Mathematical Handbook for Scientists and Engineers

1.1-1 This chapter deals with the algebra * of real and complex numbers, i.e., with the study of those relations between real and complex numbers which involve a finite number of additions and multiplications. This is considered to include the solution of equations based on such relations, even though actual exact numerical solutions may require infinite numbers of additions and/or multiplications. The definitions and relations presented in this chapter serve as basic tools in many more general mathematical models (see also Sec. 12.1-1).

1.1-2. Real Numbers. The axiomatic foundations ensuring the self-consistency of the real-number system are treated in Refs. 1.1 and 1.5, and lead to the acceptance of the following rules governing the addition and multiplication of real numbers.

The real number 0 (zero, additive identity) has the properties

for every real a.

The (unique) additive inverse —a and the (unique) multiplicative inverse (reciprocal) a^-1 = 1/a of a real number a are respectively denned by

Division by 0 is not admissible.

In addition to the “algebraic” properties (1), the class of the positive integers 1,2, . . . has the properties of being simply ordered (Sec. 12.6-2; n is “greater than” m or n > m if and only if n = m + x where x is a positive integer) and well-ordered (every nonempty set of positive integers has a smallest element). A set of positive integers containing (1) 1 and the “successor” n + 1 of each of its elements n, or (2) all integers less than nfor any n, contains all positive integers (Principle of Finite Induction).

The properties of positive integers may be alternatively defined by Peano's Five Axioms, viz., (1) 1 is a positive integer, (2) each positive integer n has a unique successor S(n), (3) S(n) ≠ 1, (4) S(n) — S(m) implies n = m, (5) the principle of finite induction holds. Addition and multiplication satisfying the rules (1) are defined by the “recursive” definitions n + 1 = S(n), n + S(m) = S(n + m); n . 1 = n, n . S(m) = n . m + n.

Operations on the elements m — n of the class of all integers (positive, negative, or zero) are interpreted as operations on corresponding pairs (m, n) of positive integers m, n such that (m — n) + n = m, where 0, defined by n + 0 = n, corresponds to (n, n), for all n. An integer is negative if and only if it is neither positive nor zero. The study of the properties of integers is called arithmetic.

Operations on rational numbers m/n (n ≠ 0) are interpreted as operations on corresponding pairs (m, n) of integers m, n such that (m/n)n = m. m/n is positive if and only if mn is positive.

Real algebraic (including rational and irrational) numbers, corresponding to (real) roots of algebraic equations with integral coefficients (Sec. 1.6-3) and real transcendental numbers, for which no such correspondence exists, may be introduced in terms of limiting processes involving rational numbers (Dedekind cuts, Ref. 1.5).

The class of all rational numbers comprises the roots of all linear equations (Sec. 1.8-1) with rational coefficients, and includes the integers. The class of all real algebraic numbers comprises the real roots of all algebraic equations (Sec. 1.6-3) with algebraic coefficients, including the rational numbers. The class of all real numbers contains the real roots of all equations involving a finite or infinite number of additions and multiplications of real numbers and includes real algebraic and transcendental numbers (see also Sec. 4.3-1).

A real number a is greater than the real number b (a > b, b < a if and only if a = b + x, where a; is a positive real number (see also Secs. 1.1-5 and 12.6-2).

1.1-3. Equality Relation (see also Sec. 12.1-3). An equation a = b implies b = a (symmetry of the equality relation), and a + c = b + c, ac — bc [in general, f(a) = f(b) if f(a) stands for an operation having a unique result], a = b and b = c together imply a = c (transitivity of the equality relation), ab ≠ 0 implies a ≠ 0, b ≠ 0.

1.1-4. Identity Relation. In general, an equation involving operations on a quantity x or on several quantities x₁ x₂, . . . will hold only for special values of x or special sets of values x₁, x₂, . . . (see also Sec. .1.6-2). If it is desired to stress the fact that an equation holds for all values of x or of x₁ x₂ . . . within certain ranges of interest, the identity symbol = may be used instead of the equality symbol = [EXAMPLE: (x — l)(x + 1) s x² — 1], and/or the ranges of the variables in question may be indicated on the right of the equation, a = b (better a b) is also used with the meaning “a is defined as equal to b.”

