Chapter Name

21.1-1. Chapter 21 is essentially a collection of formulas relating to special functions. Refer to Chap. 7 for the relevant complex variable theory, and to Chap. 9,10, and 15 for a treatment of differential equations. References 21.3 and 21.9 deal with the less frequently encountered special transcendental functions.

21.2. THE ELEMENTARY TRANSCENDENTAL FUNCTIONS

21.2-1. The Trigonometric Functions (see also Secs. 21.1-1 and 21.2-12 and Table 7.2-1). (a) The trigonometric functions w = sin z, w = cos z are defined by their power series (Sec. 21.2-12), as solutions of the differential equation by z = arcsin w, z = arccos w (integral representation, Sec. 21.2-4), or, for real z, in terms of right-triangle geometry (goniometry, Fig. 21.2-1). The remaining trigonometric functions are defined by

(b) sin z and cos z are periodic with period 2π ; tan z and cot z are periodic with period π. sin z , tan z, and cot z are odd functions, where as cos z is an even function. Figure 21.2-2 shows graphs of sin z, cos z, tan z, and cot z for real arguments. Figure 21.2-3 shows triangles which

FIG.21.2-1. Definitions of circular measure and trigonometric functions for a given angle φ:

serve as memory aids for the derivation of function values for z = π/6 = 30 deg, π/4 = 45 deg, and π/3 = 60 deg (see also Table 21.2-1).

permit one to express trigonometric functions of any real argument in

FIG. 21.2-2. Plots of the trigonometric functions for real arguments z=o.

Table 21.2-1. Special Values of Trigonometric Functions

FIG. 21.2-3. Special triangles for deriving the trigonometric functions of 30 deg, 45 deg, and 60 deg.

Table 21.2-2. Relations betweeb Trigonometric Functions of Different Arguments

terms of function values for arguments between 0 and π/2 = 90 deg (Table 21.2-2 and Fig. 21.2-1).

21.2-2. Relations between the Trigonometric Functions (see also Sec. 21.2-6). The basic relations

21.2.3 Addition Formulas and Multiple –angle Formulas. The basic realation

21.2-4. The Inverse Trigonometric Functions (see also Table 7.2-1).^* (a) The inverse trigonometric functions w = arcsin z, w = arccos z, w = arctan z, w = arccot z are respectively defined by

z = sin w z = cos w z = tan w z =cot w

or by

Figure 21.2-4 shows plots of the inverse trigonometric functions for real arguments; note that arcsin z and arccos z are real if and only if z is real and |z| ≤1. All four functions are infinitely-many-valued because

FIG. 21.2-3. Plots of the inverse trigonometric functions.

of the periodicity of the trigonometric functions. For real arguments, the principal value of arcsin z and arctan z is that between –π/2 and π/2 (see also Fig. 21.2-4); the principal value of arccos z and arccot z is that between 0 and π (see also Fig. 7.4-1).

(b) Note

21.2-5. Hyperbolic Functions (see also Fig. 21.2-5 and Table 7.2-1). The hyperbolic functions^* w = sinh z, w = cosh z are defined by the power series (21.2-42), as solutions of the differential equation

Four additional hyperbolic functions are defined as

Geometrical Interpretation of sinh t and cosh t for Real t. If t/2 is the area bounded by the rectangular hyperbola (Sec. 2.5-2b) x² – y² = 1, the x axis, and the radius vector of the point (x, y) on the hyperbola, then y = sinh t x = cosh t. Note that, if the hyperbola is replaced by the circle x² + y² = 1, then y = sin t, x = cos t.

21.2-6. Relations between the Hyperbolic Functions (see also Sec. 21.2-8). The basic relations

^* The symbols Sin z, Cos z, Tan z, Cot z are also used.

21.2-7. Formulas Relating Hyperbolic Functions of Compound Arguments (these formulas may also be derived from the corresponding formulas for trigonometric functions by using the relations of Sec. 21.2-9).

21.2-8. Inverse Hyperbolic Functions (see also Sec. 21.2-4). The inverse hyperbolic functions w = sinh^-1 z, w= cosh^-1 z, w — tanh^-1 z are respectively defined by z = sinh w,z= cosh w, z —tanh w,^* or by integrals in the manner of Sec. 21.2-4. Note

21.2-9. Relations between Exponential, Trigonometric, and Hyperbolic Functions (see also Secs. 1.2-3 and 21.2-12 and Table 7.2-1).

21.2-10. Decomposition of the Logarithm (see also Secs. 1.2-3 and 12.2-12 and Table 7.2-1).

FIG. 21.2-5. Hyperbolic functions. (From Baumeister and Marks, Mechanical Engineers' Handbook, 6th ed., McGraw-Hill, New York, 1958.)

FIG. 21.2-6. Exponential functions and logarithms. (From Baumeister and Marks, Mechanical Engineers' Handbook, 6th ed., McGraw-Hill, New York, 1958.)

21.2-11. Relations between Inverse Trigonometric, Inverse Hyperbolic, and Logarithmic Functions.

21.2-12. Power Series and Other Expansions. Power-series expansions, as well as some product and continued-fraction expansions for the elementary transcendental functions, are tabulated in Secs. E-7 to E-9 of Appendix E. See also Secs. 20.6-1 to 20.6-5 and Tables 20.6-2 to 20.6-4 for other numerical approximations.

21.2-13. Some Useful Inequalities (see also Figs. 21.2-2 and 21.2-6).For real x,

21.3. SOME FUNCTIONS DEFINED BY TRANSCENDENTAL INTEGRALS

21.3-1. Sine, Cosine, Exponential, and Logarithmic Integrals

(see also Fig. 21.3-1). One defines

FIG. 21.3-1. sin x/x = sinc (x/π) (a), and the sine integral

(From M. Schwartz, Information Transmission, Modulation, and Noise, McGraw-Hill, New York, 1959.)

where C ≈ 0.577216 is the Euler-Mascheroni constant defined in Sec. 21.4-1b.

FIG. 21.3-2. The Fresnel integrals. (from ref.21.1)

It is customary to introduce an alternative exponential integral so that, for real x, y,

21.3-2.Fresnel Integrals and Error Function (see also Sec. 18.8-3 and Fig. 21.3-2). One defines

where the J_n/2(z) are the half-integral-order Bessel functions discussed in Sec. 21.8-le. Note

Note also

The related integrals

are also sometimes known as Fresnel integrals.

The function

is known as the complementary error function.

