Mathematical Handbook for Scientists and Engineers

associates one or more values of the complex dependent variable w with each value of the complex independent variable z in a given domain of definition.

Single-valued, multiple-valued, and bounded functions of a complex variable are defined as in Secs. 4.2-2a and 4.3-3b. Limits of complex functions and sequences and continuity of complex functions as well as convergence, absolute convergence, and uniform convergence of complex infinite series and improper integrals are defined as in Chap. 4; the theorems

Table 7.2-1. Real and Imaginary Parts, Zeros, Singularities for a Number of Frequently Used Functions f(z) = u(x,y) + iv(x,y) of a complex variable z = x + iy (see also Secs. 1.3-2 and 21.2-9 to 21.2-11)

Note

of Chapter. 4 apply to complex functions and variables unless a restriction to real quantities is specifically stated. In particular, every complex power series has a real radius of convergence r_c(0 ≤ r_c ≤ ∞ ) such that the series converges uniformly and absolutely for |z — a| < r_c and diverges for |z — a| > r_c (Sec. 4.10-2a).

7.2-2. z Plane and w Plane. Neighborhoods. The Point at Infinity (see also Sec. 4.3-5). Values of the independent variable z = x + iy are associated with unique points (x, y) of an Argand plane (Sec. 1.3-2), the z plane. Values of w = u + iv are similarly associated with points (u, v) of a w plane.

An (open) δ-neighborhood of the point z = a in the finite portion of the plane is defined as the set of points z such that |z — a| δ for some δ > 0.

The point at infinity (z = ∞) is defined as the point transformed into the origin (z = 0) by the transformation = 1/z. A region containing the exterior of any circle is a neighborhood of the point z = ∞.

7.2-3. Curves and Contours (see also Secs. 2.1-9 and 3.1-13). A continuous (arc or segment of a) curve in the z plane is a set of points z = x + iy such that

where x(y) and y(t) are continuous functions of the real parameter t. A (portion of a) continuous curve (2) is a simple curve (Jordan arc) if and only if it consists of a single branch without multiple points, so that the functions x(t) and y(t) are single-valued, and the set of equations

has no distinct solutions τ₁, τ₂ in the closed interval [t₁, t₂]. A simple closed curve (closed Jordan curve) is a continuous curve consisting of a single branch without multiple points except for a common initial and terminal point.

A simple curve or simple closed curve will be referred to as a (simple) contour if and only if it is rectifiable (Sec. 4.6-9).* The element of distance between suitable points z and z + dz on a contour (2) is .

7.2-4. Boundaries and Regions (see also Secs. 4.3-6, 7.9-1b, and 12.5-1). The geometry of the complex-number plane (including definitions of distances and angles)

* Some authors restrict the use of the term contour to regular curves (Sec. 3.1-13).

is identical with the geometry of the Euclidean plane of points (x, y) or vectors r = ix + jy for finite values of x and y, but the definition of z = ∞ (Sec. 7.2-2) introduces a topology different from that usually associated with plane geometry. The points z of the complex-number plane can be represented homeomorphically (Sec. 12.5-1) by corresponding points of a sphere with longitude arg z and colatitude 2 arccot (|z|/2), (stereographic projection), so that z = 0 and z = ∞ correspond to opposite poles. The points of every simple closed curve C separate the plane into two singly connected open regions: every continuous curve containing a point of each of the two regions contains a point of their common boundary C (Jordan Separation Theorem). If C does not contain z = ∞ one of the two regions is bounded (i.e., situated entirely in the finite portion of the plane, where |z| is bounded) and the other is unbounded; if C contains z = ∞ both regions are unbounded.

More generally, the boundary C of a given region or domain D may be a multiplicity of nonintersecting simple closed curves (multiply connected region, see also Sec. 4.3-6). In any case, the positive direction (positive sense) on the boundary curve is defined as that leaving the region D (interior of the boundary) on the left (counterclockwise for outside boundaries, see also Fig. 7.5-1). The (open) set of points on one side of a boundary curve C is an open region, and the (closed) set of points on one side and on the boundary is a closed region.

7.2-5. Complex Contour Integrals (see also Secs. 4.6-1 and 4.6-10). One defines

where the points lie on the contour C connecting z = a = z₀ and z = b = z_m. If the limit (3a) exists, then

where the real line integrals are taken over the same path as the complex integral. The integration rules of Table 4.6-1 apply; in particular, reversal of the sense of integration on the contour C reverses the sign of the integral.

If f(z) is bounded in absolute value by M, and u(x, y) and v(x, y) are of bounded variation (Sec. 4.4-8b) on a contour C of finite length L, then the integral (3) exists, and

If C contains z = ∞, or if f(z) is not bounded on C, the integral (3) can often be defined as an improper integral in the manner of Sec. 4.6-2.

7.3. ANALYTIC (REGULAR, HOLOMORPHIC) FUNCTIONS

7.3-1. Derivative of a Function (see also Sec. 4.5-1). A function w = f(z) is differentiable at the point z = a if and only if the limit

[derivative of f(z) with respect to z] exists for z = a and is independent of the manner in which Δz approaches zero. A function may be differentiable at a point (e.g., |z|² at z = 0), on a curve, or throughout a region.