1.1-5. Inequalities (see also Sees. 12.6-2 and 12.6-3). a > b implies b < a, a + c > b + c, ac > bc (c > 0), —a < —b, 1/a < 1/b (a > 0, b > 0). A real number a is positive (a > 0), negative (a < 0), or zero (a = 0). Sums and products of positive numbers are positive, a ≤ A, b ≤ B implies a + b ≤ A + B. a ≥ b, b ≥ c implies a ≥ c.

1.1-6. Absolute Values (see also Sees. 1.3-2 and 14.2-5). The absolute value |a| of a real number a is defined as equal to a if a ≥ 0 and equal to -a if a < 0. Note

1.2. POWERS ROOTS, LOGARITHMS, AND FACTORIALS. SUM AND PRODUCT NOTATION

1.2-1. Powers and Roots. The n^th power of any real number (base) a is defined as the product of n factors equal to a, where the exponent

n is a positive integer. The resulting relations

are postulated to apply for all real values of p and q and thus serve to define powers involving exponents other than positive integers:

A p^th root of the radicand a is a solution of the equation x^p = a. is the square root of a, and is its cube root. Powers and roots with irrational exponents can be introduced through limiting processes (see also Sec. E-6). In general, is not unique, and some or all of the roots of a given real radicand a may not be real numbers (Sec. 1.3-3). If a is real and positive, many authors specifically denote the real positive solution values of x² = a, x³ = a, x⁴ = a, . . . as , , , . . . . The real solutions of x² = a, x⁴ = a, . . . are then written as To emphasize a choice of the positive square root, one may write . For q ≠ 0

1.2-2. Formulas for Rationalizing the Denominators of Fractions.

1.2-3. Logarithms. The logarithm x = log_c a to the base c > 0 (c ≠ 1) of the number (numerus) a > 0 may be defined by

Refer to Table 7.2-1 and Sec.21.2-10 for a more general discussion of logarithms. log_c a may be a transcendental number (Sec. 1.1-2). Note

Of particular interest are the “common” logarithms to the base 10 and the natural (Napierian) logarithms to the base

e is a transcendental number. log_e a may be written In a, log a or log nat a. log₁₀ a is sometimes written log a. Note

1.2-4. Factorials. The factorial n! of any integer n ≥ 0 is defined by

Refer to Sec. 21.4-2 for approximation formulas.

1.2-5. Sum and Product Notation. For any two integers (positive, negative, or zero) n and m ≥ n

Note

Refer to Chap. 4 for infinite series; and see Sec.E-3.

1.2-6. Arithmetic Progression. If a₀ is the first term and d is the common difference between successive terms a_i, then

1.2-7. Geometric Progression. If a₀ is the first term and r is the common ratio of successive terms, then (see Sec. 4.10-2 for infinite geometric series)

1.3. COMPLEX NUMBERS

1.3-1. Introduction (see also Sec. 7.1-1). Complex numbers (sometimes called imaginary numbers) are not numbers in the elementary sense used in connection with counting or measuring; they constitute a new class of mathematical objects defined by the properties described below (see also Sec. 12.1-1).

Each complex number c may be made to correspond to a unique pair (a, b) of real numbers a, b, and conversely. The sum and product of two complex numbers c₁ ↔ (a₁, b₁) and c₂ ↔ (a₂, b₂) are defined as c₁ + c₂ ↔ (a₁ + a₂, b₁ + b₂) and C₁C₂ ↔ (a₁a₂ — b₁b₂, a₁b₂ + a₂b₁), respectively. The real numbers a are “embedded” in this class of complex numbers as the pairs (a, 0). The unit imaginary number i defined as i ↔ (0, 1) satisfies the relations

Each complex number c ↔(a, b) may be written as the sum c = a + ib of a real number a ↔ (a, 0) and a pure imaginary number ib ↔ (0, b). The real numbers a = Re (c) and b = Im (c) are respectively called the real part of c and the imaginary part of c. Two complex numbers c = a + ib and c^* = a — ib having equal real parts and equal and opposite imaginary parts are called complex conjugates.

Two complex numbers c₁ = a₁ + ib₁ and c₂ = a₂ + ib₂ are equal if and only if their respective real and imaginary parts are equal, i.e., c₁ = c₂ if and only if a₁ = a₂, b₁ = b₂. c = a + ib = 0 implies a = b = 0. Addition and multiplication of complex numbers satisfies all rules of Sees. 1.1-2 and 1.2-1, with

The class of all complex numbers contains the roots of all equations based on additions and multiplications involving complex numbers and includes the real numbers.

