b – X₂ < b – X₁ < δ implies

(b) Tests for Improper Integrals over a Real Nonnegative Function (useful also as tests for absolute convergence of real or complex improper integrals; note that absolute convergence implies convergence). Given a real function f(x) ≥ 0 on the interval of integration, the improper integral or converges if and only if is bounded for every X in the interval of integration. In particular, the integral converges if the integration interval contains a real number M such that x > M implies f(x) ≤ g(x), where g(x) is a real comparison function such that converges (Comparison Test).

Similarly, if or diverges, f(x) ≥ g(x) implies the divergence of the corresponding integral over f(x).

NOTE: and converge for λ > 1 and diverge for λ ≤ 1. The improper integral converges absolutely if f(x) = 0(l/x^λ)(λ > 1) as x → ∞ ; converges absolutely if f(x) = 0[1/(x - b)^λ](λ > 1) as x → b – 0 (see also Sec. 4.4-3).

(c) The improper integral or converges if the corresponding improper integral or converges absolutely, and α(x) is bounded and integrable on every finite subinterval of the interval of integration.

(d) The improper integral converges if, for every X > a, α(X) is bounded and monotonic, f(x) is bounded and integrable and has a finite number of sign changes on [a, X], and

1. converges (analog of Abel's test, Sec. 4.9-1d) or

2. is bounded and (analog of Dirichlet's test, Sec. 4.9-ld).

(e) See Sec. 8.2-4 for a number of applications.

4.9-4. Tests for Uniform Convergence of Improper Integrals (see also Sec. 4.6-2c). (a) Necessary and Sufficient Conditions for Uniform Convergence. The Cauchy convergence test of Sec. 4.9-3a implies uniform convergence of an improper integral or on every set S of values of y such that the bound M or δ is independent of y (see also Sec. 4.9-2a).

(b) The improper integral or converges uniformly and absolutely on every set S of values of y such that |f(x, y)| ≤ g(x) on the interval of integration, where g(x) is a real comparison function whose integral converges (analog of Weierstrass's test, Sec. 4.9-2b).

(c) The improper integral converges uniformly on every set S of values of y such that, for x> a, α(x, y) decreases monotonically and uniformly to zero as x → ∞, ∂α/∂x exists and is continuous, and is bounded by a constant A independent of x and y (analog of Dirichlet's test,Sec. 4.9-2c).

4.10. REPRESENTATION OF FUNCTIONS BY INFINITE SERIES AND INTEGRALS. POWER SERIES AND TAYLOR'S EXPANSION

4.10-1. Representation of Functions by Infinite Series and Integrals. A function f(x) is often represented by a corresponding infinite series because

1. A sequence of partial sums (or arithmetic means, Secs. 4.8-6c and 4.11-7) may yield useful numerical approximations to f(x).

2. It may be possible to describe operations on f(x) in terms of simpler operations on the functions φ_k(x) or on the coefficients α_k (transform methods, see also Secs. 4.11-5b and 8.6-5). The functions φ_k(x) and the coefficients α_k may have an intuitive (physical) meaning (Sec. 4.11-4b).

Similar advantages apply to representations of functions by (usually improper) integrals (see also Secs. 4.11-4c, 4.11-5c, and Chap. 8).

Realization of either or both advantages listed above often requires convergence or uniform convergence of the series or integral (hence the importance of the convergence criteria of Secs. 4.9-1 to 4.9-4), but this is not always true (Secs. 4.8-6, 4.11-5, and 4.11-7).

4.10-2. Power Series. (a) A power seriesin the (real or complex) variable x is a series of the form

where the coefficients a₀, a₁, a₂, … are real or complex numbers. Given any power series (1), there exists a real number r_c (0 ≤ r_c ≤ ∞) such that the power series converges absolutely and uniformly for |x| < r_c and diverges for |x| > r_c. r_c is called the radius of convergence of the power series.

Convergence for x = x₀ implies convergence for |x| < |X₀|, and divergence for x = x₀ implies divergence for |x| > |X₀|.

(b) Convergent power series can be added according to Eq.(4.8-2) and multiplied according to Cauchy's rule (4.8-4). For |x| < r_c, the power series (1) is a continuous and repeatedly differentiable function of x; the series may be differentiated and integrated term by term, and the resulting power series have the radius of convergence r_c.

(c) If there exists a positive real number r such that two power series, and , converge to the same sum f(x) for all real x such that |x| < r, then a₀ = b₀, a₁ = b₁, a₂ = b₂, …( Uniqueness Theorem).