21.4. THE GAMMA FUNCTION AND RELATED FUNCTIONS

21.4-1. The Gamma Function, (a) Integral Representations. The gamma function T(z) is most frequently defined by

or for Re (z) < 0 by

where the contour C_o starts at – ∞, skims below the negative x axis, surrounds the origin, and returns just above the negative xaxis.The

FIG. 21.4-1. Г(n + 1) vs. n for real n. Note Г(n + 1) = n!for n = 0,1, 2, . . . , and the alternating maxima and minima given approximately by Г (1.462) = 0.886, Г(-0.5040) = -3.545, Г(-1.573) - 2.302, Г(-2.611) - -0.888, . . . .

definition can be extended by analytic continuation (Sec. 7.8-1). The only singularities of Г(z) in the finite portion of the z plane are simple poles with residues (– l)ⁿ/n\ for z = –n (n = 0, 1, 2, . . .); 1/Г(z) is an integral function.

Figure 21.4-1 shows a graph of Г(x) vs. x for real x. Note

(b) Other Representations of Г(z).

C is the Euler-Mascheroni constant denned by

(c) Functional Equations.

21.4-2. Stirling's Expansions for Г(z) and n!(see also Secs. 4.4-3, 4.8-66, and 21.5-4).

Stirling's series is especially useful for large \z\; for real positive z, the absolute value of the error is less than that of the last term used. Note, in particular,

The fractional error in Stirling's formula is less than 10 per cent for n = 1 and decreases as n increases; this asymptotic formula applies particularly to computations of the ratio of two factorials or gamma functions, since in such cases the fractional error is of paramount interest.

      More specifically

21.4-3. The Psi Function (Digamma Function).

21.4-4. Beta Functions. The (complete) beta function is defined as

or by analytic continuation of

21.4-5. Incomplete Gamma and Beta Functions. The incomplete gamma function Г_z(p) and the incomplete beta function B,(p, q) are respectively defined by analytic continuation of

I_z(p,q) ≡ B_z(p,q)/B(p, q) is called the incomplete-beta-function ratio.

See Appendix E and Ref. 20.6 for additional definite and indefinite integralsrelated to the gamma function.

21.5. BINOMIAL COEFFICIENTS AND FACTORIAL POLYNOMIALS. BERNOULLI POLYNOMIALS AND BERNOULLI NUMBERS

21.5-1. Binomial Coefficients and Factorial Polynomials (see also Sec. 1.4-1). Table 21.5-1 summarizes the definition and principal properties of the binomial coefficients The expression

is called a factorial polynomial of degree n. The coefficients S_k⁽ⁿ⁾are known as Stirling numbers of the first kind and can be obtained with the aid of the recursion formulas

or from Eq. (17). Note

If one defines x^[0] = 1, x^[r] = 1/(x+1) (x+2). . . (x-r) (r = -1, -2, . . . ), then the relation

x^[r] (x-r)^[m] = x^[r+m]

and Eq. (20.4-12) hold for all integral pseudo exponents r, m. Note 0^[0] = 1.

21.5-2. Bernoulli Polynomials and Bernoulli Numbers (see also Sec. 21.2-12). (a) Definitions. The Bernoulli polynomialsB_k⁽ⁿ⁾(x) of order n = 0, 1, 2, . . . and degree k are defined by their generating function (Sec. 8.6-5)

Note

The B_k⁽ⁿ⁾(0) = B_k⁽ⁿ⁾ are called Bernoulli numbers of order n, with

Table 21.5-1. Definition and Properties of the Binomial Coefficients

(see also Secs. 1.4-1 and 21.5-4)

Bernoulli numbers of order 1 are often simply called Bernoulli numbers B_k⁽¹⁾=B_k;B_k = 0 for all odd k > 1, and

with alternating signs. The Bernoulli numbers may also be obtained from the recursion formulas

or in determinant form by solution of the linear equations (6).

(b) Miscellaneous Properties. Note

21.5-3. Formulas Relating Polynomials and Factorial Polynomials. Bernoulli polynomials and Bernoulli numbers relate factorial polynomials (Sec. 21.5-1) to powers of x; such relations aid in the solution of difference equations and, in particular, in the summation of series (Sec. 4.8-5d). Note

21.5-4. Approximation Formulas for (see also Sec. 21.4-2). If N is a positive integer and then

If n ≪ N, use

For large valuse of N, n, and N - n, use Stirling’s formula (21.4-11).

21.6. ELLIPTIC FUNCTIONS, ELLIPTIC INTEGRALS, AND RELATED FUNCTIONS

21.6-1. Elliptic Functions: General Properties. A function w = f(z) of the complex variable z is an elliptic function if and only if

1. f(z) is doubly periodic with two finite primitive periods (smallest periods) ω₁, ω₂ whose ratio is not a real number, i.e.,

2.The only singularities of f(z) in the finite portion of the z plane are poles (see also Secs. 7.6-7 to 7.6-9).

A doubly periodic function f(z) repeats the values it takes in any period parallelogram, say that defined by the points 0, coi, co₂, on + co₂, where the two sides joining the latter three points are excluded as belonging to adjacent parallelograms. The order of an elliptic function is the number of poles (counting multiplicities) in any period parallelogram.

A doubly periodic integral function is necessarily a constant. The residues in any period parallelogram add up to zero; hence the simplest nontrivial elliptic function is of order 2. An elliptic function f(z) of order r assumes every desired value w exactly r times in each period parallelogram, if one counts multiple roots of the equation f(z) — w = 0 (Liouville's Theorems, see also Sec. 7.6-5).

Elliptic functions are usually encountered in connection with integrals or differential equations which involve square roots of third- or fourth-degree polynomials (e.g., arc length of an ellipse, equation of motion of a pendulum; see also Secs. 4.6-7 and 21.6-4). Weierstrass's elliptic functions and normal elliptic integrals are constructed in terms of simpler functions with known singularities for theoretical simplicity (Secs. 21.6-2, 21.6-3, and 21.6-56). For numerical calculations, one prefers Jacobus elliptic functions (Sec. 21.6-7), which may be regarded as generalizations of trigonometric functions; Legendre's normal elliptic integrals, which are closely related to the inverses of Jacobi's functions, are widely tabulated (Secs. 21.6-5 and 21.6-6).

21.6-2. Weierstrass's Function. (a)is an even elliptic function of order 2 with periods ω₁, ω₂ and 2^nd-order poles at z = mω₁ + nω₂(n,m = 0, ±1, ±2, . . . ); in terms of partial fractions (Sec. 7.6-8),

where the summation ranges over all integer pairs m, n except for 0, 0. ω= (± z – C/ω₁ , ω₂) satisfies the differential equation

The parameters g₂, g₃ determine the constants ω₁, ω₂ associated with each function and are known as invariants of (z) ≡ (z\ω₁, ω₂) ≡ (z; g₂, g₃); note

The points w = e₁, e₂, e₃ and w = ∞ are the branch points of the inverse function

Note the series expansions

and the addition formula

(b) Every elliptic function f{z) with periods ω₁, ω₂ can be represented as a rational function of (z; ω₁ , ω₂ ) and (z; ω₁ , ω₂ ). More specifically, f(z) can be written in the form

where R₁ and R₂ are rational functions; {z) is an odd elliptic function of order 3.