1.3-2. Representation of Complex Numbers as Points or Position Vectors. Polar Decomposition (see also Sec. 7.2-2). Complex

FIG. 1.3-1. Representation of complex numbers as points or position vectors. The x axis and the y axis are called the real axis and the imaginary axis, respectively.

numbers z = x + iy are conveniently represented as points (z) = (x, y) or corresponding position vectors (Sees.2.1-2 and 3.1-5) in the Argand or Gauss plane (Fig. 1.3-1). The (rectangular cartesian, Sec.2.1-3) x axis and y axis are referred to as the real axis and imaginary axis, respectively. The abscissa and ordinate of each point (z) respectively represent the real part x and the imaginary part y of z. The corresponding polar coordinates (Sec. 2.1-8)

are, respectively, the absolute value (norm, modulus) and the argument (amplitude) of the complex number z Note

The absolute values of complex numbers satisfy the relations (1.1-4) to (1.1-6); if z is a real number, the definition of |z| reduces to that of Sec. 1.1-6.

For any two sets of (real or) complex numbers a_h a₂, . . . , a_n; β₁, β₂, . . . , β_n (see also Sees. 14.2-6 and 14.2-6a),

1.3-3. Representation of Addition, Multiplication, and Division. Powers and Roots. Addition of complex numbers corresponds to addition of the corresponding position vectors (see also Sees. 3.1-5 and 5.2-2). Given z₁ = r₁(cos φ₁ + i sin φ₁), z₂ = r₂(cos φ₂ + i sin φ₂),

(See Sec. 21.2-9 for the case of complex exponents.) All formulas of Sees. 1.2-1 to 1.2-7 hold for complex numbers (see also Sees. 1.4-1 to 1.4-3).

Note

In particular,

1.4. MISCELLANEOUS FORMULAS

1.4-1. The Binomial Theorem and Related Formulas. If a, b, c are real or complex numbers,

(Binomial Theorem for integral exponents n; see also Sec. 21.2-12). The binomial coefficientsare discussed in detail in Sec. 21.5-1.

If n is an even positive integer,

If n is an odd positive integer,

Note also

1.4-2. Proportions. a:b = c:d or a/b — c/d implies

In particular,

1.4-3. Polynomials. Symmetric Functions. (a) A polynomial in (integral rational function of) the quantities x₁, x₂, . . . , x_n is a sum involving a finite number of terms of the form ax₁^klx₂^k2 . . . x_n^kn, where each k_i is a nonnegative integer. The largest value of k₁+ k₂ + . . . + k_n occurring in any term is the degree of the polynomial. A polynomial is homogeneous if and only if all its terms are of the same degree (see also Sec. 4.5-5).

(b) A polynomial in x₁, x₂, . . . , x_n (and more generally, any function of x₁, x₂, . . . , x_n) is (completely) symmetric if and only if its value is unchanged by permutations of the x₁, x₂, . . . , x_n for any set of values x₁, x₂, . . . , x_n. The elementary symmetric functions S₁, S₂ . . . ,S_n of x₁, x₂, . . . , x_n are the polynomials

where S_k is the sum of all products combining k factors x_i without repetition of subscripts (see also Table C-2). Every polynomial symmetric in x₁, x₂, . . . , x_n can be rewritten as a unique polynomial in S₁, S₂ , . . . , S_n; the coefficients in the new polynomial are algebraic sums of integral multiples of the given coefficients.

Every polynomial symmetric in x₁, x₂, . . . , x_n can also be expressed as a polynomial in a finite number of the symmetric functions

The symmetric functions (9) and (10) are related by Newton's formulas

where one defines S_k = 0 for k > n and k < 0, and S_o = 1 (see also Ref. 1.2 for explicit tabulations of Newton's formulas; and see also Sec. 1.6-4). Note that the relations (11) do not involve n explicitly.

1.5. DETERMINANTS

1.5-1. Definition. The determinant

of the square array (matrix, Sec. 13.2-1) of n² (real or complex) numbers (elements) a_ik is the sum of the n! terms (-l)^ra_1k₁a_2k₂ . . . a_nk_n each corresponding to one of the n! different ordered sets k₁, k₂, . . . , k_n obtained by r interchanges of elements from the set 1, 2, . . . , n. The number n is the order of the determinant (1).