EXAMPLE: The infinite geometric series

(see also Secs. 1.2-7 and 1.7-2) converges absolutely and uniformly for |x| < 1 and diverges for |x| > 1.

4.10-3. Abel's and Tauber's Theorems. (a) Given a positive real number r such that converges to the finite limit f(r), converges uniformly for 0 ≤ x ≤ r and (Abel's Theorem).

(b) Given a positive real number r such that converges for |x| < r, let . Then (Tauber's Theorem); the last condition can be weakened to k|a_k|r^k ≤ c < ∞ (Karamata's Theorem).

NOTE: The special cases r = 1 and r = r_c are of particular interest.

4.10-4. Taylor's Expansion (see also Sec. 7.5-2). (a) Given a real function f(x) such that f(ⁿ)(x) exists for a ≤ x < b,

More specifically, there exists a real number X = a + ϑ(x - a) such that a < X < x (or 0 < ϑ < 1) and

X and ϑ depend on a, x and n (see also Sec. 4.7-1).

(b) Given a function f(x) such that all derivatives f ^(k) (x) exist and for a ≤ x < b,

and the series converges uniformly to f(x) on every closed subinterval in a ≤ x < b [Taylor-series expansion of f(x) about x = a].

Equation (5) may be written with Δx = x – a (Secs. 4.5-3 and 11.2-1). For a = 0, Taylor's series reduces to MacLaurin's series .

(c) Every convergent series expansion of f(x) in powers of (x - a) is necessarily identical with Eq. (5) (Sec. 4.10-2c). If f(x) is a rational function, series expansions in powers of x or 1/x are often obtained by continued long division [Sec. 1.7-2; see also Eq. (2)]. Refer to Sec. 21.2-12 for examples of power-series expansions.

Power-series expansions permit term-by-term differentiation and integration (Sec. 4.10-2b) and are thus useful for the integration of f(x) and for the solution of differential equations (Sec. 9.2-5b). Note, however, that the n^th partial sum of a Taylor series is not necessarily the most useful n^th-degree polynomial approximation to f(x) (see also Sec. 20.5-1).

A function f(x) which can be represented by a convergent power series throughout a neighborhood of x = a is called analytic at x = a (see also Sec. 7.3-3).

4.10-5. Multiple Taylor Expansion. (a) Given a real function f(x₁, x₂, … , x_n) such that all partial derivatives of order m exist and are continuous for a_i ≤ x_i b_i(i = 1, 2, … , n) one has

The remainder R_m(x₁, x₂, … , x_n) satisfies a relation analogous to Eq. (4); in terms of differentials (Sec. 4.5-3)

where all differentials are computed at the point (a₁, a₂, … , a_n), and

(b) If all derivatives of f(x₁, x₂, … , x_n) exist and for a_i ≤ x_i < b_i then Eq. (6) yields a multiple power-series expansion for f(x₁, x₂, … , x_n) (multiple Taylor series). A function f(x₁, x₂, … , x_n) which can be represented by a convergent multiple power series throughout a neighborhood of the point (x₁, x₂, … , x_n) is called analytic at (x₁, x₂, … , x_n).

4.11. FOURIER SERIES AND FOURIER INTEGRALS

4.11-1. Introduction. Fourier series and Fourier integrals are used to represent and/or approximate functions (Sec. 4.10-1) in many important applications. Fourier expansions are special instances of expansions in terms of orthogonal functions (refer to Secs. 15.2-3 to 15.2-6).

4.11-2. Fourier Series. (a) Given the interval of expansion -π < t < π, the Fourier series generated by a real function f(t) such that exists* is the infinite trigonometric series

whose coefficients are defined by the Euler-Fourier formulas

* In many applications, it suffices to consider the integrals in this section as Riemann integrals (Sec. 4.6-1), but the use of Lebesgue integrals (Secs. 4.6-15) makes the theory more generally applicable. Refer to Secs. 15.2-3 to 15.2-6 for a number of theorems stated specifically in terms of Lebesgue integrals.

The a_k and b_k are real numbers, and the c_k are, in general, complex numbers (Sec. 4.11-2d).

(b) Given the finite interval of expansion -T /2 < t < T /2, the Fourier series generated by a real function f(t) such that exists is the infinite trigonometric series

with

Equation (2) reduces to Eq. (1) in the special case T = 2π. If (a, a+ T) is chosen as the interval of expansion, the integrals in Eq. (2b) must be taken between a and a + T instead of between -T /2 and T /2.