7.3-2. The Cauchy-Riemann Equations. f(z) ≡ u(x, y) + iv(x, y) is differentiable at the point z = x + iy if and only if u(x, y) and v(x, y) are continuously differentiable throughout a neighborhood of z, and

at the point z, so that

7.3-3. Analytic Functions. (a) A single-valued function f(z) shall be called analytic (regular, holomorphic)* at the point z = a if and only if f(z) is differentiable throughout a neighborhood of z = a. f(z) is analytic at z = a if and only if f(z) can be represented by a power series convergent throughout a neighborhood of z = a (alternative definition). Refer to Secs. 7.4-1 to 7.4-3 for an extension of the definition to multiple-valued functions.

(b) f(z) is analytic at infinity if and only if is analytic at One defines.

f(z) is analytic at infinity if and only if f(z) can be expressed as a convergent series of negative powers, for sufficiently large values of |z| (see also Sec. 7.5-3).

7.3-4. Properties of Analytic Functions. Letf(z) be analytic throughout an open region D. Then, throughout D,

* The terms differentiable, analytic, regular, and holomorphic are used interchangeably by some authors.

The Cauchy-Riemann equations (2) are satisfied (the converse is true),

u(x, y) and v(x, y) are conjugate harmonic functions (Sec. 15.6-8).

All derivatives of f(z) with respect to z exist and are analytic (see also Sec. 7.5-1).

If the open region D is simply connected (this applies, in particular, to the exterior of a bounded simply connected region),

      4. The integral is independent of the path of integration, provided that C is a contour of finite length situated entirely in D; the integral is a single-valued analytic function of z, and its derivative is f(z) (see also Sec. 7.5-1).

      5. The values of f(z) on a contour arc or a subregion in D define f(z) uniquely throughout D.

All ordinary differentiation and integration rules (Secs. 4.5-4 and 4.6-1) apply to analytic functions of a complex variable. If f(z) is analytic at z = a and ƒ′(a) ≠ 0, then f(z) has an analytic inverse function (Sec. 4.2-2a) at z = a. If W = F(w) and w = f(z) are analytic, then W is an analytic function of z. If a sequence (or an infinite series, Sec. 4.8-1) of functions f_i(z) analytic throughout an open region D converges uniformly to the limit f(z) throughout D, then f(z) is analytic, and the sequence (or series) of the derivatives ƒ′(z) converges uniformly to ƒ′(z) throughout D. The sequence (or series) of contour integrals over any contour C of finite length in D converges uniformly to

7.3-5. The Maximum-modulus Theorem. The absolute value |f(z)| of a function f(z) analytic throughout a simply connected closed bounded region D cannot have a maximum in the interior of D. If |f(z)| ≤ M on the boundary of D, then |f(z)| < M throughout the interior of D unless f(z) is a constant (see also Sec. 15.6-4).

7.4. TREATMENT OF MULTIPLE-VALUED FUNCTIONS

7.4-1. Branches. One extends the theory of analytic functions to suitable multiple-valued functions by considering branches f₁(z), f₂(z), ... of f(z) each defined as a single-valued continuous function throughout its region of definition. Each branch assumes one set of the function values of f(z) (see also Sec. 7.8-1).

7.4-2. Branch Points and Branch Cuts. (a) Given a number of branches of f(z) analytic throughout a neighborhood D of z = a, except possibly at z = a, the point z = a is a branch point involving the given branches* of f(z) if and only if f(z) passes from one of these branches to another as a variable point z in D describes a closed circuit about z = a. The order of the branch point is the number m of branches reached by f(z) before returning to the original or m + l^st branch as z describes successive closed circuits about z = a. If f(z) is defined at a branch point z = a the function value ƒ(a) is common to all branches “joining” at z = a (EXAMPLE: has a branch point of order 2 at z = 0).

The point z = ∞ is a branch point of f(z) if and only if the origin is a branch point of f(1/z) (see also Sec. 7.6-3).

Given a function w = f(z) whose inverse function Φ (w) exists and is single-valued throughout a neighborhood of w = ƒ(a), the point z = a ≠ ∞ is a branch point of order m of f{z) whenever Φ'(w) has a zero of order m (Sec. 7.6-1) or a pole of order m + 2 (Sec. 7.6-2) at w = f(a). (EXAMPLES:

Similarly, if Φ(w) exists and is single-valued throughout a neighborhood of w =/(∞), the point z = ∞ is a branch point of order m of f(z) if Φ'(w) has a zero of order m + 2 or a pole of order m at w =f(∞) (EXAMPLES:

(b) The individual single-valued branches of ƒ(z) are defined in regions bounded by branch cuts, which are simple curves chosen so that no closed circuit around a branch point lies within the region of definition of a single branch. The choice of branches and branch cuts for a given function f(z) is not unique, but the branch points and the number of branches are uniquely defined (Fig. 7.4-1).

All branches of a monogenic analytic function may be obtained by analytic continuation of successive elements (Sec. 7.8-1).