The actual computation of a determinant in terms of its elements is simplified by the use of Secs. 1.5-2 and 1.5-5a. Note that

1.5-2. Minors and Cofactors. Expansion in Terms of Cofactors.

The (complementary) minor D_ik of the element a_ik in the n^th-order determinant (1) is the (n - l)^st-order determinant obtained from (1) on erasing the i^th row and the k^th column. The cofactor A_ik of the element a_ik is the coefficient of a_ik in the expansion of D, or

A determinant D may be represented in terms of the elements and cofactors of any one row or column as follows:

Note also that

1.5-3 Examples: Second- and Third-order Determinants.

1.5-4. Complementary Minors. Laplace Development. The m^th-order determinant M obtained from the nth-order determinant D by deleting all rows except for the m rows labeled i₁, i₂, . . . , i_m (m ≤ n), and all columns except for the m columns labeled k₁, k₂, . . . , k_m is an m-rowed minor of D. The m-rowed minor M and the (n – m)-rowed minor M' of D obtained by deleting the rows and columns conserved in M are complementary minors; in the special case m = n, M′ = 1. The algebraic complement M″ of M is defined as (—l)^{i₁+i₂ . . . +i_m+k₁+k₂ . . . +k_m}M′. Given any m rows (or columns) of D, D is equal to the sum of the products MM″ of all m-rowed minors M using these rows (or columns) and their algebraic complements M″ (Laplace development by rows or columns).

An n^th-order determinant D has m-rowed principal minors whose diagonal elements are diagonal elements of D.

1.5-5. Miscellaneous Theorems. (a) The value D of a determinant (1) is not changed by any of the following operations:

1. The rows are written as columns, and the columns as rows [interchange of i and k in Eq. (1)].

2. An even number of interchanges of any two rows or two columns.

3. Addition of the elements of any row (or column), all multiplied, if desired, by the same parameter a, to the respective corresponding elements of another row (or column, respectively).

EXAMPLES:

(b) An odd number of interchanges of any two rows or two columns is equivalent to multiplication of the determinant by — 1.

(c) Multiplication of all the elements of any one row or column by a factor a is equivalent to multiplication of the determinant by a.

(d) If the elements of the j^th row (or column) of an n^th-order determinant D are represented as sums , D is equal to the sum of m n^th-order determinants D_r. The elements of each D_r are identical with those of D, except for the elements of the j^th row (or column, respectively), which are c_r1, c_r2, . . . , c_rn.

EXAMPLE:

(e) A determinant is equal to zero if

1. All elements of any row or column are zero.

2. Corresponding elements of any two rows or columns are equal, or proportional with the same proportionality factor.

1.5-6. Multiplication of Determinants (see also Sec. 13.2-2). The product of two n^th-order determinants det [a_ik] and det [b_ik] is

1.5-7. Changing the Order of Determinants. A given determinant may be expressed in terms of a determinant of higher order as follows:

where the on are arbitrary. This process can be repeated as desired.

The order of a given determinant may sometimes be reduced through the use of the relation

1.6. ALGEBRAIC EQUATIONS: GENERAL THEOREMS

1.6-1. Introduction. The solution of algebraic equations is of particular importance in connection with the characteristic equations of linear systems in physics (see also Secs. 9.4-1, 9.4-4, and 14.8-5). The general location of the roots needed (e.g., for stability determinations) may be investigated by the methods of Sec. 1.6-6 and/or Sec. 7.6-9. Numerical solutions are discussed in Secs. 20.2-1 to 20.2-3.

1.6-2. Solution of an Equation. Roots. To solve an equation (see also Sec. 1.1-3)

for the unknown x means to find values of x [roots of Eq. (1), zeros of f(x)] which satisfy the given equation. x = x₁ is a root (zero) of order (multiplicity) m (multiple root if m < 1; see also Sec. 7.6-1) if and only if, for x = x₁, f(x)/(x - x₁)^m–1 = 0 and f(x)/(x - x₁)^m ≠ 0. A complete solution of Eq. (1) specifies all roots together with their orders. Solutions may be verified by substitution.

1.6-3. Algebraic Equations. An equation (1) of the form

where the coefficients a_i are real or complex numbers, is called an algebraic equation of degree n in the unknown x. f(x) is a polynomial of degree n in. x (rational integral function; see also Secs. 4.2-2d and 7.6-5). a_n is the absolute term of the polynomial (2).