(c) If exists, the Fourier series (2) is that trigonometric series (2a) whose coefficients a_k, b_k minimize each mean-square error

when successive partial sums (Sec.4.8-1)

are used as approximations to f(t) (see also Sec. 15.2-6).

If a trigonometric series (2a) converges uniformly to f(t) in (-T/2, T/2), then its coefficients are necessarily the Fourier coefficients (2b) of f(t) (Euler's Theorem).

(d) If |f(t)| is integrable over the interval of expansion, the Fourier coefficients a_k and b_k exist and tend to zero as k → ∞ (Riemann-Lebesgue Theorem). More specifically, if, throughout the closed interval of expansion, f^(m-1)(t) exists and is piecewise continuously differentiable, and f^(m)(t) is differentiable wherever it is continuous, then a_k b_k are of the order of k^-(m+1) as k → ∞.

(e) The real coefficients a_k, b_k and the complex coefficients c_k are related by

where b₀ = 0. The Fourier series (2) of an even or odd function f(t) (Sec. 4.2-2b) reduces to a Fourier cosine series or a Fourier sine series, respectively.

4.11-3. Fourier Integrals and Fourier Transforms. (a) The Fourier Integral generated by a real function f(t) whose absolute value |f(t)| is integrable* over the interval – ∞ < t < ∞ (interval of expansion) is defined as

with

The function c(ν) ≡ F_F(iω)(ω = 2πν) will be called the Fourier transform F[f(t)] of f(t). Some authors, especially physicists, refer to as the Fourier transform of f(t) instead.

(b) The Fourier cosine and sine integrals generated by a real function f(t) whose absolute value |f(t)| is integrable over the interval of expansion 0 < t < ∞ are respectively defined as

* See footnote to Sec. 4.11-2.

with

The (real) functions c_c(ν) ≡ F_c(ω) ≡ _c[f(t)] and c_s(ν) ≡ F_s(ω) ≡ _s[f(t)] will be called, respectively, the Fourier cosine transform and the Fourier sine transform of f(t). Some authors refer to c_c(ω) and c_s(ω) as Fourier cosine and sine transforms, instead.

4.11-4. Functions Which Are Actually Equal to Their Fourier Series or Fourier Integrals. Fourier Analysis (see Appendix D for examples). (a) A Fourier series or a Fourier integral generated by a real function f(t) whose absolute value is integrable* over the corresponding expansion interval I

1.Converges uniformly to f(t) throughout every open subinterval of I where f(t) is continuous and of bounded variation (Sec. 4.4-8b)

2. Converges to 1/2[f(t - 0) + f(t + 0)] throughout every open subinterval of I in which f(t) is of bounded variation (J ordan's Test; see also Sec. 4.4-8b)

Remember that f(t) is of bounded variation in every finite open interval where f(t) is bounded and has a finite number of relative maxima, minima, and discontinuities (Dirichlet's conditions, Sec. 4.4-8b).

(b) Fourier Analysis (Harmonic Analysis) of Periodic Functions. Let f(t) be a real periodic function with period T (Sec. 4.2-2b) and such that exists. Then

* See footnote to See. 4.11-2.

throughout every open interval where f(t) is of bounded variation, if one defines f(t) = 1/2[f(t – 0) + f(t – 0)] at each discontinuity (see also Sec. 4.4-8b). f(t) is thus expressed as the sum of

1. A constant term a₀/2 = c₀ [average value of f(t), see also Secs. 4.6-3 and 18.10-7], and

2. A set of sinusoidal terms (sinusoidal components) of respective frequencies ν₀ = 1/T (fundamental frequency), 2ν₀ = 2/T(2^nd-harmonic frequency), 3ν₀ = 3/T (3^rd-harmonic frequency), …

The k^th-harmonic component has the frequency kν₀ = k/T, the circular frequency kω₀ = 2πkν₀ = 2πk/T, the amplitude , and the “phase angle” arg c_k = – arctan (b_k/a_k).

The odd-harmonic terms in the expansion (6) describe the antiperiodic part of f(t) (Sec. 4.2-2b). Note that

whenever the integral on the left exists (Parseval's Theorem, see also Sec. 15.2-4). Table D-l of Appendix D lists the Fourier coefficients and the mean-square values (7) for a number of periodic functions. Refer to Sec. 20.6-6 for numerical harmonic analysis.

throughout every open interval where f(t) is of bounded variation, if one defines f(t) = 1/2[f(t – 0) – f(t + 0)] at each discontinuity. A function f(t) having the Fourier transform c(ν) will be called an inverse Fourier transform ^-1[c(ν)] of c(ν); under the given conditions, Eq. (8) defines ^-1[c(ν)] uniquely wherever f(t) is continuous.