21.6-3. Weierstrass's ζ and σ Functions, (a) Weierstrass's ζ and σ functions are not elliptic functions but may be used to construct elliptic functions with easily recognizable singularities. They are defined by

where the sums and products range over all integer pairs m, nexcept for 0, 0. ζ(z) has simple poles, and σ(z) has simple zeros at the points z = mω₁ + mω₂ , and

Equations (8) and (13) yield Laurent expansions of ζ (z) and σ(z) about z=0. Note also

and the addition theorem

where the positive sign applies only if both m and n are even integers.

(b) Every elliptic function f(z) of order r (Sec. 21.6-1) can be represented in the form

21.6-4. Elliptic Integrals (see also Sec. 4.6-7). The function

is an elliptic integral whenever f(z)is a rational function of z and the square root of a polynomial

without multiple zeros; one includes the case of third-degree polynomials G(z) ≡ G₃(z) as well as fourth-degree polynomials G(z) ≡ G₄(z) by introducing α₄ = ∞ whenever a₀ = 0, so that formally a₀(z – α₄) = a₁ Every elliptic integral is a multiple-valued function of z; different integration paths yield an infinite number of function values. The points z= α₁ z = α₂, z= α₃, z = α₄ are branch points. One joins α₁ , α₂ and α₃, α₄ by two suitably defined branch cuts to obtain a Riemann surface (Sec. 7.4-3) connected like the surface of a torus.

An elliptic integral of the first kind is finite for all z; its only singularities are branch points at z =α₁, α₂, α₃ , α₄. An elliptic integral of the second kind is analytic throughout the z plane except for branch points at α₁ , α₂ , α₃ , α₄ and a finite number of poles. An elliptic integral of the third kind has a logarithmic singularity (see also Secs. 21.6-5 and 21.6-6).

21.6-5. Reduction of Elliptic Integrals. The following procedures reduce every elliptic integral ∫ f(z) dz to a weighted sum of elementary functions and three so-called normal elliptic integrals (see also Refs. 21.2, 21.11, and 21.12 for alternative procedures; Ref. 21.2 contains a very comprehensive collection of explicit formulas expressing elliptic integrals in terms of normal elliptic integrals).

(a) Formal Reduction Procedure. Note that even powers of are polynomials in z, and rewrite

where the P_i(z) are polynomials, and R₁(z), R₂{z) are rational functions.R₁(z) can be integrated to yield elementary functions (Sec. 4.6-6c).

Partial-fraction expansion of the rational function R₂(z) (Sec. 1.7-4)

reduces the evaluation ofto that of integrals of the form

Every one of these integrals can be expressed in terms of I₀ , I₁ , I₂ and I_-1 alone with the aid of the recursion formula

where the coefficients b_k are defined by the identity

In addition, (22) yields I₂ explicitly in terms of I₀, I₁ , and I_-1 if a₀ =0, or if c is a root of G(z) = 0 (i.e., b₄ = 0). Even if c is not such a root, one can rewrite I₂ as

where z = c′ is a root of G(z)=0. Hence every elliptic integral can be expressed as a weighted sum of elementary functions and three relatively simple types of elliptic integrals of the first, second, and third kinds (Sec.21.6-4), viz.,

The first of these integrals is usually referred to as a normal elliptic integral of the first kind; it is often convenient not to use the other two integrals (25) directly but to introduce suitable linear combinations as normal elliptic integrals of the second and third kind (Secs. 21.6-2, 21.6-3, and 21.6-6).

(b) Change of Variables. Weierstrass's and Riemann's Normal Forms. At any convenient stage of the reduction procedure, one may introduce a new integration variable to transform the elliptic integrals (21) or (25) into new elliptic integrals involving a more con venient polynomial and, possibly, a simpler recursion formula (22).

In particular, a bilinear transformation

(Sec. 7.9-2) chosen by substitution of corresponding values of z and so as to map the branch points z = α₁ , α₂ ,α₃ , α₄ into = e₁ , e₂ , e₃ , ∞yields elliptic integrals in Weierstrass's normal form with ≡ 4³ – g₂ – g₃. These integrals are related to Weierstrass's function (Sec. 21.6-2). Again, a transformation (26) mapping z = α₁, z = α₂, z = α₃, α₄ into z = 0, 1, 1/k, – 1/k, where k is a real number between 0 and 1, yields elliptic integrals in Riemann's normal form, with ≡ (1 - )(1 - k²).

(c) Reduction to Legendre's Normal Form. More frequently, one desires to transform a real elliptic integral to Legendre's normal form with ≡ (1 — ²)(l — k²²) where k² is a real number between 0 and 1. The reduction procedure will then yield (real) Legendre's normal integrals (Sec. 21.6-6) whose values are available in published tables.

Let G(x) be a real polynomial greater than zero in (a, x); since is to be real, the integration interval cannot include a real root of G(x) = 0. Table 21.6-1 (pages 712-713) lists transformations x = x(φ) mapping the real integration interval (a, x) into a corresponding range of real angles φ between 0 and π/2 SO that

for the various possible types of real fourth-degree polynomials G(x) ≡ G₄(x) and third-degree polynomials G(x) ≡ G₃(x). The correct values of the constant parameters k² and μ are also tabulated.

In each case, the leading coefficient (a₀ or a₁) of G(x) is taken to be either 1 or –1. In the case of real roots, it is assumed that α₁ > α₂, α₃ > α₄; complex roots are denoted by b₁ ± ic₁ and b₂ ± ic₂, with b₁ ≥ b₂, c₁ > 0, c₂ > 0. The following auxiliary quantities have been introduced:

21.6-6. Legendre's Normal Elliptic Integrals (see also Secs. 26.6-4 and 26.6-5). (a) Definitions. Legendre's (incomplete) normal elliptic integrals are defined as

where z = sin φ . k is a complex number called the modulus (module) of the elliptic integral. The elliptic integrals (29) are real for real

values of the amplitude φ between – π/2 and π/2 if k² is a real number between 0 and 1, F(k, φ) and E(k, φ) have been tabulated for 0 ≤ φ ≤ π/2 and real values of k² between 0 and 1 (see also Fig. 21.6-1). c is called the characteristic of the elliptic integral (29).

FIG. 21.6-1. Variation of the elliptic integral u of the first kind with φ, the amplitude of u for three values of the modulus k. (From J. Cunningham, Introduction to Nonlinear Analysis, McGraw-Hill, New York, 1958.)

For real values of the modulus k, the modular angle

is often tabulated as an argument instead of k (Fig. 21.6-2); one writes

90° — α= arccos k is the complementary modular angle, k² — mis often called the parameter of a normal elliptical integral.