An algebraic equation of degree n has exactly n roots if a root of order m is counted as m roots (Fundamental Theorem of Algebra).

Numbers expressible as roots of algebraic equations with real integral coefficients are algebraic numbers (in general complex, with rational and/or irrational real and imaginary parts); if the coefficients are algebraic, the roots are still algebraic (see also Sec. 1.1-2). General formulas for the roots of algebraic equations in terms of the coefficients and involving only a finite number of additions, subtractions, multi-plications, divisions, and root extractions exist only for equations of degree one (linear equations, Sec. 1.8-1), two (quadratic equations, Sec. 1.8-2), three (cubic equations, Secs. 1.8-3 and 1.8-4), and four (quartic equations, Sees. 1.8-5 and 1.8-6).

1.6-4. Relations between Roots and Coefficients. The symmetric functions Sk and Sk (Sec. 1.4-3) of the roots x₁, x₂, . . . , x_n of an algebraic equation (2), are related to the coefficients a₀, a₁, . . . , a_n as follows:

where one defines a_k = 0 for k > n and k < 0. The equations (1.6-4) are another version of Newton's formulas (1.4-11). Note also

1.6-5. Discriminant of an Algebraic Equation. The discriminant Δ of an algebraic equation (2) is the product of a_o^2n-2 and the squares of all differences (x_i — x_k)(i > k) between the roots x_i of the equation (a multiple root of order m is considered as m equal roots with different subscripts),

where R(ƒ, ƒ′) is the resultant (Sec. 1.7-3) of ƒ(x) and its derivative (Sec. 4.5-1) ƒ′(x). Δ is a symmetric function of the roots x₁, x₂, . . . , x_n and vanishes if and only if ƒ(x) has at least one multiple root [which is necessarily a common root of ƒ(x) and ƒ′(x); see also Sec. 1.6-6g]. The second determinant in Eq. (6) is called Vander-monde's determinant.

1.6-6. Real Algebraic Equations and Their Roots. An algebraic equation (2) is called real if and only if all coefficients a_i are real; the corresponding real polynomial f(x) is real for all real values of x. The following theorems are useful for determining the general location of roots (e.g., prior to numerical solution, Sec. 20.2-1; see also Secs. 9.4-4 and 14.8-5). In theorems (b) through (f), a root of order m is counted as m roots.

(a)Complex Roots. Complex roots of real algebraic equations occur in pairs'of complex conjugates (Sec. 1.3-1). A real algebraic equation of odd degree must have at least one real root.

(b)Routh-Hurwitz Criterion. The number of roots with positive real parts of a real algebraic equation (2) is equal to the number of sign changes (disregard vanishing terms) in either one of the sequences

Given a₀ > 0, all roots have negative real parts if and only if T₀, T₁, T₂, . . . , T_n are all positive. This is true if and only if all ai and either all even-numbered T_k or all odd-numbered T_k are positive (Liénard-Chipart Test).

ALTERNATIVE FORMULATION. All the roots of a real n^th-degree equation (2) have negative real parts if and only if this is true for the (n — l)^st-degree equation

This theorem may be applied repeatedly and yields a simple recursion scheme useful, for example, for stability investigations. The number of roots with negative real parts is precisely equal to the number of negative multipliers -a₀^(j)/a₁^(j) (j = 0,1, 2, . . . , n - l; a₀⁽⁰⁾ = a₀ > 0, a₁⁽⁰⁾ = a₁) encountered in successive applications of the theorem. The method becomes more complicated if one of the a₁^(j) vanishes (see Ref. 1.6, which also indicates an extension to complex equations).

(c) Location of Real Roots: Descartes's Rule of Signs. The number of positive real roots of a real algebraic equation (2) either is equal to the number N_a of sign changes in the sequence ao, ai, . . . , a_n of coefficients, where vanishing terms are disregarded, or it is less than N_a by a positive even integer. Application of this theorem to f( — x) yields a similar theorem for negative real roots.

(d) Location of Real Roots: An Upper Bound (Sec. 4.3-3a) for the Real Roots. If the first k coefficients a₀, a₁, . . . , a_k–1 in a real algebraic equation (2) are nonnegative (a_k is the first negative coefficient) then all real roots of Eq. (2) are smaller than , where q is the absolute value of the negative coefficient greatest in absolute value. Application of this theorem to f(–x) may similarly yield a lower bound of the real roots.