7.4-3. Riemann Surfaces. It is frequently useful to represent a multiple-valued function f(z) as a single-valued function defined on a Riemann surface which consists of a multiplicity of z planes or “sheets” corresponding to the branches of f(z) and joined along suitably chosen branch cuts. A circuit around a branch point on such a surface transfers the variable point z between two sheets corresponding to two branches of f(z). If both w = f(z) and its inverse are multiple-valued functions, then both the z plane and the w plane may be replaced by suitable Riemann surfaces; w = f(z) will now define a reciprocal one-to-one correspondence (mapping) between the points of the two Riemann surfaces, except at branch points.

NOTE: The Riemann surface for a monogenic analytic function (obtained by analytic continuation, Sec. 7.8-1) must be connected; thus the multiple-valued function f(z) = ±1, although analytic everywhere, is not a monogenic analytic function.

Many theorems about single-valued analytic functions apply also to multiple-valued monogenic analytic functions defined on suitable Riemann surfaces without

* Note that f(z) may have other branches which do not join the given branches at z = a and which may or may not be analytic at z = a.

FIG. 7.4-1. Branch points and branch cuts for some elementary functions.

restriction to a single branch. The construction of Riemann surfaces for arbitrary functions may require considerable ingenuity ,(Refs. 7.8 and 7.15).

Throughout this handbook, statements about analytic functions refer to single-valued analytic functions or to single-valued branches of monogenic analytic functions unless specific reference to multiple-valued functions is made.

7.5. INTEGRAL THEOREMS AND SERIES EXPANSIONS

7.5-1. Integral Theorems. Let z be a point inside the boundary contour C of a region D throughout which f(z) is analytic, and let f(z) be analytic on C. Then

Figure 7.5-1 illustrates the application of the Cauchy-Goursat integral theorem (1) to multiply connected domains (see also Sec. 7.7-1).

Equation (2) yields f(z) and its derivatives in terms of the boundary values of ƒ(z). Specifically,

Note that is an analytic function of z even if f(z) is not analytic throughout D and only continuous on C; Eq. (1) holds if ƒ(z) is also analytic in D.

A continuous single-valued function f(z) is analytic throughout the bounded region D if Eq. (1) holds for every closed contour C in D and enclosing only points of D (M orera's Theorem).

7.5-2. Taylor-series Expansion (see also Sec. 4.10-4). (a) If f(z) is analytic inside and on the circle K of radius r about z = a, then there exists a unique and uniformly convergent series expansion in powers of (z — a)

The largest circle K_c or |z — a| = r_c all of whose interior points are inside the region where ƒ(z) is analytic is the convergence circle of the power series (4); r_c is the radius of convergence (Sec. 4.10-2a). A number of useful power-series expansions are tabulated in Sec. E-7.

FIG. 7.5-1. Application of Cauchy's integral theorem to a multiply connected region of the z plane (Secs. 7.2-4 and 7.5-1). A region D bounded by exterior contours C₁, C₂, . . . and interior contours C′₁, C′₂, . . . is made simply connected by cuts (shown in broken lines). The integrals over each cut cancel, and Cauchy's integral theorem becomes

where all integrals are taken in the positive (counterclockwise) direction. The same technique applies if D is not bounded.

(b) If M(r) is an upper bound of |f(z)| on K, then

(c) If Taylor's series (4) is terminated with the term a_n-1(z — a)^n-1, the remainder R_n(z) is given by

7.5-3. Laurent-series Expansion. a (a) If f(z) is analytic throughout the annular region between and on the concentric circles K₁ and K₂ centered at z = a and of radii r₁ and r₂ < r₁ respectively, there exists a unique series expansion in terms of positive and negative powers of (z — a),

The first term of Eq. (7) is analytic and converges uniformly for |z — a| ≤ r₁; the second term [principal part of f(z)] is analytic and converges uniformly for |z — a| ≥ r₂.

NOTE: The case a = ∞ is treated by using the transformation = 1/z, which transforms z = ∞ into the origin.

(b) If the first term in Eq. (7) is terminated with a_n-1(z — a)^n-1 the remainder R_n(z) is given by

If the second term in Eq. (7) is terminated with b_n-1(z — a)^-(n-1), the remainder R′_n(z) is given by

M(r₁) and M(r₂) are upper bounds of |ƒ (z)| on K₁ and K₂, respectively.

7.6. ZEROS AND ISOLATED SINGULARITIES

7.6-1. Zeros (see also Sec. 1.6-2). The points z for which f(z) = 0 are called the zeros of f(z) [roots of f(z) =0]. A functionƒ(z) analytic at z = a has a zero of order m, where m is a positive integer, at z = a if and only if the first m coefficients a₀, a₁, a₂, . . . , a_m-1 in the Taylor-series expansion (7.5-4) of f(z) about z = a vanish, so that ƒ(z)(z — a)^-m is analytic and different from zero at z = a.

The zeros of a function f(z) analytic throughout a region D are either isolated points [i.e., each has a neighborhood (Sec. 7.2-2) throughout which f(z) ≠ 0 except at the zero itself], or f(z) is equal to zero throughout D.