(b) Legendre's Complete Normal Elliptic Integrals (see also Fig. 21.6-3). The functions

are respectively known as Legendre's complete elliptic integrals of the first and second kind. k = sin α and k′ = = cos α

FIG. 21.6-2. The incomplete elliptic integral of the first kind, F(k,φ) = F(φ/a), plotted against φ (a), and against the modular angle a = arcsin k (b). (From Ref. 21.1.)

FIG. 21.6-3. The elliptic integrals K(k) = F(90°\a), K′(k) = (90°\90° - φ) (a), and E(k) ≡ E(9O°\φ), E′(k) ≡ E(90°\90° - α) (b) (k = sin α from Ref. 21.1).

are called complementary moduli. K(k) and K′(k) = K(k′) are associated elliptic integrals of the first kind; E(k) and E′(k) ≡ E(k′) are associated elliptic integrals of the second kind. Note

and K(0) = K′(l) = π/2, K(l) = K′(0) = ∞. Different values of the multiple-valued elliptic integral F(k, φ) differ by 4mK + 2niK′; different values of E(k, φ) differ by 4mE + 2ni(K′ - E′) (m, n = 0, ±1, ±2, . . . ; see also Sec. 21.6-76).

K (k) and E (k) satisfy the differential equations

so that, for real k² < 1,

whereF(a, b; c; z) is the hypergeometric function defined in Sec. 9.3-9.

(c) Transformations. Legendre's normal elliptic integrals (29) with moduli k and k = 1/k, k′, 1/k′, ik/k′, k′/ik, (1 - k′)/(l + k′), are connected by the relations listed in Table 21.6-2, where

(see also Sec. 21.6-7a). Table 21.6-3 lists similar relations for the complete normal elliptic integrals (30).

In particular,

Successive substitution of

for k′ in Eq. (35) yields (1 - k′_n)/(1 + k′_n) -> 0; since K(0) = π/2, one obtains

which may be useful for numerical computation of K(k). K(k, φ) can be obtained in analogous fashion.

Table 21.2-2. Transformation to Legendre's Normal form* All zeros of G(x) real

* From A. Erdelyi et al., Higher Transcendental Functions, vol.2, McGraw-Hill, Nre York, 1953.

Table 21.6-2. Transformations of Complete Elliptic Integrals*

*From A. Erdélyi et al., Higher Transcendental Functions, vol.2, McGraw-Hill, New York, 1953.

21.6-7, Jacobi's Elliptic Functions, (a) Definitions. Inversion of the elliptic integrals z = F(k, φ) and z = F(k , w) (Sec. 21.6-6a) yields the functions am z (amplitude of z) and sn z (sinus amplitudinis of z),i.e.,

In addition, one defines the functions en z (cosinus amplitudinis of z) and dn z (delta amplitudinis of z) by

sn z, en z, and dn z are Jacobi's elliptic functions. A given value of the parameter k is implied in each definition; if required, one writes sn (z, k), cn (z, k), dn (z, k). k′, K, K′, E, and E′ are denned as in Sec. 21.6-66. Jacobi's elliptic functions are real for real z and real k ²between

FIG. 21.6-5. The Jacobian elliptic functions sn u, cn u, and dn u for k² = 1/2. (From Ref. 21.1.)

FIG. 21.6-5. One-quarter of a complete cycle of the elliptic functions cn u, sn u, and dn u, plotted against the normalized abscissa u/K, for three values of the modulus k. {From J. Cunningham, Introduction to Nonlinear Analysis, McGraw-Hill, New York,1958.)

Table 21.6-4. Periods, Zeros, Poles, and Residues of Jaconi's Elliptic Functions^*

m and n integers

* From A. Erdelyi et at., Higher Transcendental Functions, vol. 2, McGraw-Hill, New York, 1953.

0 and 1 and reduce to elementary functions for k² = 0 and k² = 1; in particular,

Jacobi's elliptic functions can also be defined in terms of >, ζ, σ ,or ν functions by the relations of Sec. 21.6-9.

(b) Miscellaneous Properties and Special Values. Jacobi's elliptic functions are of order 2 (Sec. 26.1-1); their periods, their (simple) zeros, and their (simple) poles are listed in Table 21.6-4. sn z is an odd function, while en z and dn z are even functions of z. Table 21.6-5 lists special function values. Table 21.6-6 shows the effects of argument changes by quarter and half periods, using the convenient notation

Note also

References 21.2 and 21.11 contain additional formulas.

Table 21.6-5. Special Values of Jacobian Elliptic Funtions^* sn (1/2mK + 1/2niK')

^* From A. Erdélyi et al., Higher Transcendental Functions, vol. 2, McGraw-Hill, New Year, 1953.

Table 21.6-5. Change of the Variable by Quarter and Half Periods. Symmetry^*

sn (mK + niK' u)

cn (mK + niK' u)

dn (mK + niK' u)

^* From A. Erdélyi et al., Higher Transcendental Functions, vol. 2, McGraw-Hill, New Year, 1953.

Table 21.6-7. Transformation of the First order of Jacobi's Elliptic Functions^*

^* From A. Erdélyi et al., Higher Transcendental Functions, vol. 2, McGraw-Hill, New Year, 1953.

(c) Addition Formulas.

(d) Differntiation.

(e) Transformations. Table 21.6-7 shows relations between Jacobi's elliptic functions with moduli k and ik/k′, k′, l/k, 1/k′, k′/ik (see also Sec. 21.6-6c).

(f) Series Expansions.

21.6-8. Jacobi's Theta Functions. (a) Given a complex variable v and a complex parameter q = e^iπT such that r has a positive imaginary part, the four theta functions*

are (simply) periodic integral functions of v with the respective periods 2, 2, 1, 1. The four theta functions (49) have zeros at v = m + n_T , m+n_T+1/2 , m+(n+1/2)_T+1/2 , and m+(n+1/2)_T respectively, where m, n = 0, ±1, ±2, . . . ; these zeros yield infinite-product representations (7.6-2) (Refs. 21.3 and 21.11).

The theta functions are not elliptic functions. The very rapidly converging series (49) permit one to compute various elliptic functions and elliptic integrals with the aid of the relations of Sec. 21.6-9; in their own right, the theta functions are solutions of the partial differential equation

which is related to the diffusion equation (Sec.10.3-4b).

^* Some authors denote ₄(v) by ₀(v) or (v).