(e) Location of Real Roots: Rolle's Theorem (see also Sec.4.7-la). The derivative (Sec. 4.5-1) ƒ′(x) of a real polynomial f(x) has an odd number of real zeros between two consecutive real zeros of f(x).

f(x) = 0 has no real root or one real root between two consecutive real roots a, b of f'(x) = 0 if f(a) ≠ 0 and f(b) ≠ 0 have equal or opposite signs, respectively. At most, one real root of f(x) = 0 is greater than the greatest root or smaller than the smallest root of f'(x) = 0.

(f) Location of Real Roots: Budan's Theorem. For any real algebraic equation (2), let N(x) be the number of sign changes in the sequence of derivatives (Sec. 4.5-1) f(x), ƒ′(x), ƒ″(x), . . . , f⁽ⁿ⁾(x), if vanishing terms are disregarded. Then the number of real roots of Eq. (2) located between two real numbers a and b > a not themselves roots of Eq. (2) is either N(a) – N(b), or it is less than N(a) – N(b) by a positive even integer.

The number of real roots of Eq. (2) located between a and b is odd or even if f(a) and f(b) have opposite or equal signs, respectively.

(g) Location of Real Roots: Sturm's Method. Given a real algebraic equation (2) without multiple roots (Sec. 1.6-2), let N(x) be the number of sign changes (disregard vanishing terms) in the sequence of functions

where for i > 1 each f_i(x) is (–1) times the remainder (Sec. 1.7-2) obtained on dividing f_i–2(x) by f_i–1(x); f_n(x) ≠ 0 is a constant. Then the number of real roots of Eq. (2) located between two real numbers a and b > a not themselves roots of Eq. (2) is equal to N(a) – N(b).

Sturm's method applies even if, for convenience in computation, a function f_i(x) in the above process is replaced by F_i(x) = f_i(x)/k(x), where k(x) is a positive constant or a polynomial in x positive for a ≤ x ≤ b, and the remaining functions are based on F_i(x) instead of on f(x). Similar operations may be performed again on any of the F_j(x), etc.

If f(x) has multiple roots, f(x) and ƒ′(x) have a common divisor (Sec. 1.7-3); in this case, f_n(x) is not a constant, and N(a) – N(b) is the number of real roots between a and b, where each multiple root is counted only once.

1.7. FACTORING OF POLYNOMIALS AND QUOTIENTS OF POLYNOMIALS. PARTIAL FRACTIONS

1.7-1. Factoring of a Polynomial (see also Sec. 7.6-6). If a polynomial F(x) can be represented as a product of polynomials f₁(x), f₂(x), . . . , f_s(x), these polynomials are called factors (divisors) of F(x). If x = x₁ is a zero of order m of any factor f_i(x), it is also a zero of order M ≥ m of F(x). Every (real or complex) polynomial f(x) of degree n in x can be expressed as a product of a constant and n linear factors (x – α_k) in one and only one way, namely,

where the x_k are the zeros of f(x); a zero x_k of order m_k (Sec. 1.6-2) contributes m_k factors (x – x_k) (Factor Theorem). Pairs of factors [x – (a_k + iω_k)], [x – (a_k – iω_k)] corresponding to pairs of complex conjugate roots (see also Sec. 1.6-6a) x_k = a_k + iω_k, x_k = a_k – iω_k may be combined into real quadratic factors [(x – a_k)² + ω_k²].

1.7-2. Quotients of Polynomials. Remainder. Long Division. The quotient F(x)/f(x) of a polynomial F(x) of degree N and a polynomial f(x) of degree n < N may be expressed in the form

where the remainder x₁(x) is a polynomial of degree smaller than n. The coefficients b_k and the remainder x₁(x) are uniquely determined, e.g., by the process of long division (division algorithm) indicated in Fig. 1.7-1.

In Fig. 1.7-1, each product b₀f(x), b₁f(x), . . . is subtracted in turn, with the coefficients b₀, b₁, . . . chosen so as to eliminate the respective coefficients of x^N, x^N^–l, . . . , in successive differences until the remainder is reached. The remainder r₁(x) vanishes if and only if f(x) is a divisor (Sec. 1.7-1) of F(x).

The remainder obtained on dividing any polynomial f(x) by (x – c) is equal to f(c) (Remainder Theorem).

EXAMPLES

FIG.1.7-1.Long division.