If f₁(z) and f₂(z) are analytic throughout a simply connected bounded open region D and on its boundary contour C, and if |ƒ₂(z)| < |ƒ₁(z)| ≠ 0 on C, then f₁(z) and f₁(z) + ƒ₂(z) have the same number of zeros in the region D (Rouche's Theorem). Every polynomial of degree n has n zeros, counting multiplicities (Fundamental Theorem of Algebra, see also Sec. 1.6-3).

7.6-2. Singularities. A singular point or singularity of the function f(z) is any point where f(z) is not analytic. The point z = a is an isolated singularity of ƒ(z) if and only if there exists a real number δ > 0 such that ƒ(z) is analytic for 0 < |z — a| < δ but not for z = a. An isolated singularity of f(z) at z = a ≠ ∞ is

1. A removable singularity if and only if f(z) is finite throughout a neighborhood of z = a, except possibly at z = a itself; i.e., if and only if all coefficients b_k in the Laurent expansion (7.5-7) of ƒ(z) about z = a vanish.

2. A pole of order m (m = 1, 2, . . .) if and only if (z — a)^mf(z) but not (z — a)^m-lf(z) is analytic at z = a; i.e., if and only if in the Laurent expansion (7.5-7) of f(z) about z = a; or if and only if 1/f(z) is analytic and has a zero of order m at z = a. In this case, no matter how z approaches z = a.

3. An isolated essential singularity if and only if the Laurent expansion (7.5-7) ofƒ(z) about z = a has an infinite number of terms involving negative powers of (z — a); |f(z)| becomes indefinitely large as z approaches the value z = a for some approach paths but not for others.*

4. A branch point if f(z) is a multiple-valued function, and z = q satisfies the conditions of Sec. 7.4-2.

If m branches of a function f(z) join at a branch point z = a, that branch point is considered a point of continuity, a zero, a removable singularity, a pole, or an essential singularity of the “complex” F(z) of the m branches in question if and only if the function

has a point of continuity, a zero, a pole, or an essential singularity, respectively, at ζ = 0; F(z) = f(z) in a neighborhood of z = a, except that F(z) can take only values of the m branches joining at z = a. Note that ƒ(z) may have other branches at z = a which may or may not join at z = a and whose behavior at z = a may or may

* Sometimes the definition of an isolated essential singularity is extended to cover limit points of poles.

not be different from that of the m branches considered above. If the m branches joining at z = a are the only branches of f(z) at z = a, then F(z) = f(z) at z = a.

The behavior of different branches of a multiple-valued function ƒ(z) at a point z = a which is not a branch point ƒ(z) must be considered independently for each branch.

EXAMPLES: The function ƒ(z) defined by ƒ(z) = 0 for z = 0, ƒ(z) = i for z ≠ 0 has a removable singularity at z = 0. sin 1/z has a removable singularity at z = 0. l/(z — 2)³ has a pole of order 3 at z = 2. e^1/z has an essential singularity at z = 0. has a branch point of order 1 at z =0; this branch point is a zero of order 1 for all branches.

7.6-3. Zeros and Singularities at Infinity. ƒ(z) is analytic at infinity if and only if ƒ(1/z) is analytic at the origin. The point z = ∞ is a zero or a singularity of any of the types listed in Sec. 7.6-2 if ƒ(1/z) behaves correspondingly at the origin. The behavior of ƒ(z) at infinity may be investigated with the aid of the Laurent expansion of ƒ(1/z) about the origin.

7.6-4. Weierstrass's and Picard's Theorems. Let ƒ(z) be a single-valued function having an isolated essential singularity at z = a. Then

For any complex number w, every neighborhood of z = a contains a point z such that |w — ƒ(z)| is arbitrarily small (Weierstrass's Theorem).

Every neighborhood of z = a contains an infinite set of points z such that ƒ(z) = w for every complex number w with the possible exception of a single value of w (Picard's Theorem).

7.6-5. Integral Functions. An integral (entire) function ƒ(z) is a function whose only singularity is an isolated singularity at z = ∞. If this singularity is a pole of order m, then f(z) must be a polynomial (integral rational function) of degree m. A function f(z) analytic for all values of z is a constant (Liouville's Theorem).

An integral function ƒ(z) which is not a constant assumes every value w, except possibly one, at at least one point z, and, if ƒ(z) is not a polynomial, at infinitely many points z.

7.6-6. Product Expansion of an Integral Function. For every set of points z₀, z₁, Z₂ . . . having no limit point (Sec. 4.3-6a except possibly z = ∞, there exists an integral function f(z) whose only zeros are zeros of given orders m_k at the points z = z_k. Let z₀ = 0, Zk ≠ 0 (k > 0); if there is no zero for z = 0, then m₀ = 0. Then ƒ(z) can be represented in the form

where g(z) is an arbitrary integral function, and the r_k are (finite) integers chosen so as to make the product converge uniformly throughout every bounded region (Theorem of Weierstrass; see Sec. 21.2-13 for examples of product expansions).