(c) To find _i (and thus similarly transformed elliptic functions, Sec. 21.6-9), use

(d) The values of the four theta functions and their derivatives for v = 0 (zero-argument values) are simply denoted by ν_i = ν_i(0) , ν′_i = ν′_i(0), . . .(i = 1,2,3,4) and satisfy the relations

21.6-9. Relations between Jacobi's Elliptic Functions, Weier-strass's Elliptic Functions, and Theta Functions. If the various parameters implicit in the definitions of sn z, cn z, dn z, $(z), ζ(z), φ(z), and ν_j(z) (Secs. 21.6-2, 21.6-3, 21.6-6, 21.6-7, and 21.6-8) are related by

21.7. ORTHOGONAL POLYNOMIALS

21.7-1. Survey. The orthogonal polynomials discussed in Secs. 21.7-1 to 21.7-8 are special solutions of linear homogeneous second-order differential equations related to the hypergeometric differential equation (9.3-31) (Legendre, Chebychev, and Jacobi polynomials) or to the confluent hypergeometric differential equation (9.3-42) (Laguerre and Hermite polynomials). These special solutions are generated by special homogeneous boundary conditions: each class of orthogonal polynomials is a set of eigenfunctions for an eigenvalue problem reducible to the generalized Sturm-Liouville type (Secs. 15.4-8a andc). Only real z = x are of interest in most applications.

The polynomials φ₀(x), φ₁(x), φ₂(x) , ... of each type are, then, defined except for multiplicative constants, which are usually (but not always) chosen so that the coefficient of xⁿ in the n^th-degree polynomial φ_n(x) is unity. Successive polynomials φ₀(x), φ₁(x), φ₂(x) , ... of each type can be derived

1. In terms of the appropriate hypergeometric series (Sec. 9.3-9a) or confluent hypergeometric series (Sec. 9.3-10)

2. By means of recursion formulas derived from the differential equation

3. By successive differentiations of a generating function γ(x, s) (see also Sec. 8.6-5)

4. Through Gram-Schmidt orthogonalization of the powers 1, x, x² . . . with the appropriate inner product (Sec. 15.2-5)

5. From an integral representation (Sec. 21.7-7), which is usually related to an integral-transform solution of the differential equation or to the Taylor- or Laurent-series coefficients of the generating function

Table 21.7-1 lists the principal formulas.

Series expansions in terms of orthogonal polynomials are derived in the manner of Sec. 15.2-4a and yield useful approximations which minimize appropriately defined mean-square errors (Sec. 15.2-6; see also Secs. 20.5-1 and 20.6-3).

21.7-2. Real Zeros of Orthogonal Polynomials. All zeros of each orthogonal polynomial discussed in Secs. 21.7-1 to 21.7-8 are simple and located in the interior of the expansion domain. Two consecutive zeros of φ_n{x) bracket exactly one zero of φ_n+1(x), and at least one zero of φ_m(x) for each m > n (Ref. 21.9).

21.7-3. Legendre Functions (see also Secs. 21.8-10, 21.8-11, and 21.8-13; and Refs. 21.3, 21.9, and 21.11). The differential equation {Legendre's differential equation) and the recursion formulas for the Legendre polynomials (Table 21.7-1) are satisfied not only by the Legendre polynomials of the first kind P_n{z) of Table 21.7-1 but

also by the Legendre functions of the second kind

More generally, the method of Sec. 9.3-8b permits one to derive linearly independent solutions P_α(z), Q_α{z) of Legendre's differential equation (Legendre functions of the first and second kind) for nonintegral positive or negative or complex values of n = α solutions for n = a and n = – α – 1 are necessarily identical.

21.7-4. Chebyshev Polynomials of the First and Second Kind. The differential equation and the recursion formulas for the Chebyshev polynomials (Table 21.7-1) are satisfied not only by the Chebyshev polynomials of the first kind

of Table 21.7-1, but also by the Chebyshev “polynomials” of the second kind

While the functions U_n(x) are not polynomials in x, the functionsand are polynomials; both are also sometimes referred to as Chebyshev polynomials of the second kind. Note

21.7-5. Associated Laguerre Polynomials and Functions (see also Secs. 9.3-10 and 10.4-6). (a) The associated (or generalized) Laguerre polynomials* of degree n — k and order k,

satisfy the differential equation

* This notation is used in most physics books. Some authors use alternative associated Laguerre polynomials L_n^(k)(x) ≡ ( – l)^kL^k_n+k(x)/(n + k)! of degree n, which satisfy the differential equation

FIG. 21.7-3. Lagurerre polynomials (a), and Hermite polynomials (b). (From Ref. 21.1)

FIG. 21.7-2. Chebyshev polynomials T_n(x). (From Ref. 21.1.)

FIG. 21.7-3. Laguerre polynomials (a), and Hermite polynomials (b). (From Ref. 21.1.)

Table 21.7-1. Orthogonal Polynomials of Legendre, Chebyshev, Laguerre, and Hermite (see also secs. 21.7-1 to 21.7-7)

Table 21.7-1. Orthogonal Polynomials of Legendre, Chebyshev, Laguerre, and Hermite (see also secs. 21.7-1 to 21.7-7)(continued)

^* Some authors denote the polynomial

^† Some authors use alternative Hermite polynomials which satisfy the differential equation

Table 21.7-2. Coefficients for Orthogonal Polynomials, and for xH in Terms of Orthogonal Polynomials*

(a) Legend re Polynomials:

(b) Chebyshev Polynomials:

* Abridged from M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, National Bureau of Standards, Washington, D.C., 1964.

(c) Laguerre Polynomials:

(d) Hermite Polynomials:

for integral values n = 1, 2, . . . ; k = 0, 1, 2, . . . , n. Equation (7) reduces to the differential equation of the Laguerre polynomials (Table 21.7-1) for k = 0.

Note the generating function

and the orthogonality and normalization given by

If nonintegrai real or complex values of n and k are admitted, the differential equation (7) defines the generalized Laguerre functions. These functions, of which the polynomials (6) are special cases, are confluent hypergeometric functions (Sec. 9.3-10) with a = k - n, c = k + 1.

(b) The functions which are often referred to as associated Laguerre functions, satisfy the differential equation

and

(see also Sec.10.4-6).

21.7-6. Hermite Functions. The functions Ψ_n(x) = e^-x2/2H_n(x) (n = 0, 1, 2, . . .), which are often referred to as Hermite functions, satisfy the differential equation

and

21.7-7. Some Integral Formulas (see Refs. 21.3 and 21.11 for additional formulas).

The integration contour in Eq. (15) surrounds the point z = x.

21.7-8. Jacobi Polynomials and Gegenbauer Polynomials (see Ref. 21.11 for a detailed discussion), (a) Jacobi's (hypergeometrie) polynomials are special instances of hypergeometrie functions

(Sec.9.3-9) and satisfy the orthogonality conditions

(b) The functions

are known as Gegenbauer (ultraspherical) polynomials. They constitute a generalization of the Legendre polynomials (Table 21.7-1), to which they reduce for α = ½. The Gegenbauer polynomials satisfy the differential equation

and the orthogonality condition

The Gegenbauer polynomials can be generated as coefficients of the power series

21.8. CYLINDER FUNCTIONS, ASSOCIATED LEGENDRE FUNCTIONS, AND SPHERICAL HARMONICS

21.8-1. Bessel Functions and Other Cylinder Functions, (a) A cylinder function (circular-cylinder function) of order m is a solution w = Z_m{z) of the linear differential equation

where m is any real number; one usually imposes the recurrence relations

as additional defining conditions. The functions e^±i(kz±mφ)Z_m(iKr′) are solutions of Laplace's partial differential equation in cylindrical coordinates γ′, φ z (cylindrical harmonics. Sec. 10.4-36). Cylinder functions of nonintegral order are multiple-valued (Sec. 9.3-56); one defines the principal branch by |arg z| < π (branch cut from z = 0 to z = -∞, Sec. 7.4-2).