Equation (8) may be rewritten in different forms, e.g.,

The real functions A(ν) and – B(ν) in Eq. (9) are, respectively, the cosine transform of the even part (Sec. 4.2-2b) of f(t), and the sine transform of the odd part of f(t):

A(ν) ≡ 0 if f(t) is odd, and B(ν) ≡ 0 if f(t) is even (see also Sec. 4.11-3b).

The Fourier-integral expansion describes f(t) as a “sum” of infinitesimal sinusoidal components with frequencies ν or circular frequencies ω = 2πν (ν ≥ 0); the functions 2|c(ν)| and arg c(ν) respectively define the amplitudes and the phase angles of the sinusoidal components. Note that c( -ν) ≡ c*(ν), C( -ω) ≡ C*(ω), and that

whenever the integral on the left exists (Parseval's Theorem; see also Table 4.11-1).

(d) A more general type of function can be expressed as a sum of a function (8) and a set of periodic functions (6), so that both a “band spectrum” and a “line spectrum” exist (see also Sec. 18.10-9). The treatment of Fourier series and Fourier integrals can be formally unified through the introduction of generalized (integrated) Fourier transforms(Sec. 18.10-10).

(e) The Paley-Wiener Theorem. Given a positive real function Φ(ω), there exists a real function f(t) ≠ 0 such that exists and f(t) = 0 for either t > 0 or t <0 and

if Φ(–ω) ≡ Φ(ω) and both exist (Paley-Wiener Theorem).

4.11-5. Representation of Functions and Operations in Terms of Fourier Coefficients or Fourier Transforms (see also Secs. 4.10-1 and 8.3-1). (a) Uniqueness Theorem. A suitably integrable function f(t) uniquely defines its Fourier coefficients (2b) or its Fourier transform. Conversely, a complete set of Fourier coefficients or a Fourier transform uniquely defines the corresponding function f(t) almost everywhere (Sec.4.6-14b) in the interval of expansion; in particular, f(t) is uniquely defined at each point of continuity in the interval of expansion. This uniqueness theorem holds even if the Fourier series or Fourier integral does not converge(see also Sec. 4.11-7).

Note that not every trigonometric series (not even every convergent trigonometric series) is a Fourier series, nor is every function c(ν) a Fourie transform (see also Sec. 4.11-2c).

(b) Operations with Fourier Series. Given f(t) with the Fourier coefficients a_k, b_k, c_k, and φ(t) with the Fourier coefficients α_k, β_k, γ_k for the same interval of expansion, let λ and μ be real constants. Then the function λf(t) + μφ(t) has the Fourier coefficients λa(k)+ μα_k, λb(k) + μβ_k, λc(k) + μγ_k (term-by-term addition and multiplication by constants).

Term-by-term integration of a Fourier series (2) over an interval (t₀, t) in the interval of expansion yields a series converging to The theorem holds for all values of t₀ and t if f(t) is periodic with period T.

Note that these theorems do not require convergence of the given Fourier series. Refer to Sec. 4.8-4c for differentiation of infinite series.

(c) Properties of Fourier Transforms. Table 4.11-1 lists the most important properties of Fourier transforms (see also Sec. 8.3-1).

Note also

Table 4.11-1. Properties of Fourier Transforms(see also Sec. 4.11-3 and Table 8.3-1)

for r= 0, 1, 2, … , provided that the derivative on the left exists for 0 < t < ∞, and that all derivatives of lesser order vanish as t → ∞.

4.11-6. Dirichlet's and Féjér's Integrals (see also Sec. 4.11-7). The partial sums

and the corresponding arithmetic means (Sec. 4.8-6c) of a Fourier series (2) may be written

4.11-7. Summation by Arithmetic Means. (a) The partial sums of a Fourier series may not constitute useful approximations to f(t). This may be true, in particular, if the series diverges, or if the partial sums “overshoot” f(t) badly near a discontinuity of f(t) (nonuniform convergence near the discontinuity, Gibbs phenomenon). One may then resort to summation by arithmetic means (Sec. 4.8-6c). Every Fourier series (2) is summable by arithmetic means to the sum 1/2|f(t - 0) + f(t + 0)] for all t in (– T /2, T /2) where the latter function exists (Féjér's Theorem). The arithmetic means converge to f(t) almost everywhere in the interval of expansion; they converge uniformly f(t) on every open subinterval of (– T /2, T /2) where f(t)is continuous.