7.6-7. Meromorphic Functions. f(z) is meromorphic throughout a region D if and only if its only singularities throughout D are poles. The number of such poles in any finite region D is necessarily finite. Many authors alternatively define a function as meromorphic if and only if its only singularities throughout the finite portion of the plane are poles.

Every function meromorphic throughout the finite portion of the plane can be expressed as the quotient of two integral functions without common zeros, and thus as the quotient of two products of the type discussed in Sec. 7.6-6. A function meromorphic throughout the entire plane is a rational algebraic function expressible as the quotient of two polynomials (see also Sec. 4.2-2c).

7.6-8. Partial-fraction Expansion of Meromorphic Functions (see also Sec. 1.7-4). Let f(z) be any function meromorphic in the finite portion of the plane and having poles with given principal parts (Sec. 7.5-3) at the points z = z_k of a given finite or infinite set without limit points in the finite portion of the plane. Then it is possible to find polynomials p₁(z), P₂(z), . . . and an integral function g(z) such that

and the series converges uniformly in every bounded region where f(z) is analytic (Mittag-Leffler's Theorem).

7.6-9. Zeros and Poles of Meromorphic Functions. Let f(z) be meromorphic throughout the bounded region inside and continuous on a closed contour C on which f(z) ≠ 0. Let N be the number of zeros and P the number of poles of f(z) inside C, respectively, where a zero or pole of order m is counted m times. Then

For P = 0, Eq. (4) reduces to the principle of the argument

where Δ_cϑ is the variation of the argument ϑ ofƒ(z) around the contour C.

Equation (5) means that w = f(z) maps a moving point z describing the contour C once into a moving point w which encircles the w plane origin N = 0, 1, 2, . . . times if f(z) has, respectively, 0, 1, 2, . . . zeros inside the contour C in the z plane. Equations (4) and (5) yield important criteria for locating zeros and poles of f(z), such as the famous Nyquist criterion (Ref. 7.6).

7.7. RESIDUES AND CONTOUR INTEGRATION

7.7-1. Residues. Given a point z = a where ƒ(z) is either analytic or has an isolated singularity, the residue Res_ƒ (a) of f(z) at z = a is the coefficient of (z — a)^-1 in the Laurent expansion (7.5-7), or

where C is any contour enclosing z = a but no singularities of ƒ(z) other than z = a.

The residue Res_ƒ (∞) of ƒ(z) at z = ∞ is defined as

where the integral is taken in the negative sense around any contour enclosing all singularities of ƒ(z) in the finite portion of the plane. Note that

if the limit exists.

If f(z) is either analytic or has a removable singularity at z = a ≠ ∞, then Res_ƒ (a) = 0 [see also Eq. (7.5-1)]. If z = a ≠ ∞ is a pole of order m, then

In particular, let z = a ≠ ∞ be a simple pole of ƒ(z) ≡ p(z)/q(z), where p(z) and q(z) are analytic at z = a, and p(a) ≠ 0. Then

7.7-2. The Residue Theorem (see also Sec. 7.5-1). For every simple closed contour C enclosing at most a finite number of (necessarily isolated) singularities z₁, z₂, . . . , z_n of a single-valued function f(z) continuous on C,

One of the z_k may be the point at infinity. Note carefully that the contour C must not pass through any branch cut (see also Sec. 7.4-2).

7.7-3. Evaluation of Definite Integrals. (a) One can often evaluate a real definite integral as a portion of a complex contour integral such that the contour C includes the interval (a, b) of the real axis. The residue theorem (5) may aid in such computations and may, in particular, relate the unknown integral to one that is already known. Figure 7.7-1 illustrates typical examples.

(b) To evaluate certain integrals of the form one applies Eq. (5) to a contour C comprising the interval (—R, R) of the real axis and the arc S of the circle |z| = R in the upper half plane. The following lemmas often yield the integral over S as R → ∞ :

whenever the integral exists for all finite values of R, and zf(z) tends uniformly to zero as |z| → ∞ with y ≥ 0

Jordan's Lemma: if F(z) is analytic in the upper half plane, except possibly for a finite number of poles, and tends uniformly to zero as |z| → ∞ with y ≥ 0, then for every real number m

The contour-integration method may yield the Cauchy principal value (Sec. 4.6-2b) of even if the integral itself does not exist. Jordan's lemma is particularly useful for the computation of improper integrals of the form and, because of Eq. (21.2-28), integrals of the form sin (mx) dx (inverse Laplace and Fourier transforms, Secs. 4.11-3 and 8.2-6).

(c) If S' is any semicircular arc of the circle |z — a| = є about a simple pole z = a of f(z), then

This fact is used (1) to evaluate integrals over contours “indented” around simple poles and (2) for computing the Cauchy principal values of certain improper integrals. (d) One may apply the residue theorem (5) to integrals of the type

where Φ is a rational function of cos φ and sin φ, with the aid of the transformation

(e) The Method of Steepest Descent (Saddle-point Method). For a given or suitably deformed contour C such that |ƒ(z)| is small except for a pronounced

FIG. 7.7-1a.