(b) The most generally useful cylinder functions of order m satisfying the recurrence relations (2) are (see also Figs. 21.8-2 to 21.8-4)

The last three sums in Eq. (4) are given the value zero whenever the lower limit exceeds the upper limit; and C ≈ 0.577216 is Euler's constant (21.4-6). Note that every function N_m(z) has a singularity at the origin. *

(c) Analytic Continuation. To obtain values of the cylinder func tions for |arg z| > π, use

where one uses sin mnπ cot mπ = (-1)^mnn for m=πn; and

Note that cylinder junctions of integral order are single-valued integral functions (Sec. 7.6-5).

(d) Every cylinder function of order m can be expressed as a linear com bination of J_m(z) and N_m{z) and as a linear combination of H_m^(l)(z) and H_m2(z):

(fundamental systems, Sec. 9.3-2). J_m(z) and J-_m(z) constitute a fundamental system unless m = 0, ±1, ±2, . . . , since then J__m(z) ≡ (– l)^mJ_m(z). The three fundamental systems have the respective Wronskians (Sec. 9.3-2) 2/πZ, –4i/πz, and –2 sin (mπ)/πz) the first two Wronskians are independent of m. Note

(e) Cylinder functions with m = ±1/2 , ±3/2, . . . can be written as elementary transcendental functions (see also Sec. 21.8-8):

* The Neumann functions N_m(z) are sometimes denoted by Y_m(z); some authors refer to them as Weber's Bessel functions of the second kind.

21.8-2. Integral Formulas (see Sec. 8.6-4, Table D-7, and Refs. 21.3 and 21.11 for additional relations), (a) Integral Representations of J₀(z), J₁(z), J₂(z), . . . .

where the integration contour encloses the origin, and |arg z| < π.

FIG. 21.8-1. Integration contours for Sommerfeld's integrals; t = γ + in. (J. A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941.)

(b) Sommerfeld's and Poisson's Formulas. Referring to Fig. 21.8-1, the complex contour integral

equals H_m^(l)(z) for the contour C ≡ C₁, H_m⁽²⁾(z) for the contour C ≡ C₂, and 2J_m(z) for the contour C ≡ C₃. The contours may be deformed at will, provided that they start and terminate by approaching t = ∞ in the shaded areas indicated, t = 0 and t= π may be used as saddle points (Sec. 7.7-3e) for C₁ and C₂ in computations of Z_m(z) for large values of z (see also Sec. 21.8-9).

Note also

(c) Miscellaneous Integral Formulas Involving Cylinder Functions (see also Sec. 21.8-4c).

21.8-3. Zeros of Cylinder Functions. (a) All zeros of cylinder functions are simple zeros, except possibly z = 0. Consecutive positive or negative real zeros of two linearly independent real cylinder functions of order m alternate; z = 0 is the only possible common zero.

FIG. 21.8-2. The Bessel functions J₀(x), J₁(x), J₂(x), ... for real arguments. Note that J_m(-x)=(-1)^mJ_m(x).

(b) (See also Fig. 21.8-2). The function J_m(z) has an infinite number of real zeros; for m > – 1, all its zeros are real. For m = 0, 1/2, 1 , 3/2 , 2 , . . . and n = 1, 2, . . . , J_m(z) and J_m+_n(z) have no common zeros.

For m = 1, 2, . . . , consecutive positive or negative real zeros of J_m(z) are separated by unique real zeros of J_m+1(z) and by unique real zeros of J'_m(z); and consecutive positive or negative real zeros of z^–mJ_m(z) are separated by unique real zeros of z^–mJ_m+1(z).

21.8-4. The Bessel Functions J_o(z), J₁(z), J₂{z), ... (a) Generation by Series Expansions. Bessel functions of nonnegative integral order m = 0, 1, 2, . . . are single-valued integral functions of z. They may be “generated” (see also Sec. 8.6-5) as coefficients of the Laurent series (Sec. 7.5-3)

or as coefficients of the Fourier series

(b) Behavior for Real Arguments.For real z, J₀(z), J₁(z), J₂(z), ... are real functions of z = x; Fig. 21.8-2 illustrates their zeros, their maxima and minima, and their asymptotic behavior for x → ∞ (see also Secs. 21.8-3 and 21.8-9).

(c) Orthogonality Relations (see also Secs. 15.4-6 and 21.8-2c). Given any two real zeros x_i, x_k of J_m(z), one has the orthogonality relations

which yield an orthogonal-function expansion. Note also

where δ(x) is the impulse function introduced in Sec. 21.9-2.

Fig. 21.8-3. Bessel functions and Neumann functions. (From Ref. 21.1.)

Fig. 21.8-4. Contour lines of the modulus and phase of the Hankel function H₀⁽¹⁾(x + iy) = M₀e^iθ0. (From E. Jahnke, F. Emde, and F Losch, Tables of Higher Functions. McGraw-Hill. New York, 1960.)

Fig. 21.8-5. Modified Bessel and Hankel functions. (From Ref.21.1.)

Fig. 21.8-6. ber x, bei x, ker x, and kei x. (From Ref.21.1.)

Fig. 21.8-7. Spherical Bessel functions. (From Ref.21.1.)

Fig. 21.8-8. The associated Legendre functions P₁¹(x), P₂¹(x), P₃¹(x). (From Ref.21.1.)

21.8-5. Solution of Differential Equations in Terms of Cylinder Functions and Related Functions. The linear differential equation

Many special cases of Eq. (30) are of interest (Secs. 21.8-6 to 21.8-8, Refs. 21.6 and 21.9).

21.8-6. Modified Bessel and Hankel Functions. The modified cylinder functions of order m,

are defined with the aid of Eqs. (3) to (5); the definition is extended by means of Eqs. (6) and (7). The functions (32) are linearly independent solutions of the differential equation

(see also Sec. 10.4-36) and satisfy the recursion formulas

I_m(z) and K_m(z) are real monotonic functions for m = 0, ±1, ±2, . . . and real z.