(b) Similarly, if exists, and f(t) is continuous in every finite interval, then the arithmetic means

converge uniformly to f(t) in every finite interval as λ → ∞. The uniform convergence extends over (– ∞, ∞) if f(t) is uniformly continuous in (– ∞, ∞).

4.11-8. Multiple Fourier Series and Integrals. (a) Given an n-dimensional region of expansion defined by a_i < t_i + T_i, j = 1, 2, … , n, the multiple Fourier series generated by a function f(t₁, t₂, … , t_n) such that exists is defined as

(b) The multiple Fourier integral generated by a function f(t₁, t₂, … , t_n) such that exists is defined as

One may introduce νi = ω_i/2π as in Eq. (4). For regions of expansion defined by 0 < t_i < or - ∞ < t_i < 0 one obtains multiple Fourier sine or cosine integrals by analogy to Sec. 4.11-3b.

(c) Given the region of expansion - ∞ < t₁ < ∞, a₂ < t₂ < a₂ + T₂, one may write a “mixed” Fourier expansion for f(t₁ t₂):

One may similarly mix Fourier integrals, Fourier series (and also Fourier sine and cosine integrals) in more than two dimensions.

(d) The exponentials in Eqs. (16), (17), and (18) can be expanded into sine and cosine terms through the use of Eq. (21.2-28). All the theorems of Secs.4.11-2 through 4.11-7 can be generalized to apply to multiple Fourier series and integrals.

Functions of a complex variable Chap. 7

Divergence theorem, Stokes' theorem Chap. 5

Metric spaces, convergence Chap. 12

Orthogonal expansions Chap. 15

Numerical approximations Chap. 20

Special functions Chap. 21

4.12-2. References and Bibliography (see also Secs. 8.7-2 and 12.9-2).

4.1. Apostol, T. M.: Mathematical Analysis, Addison-Wesley, Reading, Mass., 1957.

4.2. Bartle, R. G.: The Elements of Real Analysis, Wiley, New York, 1964.

4.3. Boas, R. P.: A Primer of Real Functions, Wiley, New York, 1960.

4.4. Buck, R. C.: Advanced Calculus, 2d ed., McGraw-Hill, New York, 1965.

4.5. Churchill, R. V.: Fourier Series and Boundary Value Problems, 2d ed., McGraw-Hill, New York, 1963.

4.6. Dieudonné, J.: Foundations of Modern Analysis, Academic, New York, 1960.

4.7. Eggleston, H. G.: Introduction to Elementary Real Analysis, Cambridge University Press, New York, 1962.

4.8. Fleming, W.: Functions of Several Variables, Addison-Wesley, Reading, Mass., 1966.

4.9. Gelbaum, B. R., and J. M. H. Olmsted: Counterexamples in Analysis, Holden-Day, San Francisco, 1964.

4.10. Goffman, C: Calculus of Several Variables, Harper & Row, New York, 1965.

4.11. Goldberg, S.: Methods of Real Analysis, Blaisdell, New York, 1964.

4.12 Graves, L. M.: The Theory of Functions of Real Variables, 2d ed., McGraw-Hill, New York, 1956.

4.13. Knopp, K.: Theory and Application of Infinite Series, Blackie, Glasgow, 1951; also 5th ed. (in German), Springer, Berlin, 1964.

4.14. Natanson, I. P.: Theory of Functions of a Real Variable (2 vols.), Unger, New York, 1955/9.

4.15. Papoulis, A.: The Fourier Integral and Its Application, McGraw-Hill, New York, 1962.

4.16. Rankin, R. A.: Introduction to Mathematical Analysis, Pergamon Press, New York, 1962.

4.17. Rosser, J. B.: Asymptotic Formulas and Series, in E. F. Beckenbach, Modern Mathematics for the Engineer, 2d series, McGraw-Hill, New York, 1961.

4.18. Rudin, W.: Principles of Mathematical Analysis, 2d ed., McGraw-Hill, New York, 1964.

4.19. Wall, H. S.: Analytic Theory of Continued Fractions, Van Nostrand, Princeton, N.J., 1948.

4.20. Widder, D. V.: Advanced Calculus, 2d ed., Prentice-Hall, Englewood Cliffs, N.J., 1961.

* Many authors categorically define every function as single-valued, so that, for example, the two branches + are always regarded as two functions.

* See also Sec. 4.2.1

* The indices in x¹, x², x³ are superscripts, not exponents.