FIG. 7.7-1b.

FIG. 7.7-1c. Given 0 < a < 1,

The integrals over S and S' go to zero, and the integral over C equals 2πi Res (z = — 1) = 2πie^πi(a-1). Hence,

FIG. 7.7-1. Simple examples of contour-integral evaluation (Sec. 7.7-3; see Refs. 7.11, and 7.16 for more advanced problems).

maximum at z = zo with ƒ′(z₀) = 0 one may attempt an approximation of the form

The method applies especially to integrals of the form (see also Ref. 6.4).

7.7-4. Use of Residues for the Summation of Series. Given a contour C enclosing the points z = m, z = m + 1, z = m + 2, . . . z = n, where m is an integer, letƒ(z) be analytic inside and on C, except possibly for a number of poles a₁, a₂ . . . , a_N none of which coincide with z = m,z=m + l, z = m + 2, . . . , z = n. Then

where Res_g (a_i) is the residue of πf(z) cot πz at z = a_i; and

where Res_g (a_i) is the residue of πf(z) cosec πz at z = a_i. It is frequently possible to chose the contour C so that the integral on the right of Eq. (7) or (8) vanishes.

7.8. ANALYTIC CONTINUATION

7.8-1. Analytic Continuation and Monogenic Analytic Functions (see also Secs. 7.4-1 to 7.4-3). (a) Given a single-valued function ƒ₁(z) defined and analytic throughout a region D₁ the function ƒ₂(z) defined and analytic throughout a region D₂ is an analytic continuation of f₁(z) if and only if the intersection of D₁ and D₂ contains a simply connected open region Dc where ƒ₁(z) and ƒ₂(z) are identical.

The analytical continuation ƒ₂(z) is uniquely defined by the values of ƒ₁(z) in Dc (see also Sec. 7.3-3). Moreover, analytic continuations of f₁(z) satisfy every functional equation and, in particular, every differential equation satisfied by f₁(z) (principle of conservation of functional equations). One can thus use ƒ₂(z) to extend the region of definition of ƒ₁(z), and conversely:ƒ₁(z) and ƒ₂(z) are regarded as elements of a single analytic function ƒ(z) defined throughout D₁ and D₂. ƒ₁(z) and/or ƒ₂(z) may be capable of further analytic continuation leading to additional elements of ƒ(z).

(b) Multiple-valued Functions. Note that ƒ₁(z) and ƒ₂(z) are not necessarily identical throughout the entire intersection of their regions of definition. ƒ₂(z) may have an analytic continuation ƒ₃(z) defined on D₁ but not identical with ƒ₁(z). Analytic continuation can yield elements belonging to different branches of a multiple-valued analytic function f(z); the two values f(zo) obtained by analytic continuation of f₁(z) along two different routes C, C′ are identical if C and C′ do not enclose a branch point of f(z).

(c) The possible analytic continuations of a given element constitute a monogenic analytic function f(z) defined, except at isolated singularities, throughout the plane or on a connected region with natural boundaries. Whereas the choice of the successive elements defining f(z) is not fixed, the principle of conservation of functional equations applies to all elements, and any one element uniquely defines all branches, isolated singularities, and natural boundaries of f(z).

7.8-2. Methods of Analytic Continuation. (a) The standard method of analytic continuation starts with a function f(z) defined by its power-series expansion (7.5-4) inside some circle |z — a| = r. For any point z = b inside the circle, f(b), f'(b), . . . are then known and yield a Taylor-series expansion about z = b. The new power series converges inside a circle |z — b| = r′ which intersects the first circle but may contain a region not inside the first circle. This process may be continued up to the natural boundaries of the function; each power series is an element of f(z).

NOTE: A function defined as a power series with a finite radius of convergence has at least one singularity on the circle of convergence.

(b) Two functions f₁(z) and f₂(z) defined and analytic throughout the respective open regions D₁ and D₂ separated by a contour arc C are analytic continuations of each other (elements of the same monogenic analytic function) if they are equal and uniformly continuous on C.

(c) The Principle of Reflection. Let f(z) be defined and analytic throughout a region D intersected by the real axis, and let f(z) be real for real z. Then, for every value of a in D, f(z) is defined and analytic at z = a*, and

More generally, let f₁(z) be defined and analytic throughout a region D₁ bounded in part by a straight-line segment S_z, where f₁(z) is continuous; and let w = f₁(z) map S_z onto a corresponding straight-line segment S_w in the w plane. Then the function w = f₂(z) mapping the reflection in S_z of every point z in D into the reflection of w = fi(z) in S_w is an analytic continuation of f₁(z).

7.9. CONFORMAL MAPPING

7.9-1. Conformal Mapping. (a) A function w = f(z) maps points of the z plane (or Riemann surface, Sec. 7.4-3) into corresponding points of the w plane (or Riemann surface). At every point z such that f(z) is analytic and f′(z) ≠ 0 the mapping w = f(z) is conformal; i.e., the angle between two curves through such a point is reproduced in magnitude and sense by the angle between the corresponding curves in the w plane.