21.8-7. The Functions ber_m z, bei_m z, her_m z, hei_m z, ker_m z, kei_m z . The functions ber_m z, bei_m z, her_m z, hei_m z, ker_m z, kei_m z, defined by

are real for real values of z. The subscript m is omitted for m = 0, e.g., ber ₀ z = ber z. Note that

ber z, bei z, her z, and hei z as well as J₀(i^±3/2z) and H₀⁽¹⁾(i^3/2z) are solutions of the differential equation

In some applications, it is convenient to introduce \J_m(i^3/2z)\, \K_m(i^3/2z)\, arg J_m(i^3/2z), and arg [i^–mK_m(i^3/2z)] as special functions instead of or together with ber_w z, bei_m z, ker_m z, and kei_m z (Ref. 21.6). All these functions are real for real values of z.

21.8-8. Spherical Bessel Functions. The spherical Bessel functions of the first, second, third, and fourth kind

For integral values of j, the spherical Bessel functions are elementary transcendental functions (see also Sec. 21.8-le):

21.8-9. Asymptotic Expansion of Cylinder Functions and Spherical Bessel Functions for Large Absolute Values of z (see also Secs.4.4-3 and 4.8-6). As z → ∞,

where A_m(z) and B_m(z) stand for the asymptotic series

Substitution of Eqs. (44) and (45) into Eq. (5) yields corresponding asymptotic expansions of H_m^(l)(z) and H_m⁽²⁾(z).

For \z\ » \m\ as z → ∞,

Note that the asymptotic relationship of J_m(z), N_m(z), H_m¹(z), and H_m⁽²⁾(z) is analogous to the relationship of the more familiar trigonometric and exponential functions.

21.8-10. Associated Legendre Functions and Polynomials (see also Secs. 21.7-1 and 21.7-3). (a) The associated Legendre functions of degree j and order m are solutions of the differential equation

where j and m are complex numbers; Eq. (49) reduces to Legendre's differential equation (Table 21.7-1 and Sec. 21.7-3) for m = 0. The general theory of the associated Legendre functions is found in Refs. 21.3 and 21.9. In the most important applications (Secs. 10.4-3c and 21.8-12), j and m are effectively restricted to the real integral values 0, 1, 2, . . . , while z = x is the cosine of a real angle υ and hence a real number

between – 1 and 1. Under these conditions, Eq. (49) is satisfied by the associated Legendre “polynomials” of the first kind*

(see also Sec. 9.3-9), with , and for m > j.

In particular,

where cos υ = x (see also Sec. 21.8-12).

(b) The associated Legendre “polynomials” defined by Eq. (50) satisfy the following recurrence relations (–1 < x < 1) :

* Some authors reserve the symbol for the functions here denoted by or by . Note that not all are actually polynomials in x.

(c) Asymptotic Behavior. As j→ ∞ ,

21.8-11. Integral Formulas Involving Associated Legendre Polynomials (see also Sec. 21.7-7).

21.8-12. Spherical Harmonics. Orthogonality (see also Secs. 10.4-3c, 14.10-76, and 15.2-6). (a) Solutions Φ(r, υ, φ) of Laplace's par–tial differential equation in spherical coordinates (10.4-15) are known as spherical harmonics. Spherical surface harmonics of degree j are solutions Y_j(υ, φ) of the partial differential equation

obtained on separation of variables in Eq. (10.4-15). If one imposes the “boundary” conditions of regularity and uniqueness for 0 ≤ υ ≤ π, 0 ≤ φ ≤ 2π together with Y_j (υ, φ + 2π) ≡ Y_j (υ , φ) the problem becomes an eigenvalue problem (Sec. 15.4-5) admitting only integral values of j. One may disregard negative values of j, since –j – 1 and j yield identical values of j(j + 1). The resulting real eigenfunctions

are known as tesseral spherical harmonics of degree j and order m; they are periodic on the surface of a sphere and change sign along “node lines” υ = constant and φ = constant (Fig. 21.8-9). For m = j, one has sectorial spherical harmonics, and for m = 0, zonal spherical harmonics.

Both the functions (65) and the frequently more convenient complex functions

are orthonormal sets of eigenfunctions in terms of the inner product

(Sec. 15.4-6b). (f , h)= 0 for every pair of functions (65) or of functions (66) unless f = h, in which case the inner product equals one. There are exactly 2j + 1 linearly independent spherical surface harmonics of degree j.

(b) Every twice continuously differentiable, suitably periodic real function Φ(υ , φ) defined on the surface of a sphere admits the absolutely convergent expansion

Expansions of the form (68) can be physically interpreted as multipole expansions of potentials (Secs. 15.6-5a and c).

21.8-13. Addition Theorems, (a) Addition Theorem for Cylinder Functions. Let P₁ and P₂ be two points of a plane, with polar coordi–nates (r ₁, φ₁), (r ₂ φ , φ ₂). Referring to Fig. 21.8-10a, let r₁ > r₂, so that 0≤ \ ψ\ < ) π/2 , and

FIG. 21.8-9. Nodes of the function (cos υ) sin 3φ on the developed surface of a sphere. The function is negative on the shaded areas. (J. A. Stratton, Electro–magnetic Theory, McGraw-Hill, New York, 1941.)

FIG. 21.8-10. Geometry for addition theorems. The addition theorems are useful for expressing effects at P₂ of a source of potential, radiation, etc., at P₁, or vice versa (see also Secs. 15.6-5 and 15.6-6).

Then, for every cylinder function Z_m(z) satisfying Eqs. (1) and (2),

where α is any complex number. In particular, φ ₁ – φ₂ = π yields

(b) Addition Theorems for Spherical Bessel Functions and Legendre Polynomials. Let P₁ and P₂ be two points in space, with spherical coordinates (r₁ , υ₁ , φ₁), (r₂, υ₂, φ₂). Referring to Fig. 21.8-10b, let υ₁ + υ₂ < π. One has

and

21.9. STEP FUNCTIONS AND SYMBOLIC IMPULSE FUNCTIONS

21.9-1. Step Functions (see also Secs. 4.6-17c, 18.3-1, and 20.4-5c). (a) A step function of the real variable x is a function which changes its value only on a discrete set of discontinuities (necessarily of the first kind, Sec. 4.4-7b). The function values at the discontinuities may or may not be defined . The most frequently useful step functions are*

(See also Fig. 21.9-1).

* The notations employed to denote the various unit-step functions vary ; use caution when referring to different texts.

FIG. 21.9-1. The init-step functions U(x) and U + (x) and approximations to the impulse functions δ(x), δ₊(x), δ₊(x).

Every step function can be expressed (except possibly at its discontinuites x=x_k) as a sum of the form PLES: Sgn x = 2U(x)-1; and the "jump" functions of Sec.20.4-5c).

(b) Approximation of Step Functions by Continuous Functions.