1.7-3. Common Divisors and Common Roots of Two Polynomials. If a polynomial g(x) is a common divisor (factor) of F(x) and f(x), its zeros are common zeros of F(x) and f(x). In the quotient (2), any common divisor may be factored out and canceled as with numerical fractions.

F(x) and f(x) have at least one common root (and thus a common divisor of degree greater than zero) if and only if the determinant of order N + n

[resultant of F(x) and f(x)] is equal to zero; otherwise, F(x) and f(x) are relatively prime.

The greatest common divisor (common factor of greatest degree) of F(x) and f(x) is uniquely defined except for a constant factor and may be obtained as follows: Divide r₁(x) into f(x); divide the resulting remainder r₂(x) into r₁(x), and continue until some remainder, r_k(x), say, vanishes. Then any constant multiple of r_k–1(x) is the desired greatest common divisor.

1.7-4. Expansion in Partial Fractions. Any quotient g(x)/f(x) of a polynomial g(x) of degree m and a polynomial f(x) of degree n > m, without common roots (Sec. 1.7-3) can be expressed as a sum of n partial fractions corresponding to the roots x_k (of respective orders m_k) of f(x) = 0 as follows:

The coefficients b_kj are obtained by one of the following methods, or by a combination of these methods:

1. If m_k = 1 (x_k is a simple root), then b_k1 = g(x_k)/ƒ′(x_k).

2. Multiply both sides of Eq. (4) by f(x) and equate coefficients of equal powers of x on both sides.

3. Multiply both sides of Eq. (4) by f(x) and differentiate successively. Let φ_k(x) = f(x)/(x – x_k)^m_k. Then obtain b_{km_k}, b_{km_k–1}, . . . successively from

The partial fractions corresponding to any pair of complex conjugate roots a_k + iw_k, a_k — iw_k of order m_k are usually combined into

The coefficients a_kj and d_kj may be determined directly by method 2 above. If g(x) and f(x) are real polynomials (Sec. 1.6-6), all coefficients b_kj, c_kj; d_kj in the resulting partial-fraction expansion are real.

Every rational function of x (Sec. 4.2-2c) can be represented as a sum of a polynomial and a finite set of partial fractions (see also Sec. 7.6-8). Partial-fraction expansions are important in connection with integration (Sec. 4.6-6c) and integral transforms (Sec. 8.4-5).

1.8. LINEAR, QUADRATIC, CUBIC, AND QUARTIC EQUATIONS

1.8-1. Solution of Linear Equations. The solution of the general equation of the first degree (linear equation)

1.8-2. Solution of Quadratic Equations. The quadratic equation

has the roots

The roots x₁ and x₂ are real and different, real and equal, or complex conjugates if the discriminant (Sec. 1.6-5) D = b² — 4ac is, respectively, positive, zero, or negative. Note x₁ + x₂ = –b/a, x₁x₂ = c/a.

1.8-3. Cubic Equations: Cardan's Solution. The cubic equation

is transformed to the “reduced” form

through the substitution x = y — a/3. The roots y₁, y₂, y₃, of the “reduced” cubic equation (6) are

where the real values of the cube roots are used. The cubic equation has one real root and two conjugate complex roots, three real roots of which at least two are equal, or three different real roots, if Q is positive, zero, or negative, respectively. In the latter case (“irreducible” case), the method of Sec. 1.8-4a may be used. Note that the discriminants (Sec. 1.6-5) of Eq. (5) and Eq. (6) are both equal to - 108Q.

1.8-4. Cubic Equations: Trigonometric Solution, (a) If Q < 0 (“irreducible case”)

The real value of the cube root is used.

1.8-5. Quartic Equations: Descartes-Euler Solution. The quartic equation (biquadratic equation)

is transformed to the “reduced” form

through the substitution x = y — a/4. The roots y₁, y₂, y₃, y₄ of the “reduced” quartic equation (11) are the four sums

with the signs of the square roots chosen so that

where z₁, z₂, z₃ are the roots of the cubic equation

1.8-6. Quartic Equations: Ferrari's Solution. Given any root y₁ of the resolvent cubic equation corresponding to Eq. (10)

the four roots of the quartic equation (10) are given as roots of the two quadratic equations

where the radicand on the right is a perfect square. Note that the discriminants (Sec. 1.6-5) of Eq. (10) and Eq. (15) are equal.