Infinitely small triangles around such points z are mapped onto similar infinitely-small triangles in the w plane; each triangle side is “stretched” in the ratio |f'(z)|:l and rotated through the angle arg f'(z). The superficial magnification (local magnification of small areas) due to the mapping w = f(z) = u(x, y) + iv(x, y) is

at every point z where the mapping is conformal. A conformal mapping transforms the lines x = constant, y = constant into families of mutually orthogonal trajectories in the w plane. Similarly, the lines u(x, y) = constant, v(x, y) = constant correspond to orthogonal trajectories in the z plane (see also Table 7.2-1).

A region of the z plane mapped onto the entire w plane by w = ƒ(z) is called a fundamental region of the function ƒ(z). Points where ƒ′(z) = 0 are called critical points of the transformation w = ƒ(z).* A mapping which preserves the magnitude but not necessarily the sense of the angle between two curves is called isogonal (example of an isogonal but not conformal mapping: w = z*).

(b) The mapping w = f(z) is conformal at infinity if and only if w = ƒ(1/) = F() maps the origin = 0 conformally into the w plane. Two curves are said to intersect at an angle γ at z = ∞ if and only if the transformation = 1/z results in two corresponding curves intersecting at an angle γ at = 0. Similarly, w = f(z) maps the point z = a conformally into w = ∞ if and only if = l/f(z) maps z = a conformally into the origin = 0 (see also Sec. 7.2-3).

7.9-2. Bilinear Transformations. (a) A bilinear transformation (linear fractional transformation, Moebius transformation)

establishes a reciprocal one-to-one correspondence between the points of the z plane and the points of the w plane. In particular, each of the two invariant points

(which may or may not be distinct) is mapped onto itself. The mapping is conformal everywhere except at the point z = — d/c, which corresponds

* Some authors also refer to the (singular) points where l/f'(z) = 0 as critical points of ƒ(z).

to w = ∞. Straight lines and circles in the z plane correspond to straight lines or circles in the w plane, and conversely; in this connection, every straight line is regarded as a circle of infinite radius through the point at infinity.

For any bilinear transformation mapping the four points z₁, z₂, z₃, z, respectively, into w₁, w₂, w₃, w

(invariance of the cross ratio or anharmonic ratio*). Equation (4) defines the unique bilinear transformation mapping three given points z₁, z₂, z₃, respectively, into three given points w₁, w₂, w₃. There exists a bilinear transformation which will transform a given circle or straight line in the z plane into a given circle or straight line in the w plane (see also Table 7.9-2 and Sec. 7.10-1).

SPECIAL CASES. The transformation

where A and B are complex numbers, corresponds to a rotation through the angle arg A together with a stretching or contraction by a factor |A|, followed by a translation through the vector displacement B. The linear transformation (5) is the most general conformal mapping which preserves similarity of geometrical figures.

The transformation

represents a geometrical inversion of the point z with respect to the unit circle about the origin, followed by a reflection in the real axis. The transformation (6) maps

Straight lines through the origin into straight lines through the origin

Circles through the origin into straight lines which do not contain the origin, and conversely

Circles which do not pass through the origin into circles which do not pass through the origin

(b) The bilinear transformations (2) constitute a group; inverses and products of bilinear transformations are bilinear transformations (Sec. 12.2-7). Every bilinear transformation (2) may be expressed as the result (product) of three successive simpler bilinear transformations (see also Sec. 7.9-2a):

* The cross ratio (4) is real (and admits a geometrical interpretation, Ref. 7.2) if and only if the points z₁, z₂, z₃, z (and hence also the points w₁, w₂, w₃, w) lie on a circle or straight line.

7.9-3. The Transformation w = 1/2(z + l/z). The transformation

is equivalent to

The transformation (8) is conformal except at the critical points z = 1 and z = — 1. Both the exterior and the interior of the unit circle |z| = 1 are mapped onto the entire plane with the exception of the straight-line segment (u = — l,u = 1), which corresponds to the unit circle |z| = 1 itself. Some important properties of this transformation are outlined in Table 7.9-1.

7.9-4. The Schwarz-Christoffel Transformation. The Sehwarz-Christoffel transformation

maps the upper half-plane y > 0 conformally onto the interior of a polygon in the w plane; the polygon corresponds to the x axis, the vertices w₁, w₂, . . . , w_n correspond to different points x₁, x₂, . . . , x_n on the x axis, and the exterior angle of the polygon at the vertex w_j equals α_j (j = 1, 2, . . . , n). For any given polygon in the w plane, three of the x_j′ can be chosen arbitrarily; the other x_j′ and the parameters A and B are then uniquely determined.

If x_n is chosen to be infinitely large Eq. (9) reduces to

where A′ and B′ are constant parameters, and x′₁, x′₂, . . . , x′_n-1 are new points on the x axis.

Table 9.9-1. Properties of the Transformation (see also Tables 7.9-2 and 7.10-1)

The actual determination of the x_i or x′_i is quite complicated, except in certain “degenerate” cases where one or more of the angles α_i vanish (see also Ref. 7.4). Application of the Schwarz-Christoffel transformation to parallelograms and rectangles in the w plane yields elliptic functions z of w (Sec. 21.6-1).