1.9. SYSTEMS OF SIMULTANEOUS EQUATIONS

1.9-1. Simultaneous Equations. To solve a suitable set (system) of simultaneous equations

for the unknowns x₁, x₂, . . . means to determine a set of values of x₁, x₂, . . . which satisfy the equations (1) simultaneously. The solution is complete if all such sets are found. One can frequently eliminate successive unknowns x_j from a system (1), e.g., by solving one equation for x_j and substituting the resulting expression in the remaining equations. The number of equations and unknowns is thus reduced until a single equation remains to be solved for a single unknown. The pro-cedure is then repeated to yield a second unknown, etc. Solutions may be verified by substitution.

To eliminate x₁, say, from two equations f₁(x₁, x₂) = 0,f₂(x₁, x₂) = 0 where f₁(x₁, x₂) and f₂(x₁, x₂) are polynomials in x₁ and x₂ (Sec. 1.4-3), consider both functions as polynomials in x₁ and form their resultant R(Sec. 1.7-3). Then x₂ must satisfy the equation R = 0 (Sylvester's Dialytic Method of Elimination).

1.9-2. Simultaneous Linear Equations: Cramer's Rule. Consider a set (system) of n linear equations in n unknowns x₁, x₂, . . . , x_n

such that at least one of the absolute terms h is different from zero. If the system determinant

differs from zero, the system (2) has the unique solution

where D_k is the determinant obtained on replacing the respective elements a_1k, a_2k, . . . , a_nk in the k^th column of D by b₁, b₂, . . . , b_n, or

where A_ik is the cofactor (Sec. 1.5-2) of a_ik in the determinant D (see also Secs. 13.2-3 and 14.5-3).

1.9-3. Linear Independence (see also Sees. 9.3-2, 14.2-3 and 15.2-la).

(a) m equations f_i(x₁, x₂, . . . , x_n) = 0 (i = 1, 2, . . . , m), or m functions f_i(x₁, x₂, . . . , x_n) are linearly independent if and only if

Otherwise the m equations or functions are linearly dependent; i.e., at least one of them can be expressed as a linear combination of the others.

As a trivial special case, this is true whenever one or more of the equations f_i (x₁, x₂, . . . , x_n) ≠ 0 is satisfied identically.

n homogeneous linear functionsare linearly independent if and only if det [a_ik] ≠ 0 (see also Sec. 1.9-5).

More generally, m homogeneous linear functionsare Hnearly independent if and only if the m X n matrix [and is of rank m (Sec. 13.2-7).

(b) m sets of n numbers x₁⁽¹⁾, x₂⁽¹⁾, . . . , x_n⁽¹⁾; x₁⁽²⁾, x₂⁽²⁾, . . . , x_n⁽²⁾; . . . ; x₁^(m), x₂^(m), . . . , x_n^(m) (e.g., solutions of simultaneous equations, or components of m n-dimensional vectors) are linearly independent if and only if

This is true if and only if them X n matrix [x_j⁽ⁱ⁾] is of rank m (Sec. 13.2-7).

1.9-4. Simultaneous Linear Equations: General Theory (see also Sec. 14.8-10). The system of m linear equations in n unknowns x₁ , x₂, . . . , x_n

possesses a solution if and only if the matrices

(system matrix and augmented matrix) are of equal rank (Sec. 13.2-7). Otherwise the equations are inconsistent.

The unique solution of Sec. 1.9-2 applies if r = m = n. If both matrices (9) are of rank r < m, the equations (8) are linearly dependent (Sec. 1.9-3a); m — r equations can be expressed as linear combinations of the remaining r equations and are satisfied by their solution. The r independent equations determine r unknowns as linear functions of the remaining n — r unknowns, which are left arbitrary.

1.9-5. Simultaneous Linear Equations: n Homogeneous Equations in n Unknowns. In particular, a system of n homogeneous linear equations in n unknowns,

has a solution different from the trivial solution x\ = x₂ = * * * = x_n — 0 if and only if D — det [a^] = 0 (see also Sec. 1.9-3a).

In this case, there exist exactly n — r linearly independent solutions x₁⁽¹⁾, x₂⁽¹⁾, . . . , x_n⁽¹⁾; x₁⁽²⁾, x₂⁽²⁾; . . . ; x₁^(n-r), x₂^(n-r), . . . ,x_n^(n-r), where r is the rank of the system matrix (Sec. 1.9-4). The most general soution is, then,