The transformations 26 to 30 in Table 7.9-2 are special cases of the Schwarz-Christoffel transformation.

7.9-5. Table of Transformations. Table 7.9-2 illustrates a number of special transformations of interest in various applications.

Table 7.9-2. Table of Transformations of Regions*

FIG. 1. w = z².

FIG. 2. w = z².

FIG. 3. w = z²;

FIG. 4. w = 1/z.

*From R. V. Churchill, Introduction to Complex Variables and Applications, 2d ed., McGraw-Hill, New York, 1960.

FIG. 5. w = 1/z.

FIG. 6. w = e².

FIG. 7. w = e².

FIG. 8. w = e².

FIG. 9. w = sin z.

FIG. 10. w = sin z.

FIG. 11.

FIG. 12.

FIG. 13.

FIG. 14.

FIG. 15.

FIG. 16. w = z + 1/z.

FIG. 17. w = z + 1/z.

FIG. 18.

FIG. 19.

FIG. 20.

FIG. 21. centers of circles at z = coth c_n(n = 1, 2).

FIG. 22.

FIG. 23.

FIG. 24.

FIG. 25.

FIG. 26. w = πi + z - log_e z.

FIG. 27.

FIG. 28.

FIG. 29.

FIG. 30.

Table 7.10-1. Table of Transformations Mapping a Specified Region D Conformally onto the Unit Circle (|w| ≤ 1)

7.10. FUNCTIONS MAPPING SPECIFIED REGIONS ONTO THE UNIT CIRCLE

7.10-1. Riemann's Mapping Theorem. For every simply connected open region D in the z plane, with the exception of the entire z plane and the entire z plane minus one point, there exists a conformal mapping w = f(z) which establishes a reciprocal one-to-one (biunique) correspondence between all points of D and all interior points of the unit circle |w| = 1. The analytic function f(z) is uniquely determined if the mapping of a point in D and a direction through that point is specified. If D is bounded by a regular curve (Sec. 3.1-13) C, then f(z) is continuous on C and establishes a reciprocal one-to-one correspondence between all points of C and all points on the unit circle |w| = 1.

NOTE: (1) The transformation thus specified defines the analytic function f(z); and (2) with the trivial exceptions noted above, it is possible to map every region D bounded by a simple contour conformally onto any other region D' bounded by a simple contour. The problem of mapping a region D conformally onto the unit circle is closely related to the solution of Dirichlet's boundary-value problem for the region D (Sec. 15.6-9).

Frequently, a desired conformal mapping can be obtained through successive relatively simple transformations.

Table 7.10-1 lists a number of transformations mapping a given region D conformally onto the unit circle.

7.11. RELATED TOPICS, REFERENCES, AND BIBLIOGRAPHY

7.11-1. Related Topics. The following topics related to the study of functions of a complex variable are treated in other Chapters of this handbook:

Complex numbers Chap. 1

Roots of polynomials Chap. 1

Functions, limits, differentiation, integration, infinite series Chap. 4

Laplace transformation Chap. 8

Two-dimensional potential theory, conjugate harmonic functions Chap. 15

Special functions Chap. 21

7.11-2. References and Bibliography.

      7.1. Ahlfors, L. V.: Complex Analysis, 2d ed., McGraw-Hill, New York, 1966.

      7.2. Bieberbach, L.: Einführung in die Konforme Abbildung, De Gruyter, Berlin, 1927.

      7.3. Carrier, G. F., et al.: Functions of a Complex Variable, McGraw-Hill, New York, 1966.

      7.4. Churchill, R. V.: Complex Variables and Applications, 2d ed., McGraw-Hill, New York, 1960.

      7.5. Copson, E. T.: Theory of Functions of a Complex Variable, Oxford, New York, 1960.

      7.6. Cunningham, J.: Complex-variable Methods in Science and Technology, Van Nostrand, Princeton, N.J., 1965.

      7.7. Hille, E.: Analytic Function Theory, 2 vols., Blaisdell, New York, 1959/62.

      7.8. Hurwitz, A., and R. Courant: Allgemeine Funktionentheorie und elliptische Funktionen, Springer, Berlin, 1964.

      7.9. Knopp, K.: Funktionentheorie, translated by F. Bagemihl, Dover, New York, 1947.

      7.10. Kober, H.: Dictionary of Conformal Representations, Dover, New York, 1952.

      7.11. McLachlan, N. W.: Complex Variable and Operational Calculus with Applications, Macmillan, New York, 1946.

      7.12. Nehari, Z.: Conformal Mapping, McGraw-Hill, New York, 1952.

      7.13. : Introduction to Complex Analysis, Allyn and Bacon, Boston, 1961.

      7.14. Pennisi, L. L.: Elements of Complex Variables, Holt, New York, 1963.

      7.15. Springer, G.: Introduction to Riemann Surfaces, Addison-Wesley, Reading, Mass., 1957.

      7.16. Whittaker, E. T., and G. N. Watson: A Course in Modern Analysis, Cambridge, New York, 1958.