(c) Fourier-integral Representations (see also Sec. 4.11-6). The complex con–tour integral is respectively equal to U(t) or –U( – t) if the integration contour passes below or above the origin. The Cauchy principal value of the integral (Sec. 4.6-2b) equals U(x) - 1/2. Note also

21.9-2. The Symbolic Dirac Delta Function, (a) The symmetrical unit-impulse function or Dirac delta function δ(x) of a real variable x is “defined ” by

where f(x) is an arbitrary function continuous for x = X. More gen–erally, one “defines” b(x) by

where f{x) is an arbitrary function of bounded variation in a neighborhood of x = X. δ{x) is not a true function, since the “definition” (10) implies the inconsistent relations

δ(x) is a “symbolic function” permitting the formal representation of the functional identity transformation f(ξ) → f(x) as an integral transformation (Sec. 15.3-la). The “formal” use of δ(x) furnishes a con- venient notation permitting suggestive generalizations of many mathematical relations (see also Secs. 8.5-1, 15.3-la, 15.5-1, and 18.3-1). Although no functions having the exact properties (10) exist, it is possible to “approximate” δ(x) by functions exhibiting the desired properties to any degree of approximation (Sec. 21.9-4).

One can usually avoid the use of impulse functions by introducing Stieltjes integrals (Sec. 4.6-17); thus

It is possible to introduce a generalizing redefinition of the concepts of “function” and “differentiation” (Schwarz's theory of distributions, Refs. 21.13 to 21.18). Other–wise, mathematical arguments involving the use of impulse functions should be regarded as heuristic and require rigorous verification (see also Sec. 8.5-1).

(b) “Formal” Relations Involving 5(x).

21.9-3. “Derivatives” of Step Functions and Impulse Functions

(see also Sec. 8.5-1). Equations (10) and (19) and also the relation (a > 0) suggest the symbolic relation-ship

The impulse functions δ′(x), δ′′(x) are “defined” by

for an arbitrary function f(x) such that the unilateral limits f^(r)(X – 0) and f^(r)(x + 0) exist. The functions δ^(r)(ξ – X) are kernels of linear integral transformations (Sec. 15.2-1) representing repeated differentiations. Note also the symbolic relation

21.9-4. Approximation of Impulse Functions (see also Fig. 21.9-1). (a) Continuously Differentiable Functions Approximating δ(x). One can approximate δ(x) by the continuously differentiable functions

Integration of the approximating functions (18) yields the corresponding step-function approximations (4) and (5). (a > 0) converges to as α → ∞ for each function (18).

(b) Discontinuous Functions Approximating δ(x). δ(x) is often approximated by the central-difference coefficient (Sec. 20.4-3)

if f(x) is of bounded variation in a neighborhood of x = X (see also Sec. 4.11-6).

(c) Functions Approximating δ′(x), δ′′(x), .., δ^(r)(x) Successive differ–entiations of Eq. (18a) yield the approximation functions

21.9-5. Fourier-integral Representations. Note the formal relations

21.9-6. Asymmetrical Impulse Functions (see also Secs. 8.5-1 and 9.4-3). (a) The asymmetrical impulse functions are “defined” by

One way to obtain approximation functions for δ₊(x) is to substitute the approximation functions of Sec. 21.9-4 into one of the relations (30), e.g.,

(b) One may introduce δ₊(–x) ≡ δ_(x) as a second asymmetrical impulse function corresponding to the “derivative” of the asymmetrical step function U_(x) (Sec. 21.9-1).

21.9-7. Multidimensional Delta Functions (see also Sec. 15.5-1). For an n-di-mensional space of “points” (x¹, x², . . . , xⁿ) with a volume element defined as

(See. 16.10-10), the n-dimensinal delta function

for every “point” (X¹, X², . . . , Xⁿ) in V where f(x¹, x², . . . , xⁿ) is continuous. Note that the definition of δ (x¹; ξ¹; x², ξ²; . . . ; xⁿ, ξⁿ) depends on the coordinate system used and is meaningless wherever dV = 0. In particular, for rectangular cartesian coordinates x, y, z, one has dV = dx dy dz, and

21.10. REFERENCES AND BIBLIOGRAPHY

      21.1. Abramowitz, M., and I. A. Stegun (eds.): Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series 55, Washington, D.C., 1964.

      21.2.Byrd, P. F., and M. D. Friedman: Handbook of Elliptic Integrals, Springer, Berlin, 1954.

      21.3.Erdelyi, A.: Higher Transcendental Functions, vols. 1 and 2 (Bateman Project), McGraw-Hill, New York, 1953.

       21.4.Jahnke, E., and F. Emde: Tables of Functions with Formulae and Curves, Dover, New York, 1945.

       21.5.Hurwitz, A., and R. Courant: Vorlesungen iiber allgemeine Funktionentheorie und elliptische Funktionen, 4th ed., Springer, Berlin, 1964.

       21.6.McLachlan, N. W.: Bessel Functions for Engineers, Oxford, Fair Lawn, N.J., 1946.

       21.7.Schafke, F. W.: Einfuhrung in die Theorie der speziellen Funktionen der mathematischen Physik, Springer, Berlin, 1963.

       21.8.Sneddon, I. N.: The Special Functions of Physics and Chemistry, Oliver & Boyd, Edinburgh, 1956.

       21.9.Whittaker, E. T., and G. N. Watson: Modern Analysis, Macmillan, New York, 1943.

       21.10. — : A Course in Modern Analysis, Cambridge, New York, 1946.

       21.11.Oberhettinger, F., and W. Magnus: Formulas and Theorems for the Functions of Mathematical Physics, Chelsea, New York, 1954; 3d ed., Springer, Berlin, 1966,

       21.12.Tricomi, F. G.: Elliptische Funktionen, Akademische Verlagsgesellschaft, Leipzig, 1948.

(See also the articles by J. Lense and J. Meixner in vol. I of Handbuch der Physik, Springer, Berlin, 1956; and see also Sec. 10.6-2).

Generalized Functions and the Theory of Distributions

      21.13. Arsac, J.: Fourier Transforms and the Theory of Distributions, Prentice-Hall, Englewood Cliffs, N.J., 1966.

      21.14. Friedman, A.: Generalized Functions and Partial Differential Equations, Prentice-Hall, Englewood Cliffs, N.J., 1956.

      21.15. Gelfand, I. M., et al.: Generalized Functions, Academic, New York, 1964.

      21.16. Lighthill, M. J.: Introduction to Fourier Analysis and Generalized Functions, Cambridge, New York, 1958.

      21.17. Schwartz, L.: Theorie des Distributions, 2d ed., Hermann & Cie, Paris, 1957.

      21.18. Zemanian, A. H.: Distribution Theory and Transform Analysis, McGraw-Hill, New York, 1965.

*811; The functions arcsin z, arccos z, arctan z, and arccot z are often denoted by sin^-1 z, cos^-1 z, tan^-1 z, and cot^-1 z, respectively. This notation tends to be mislead-ing and is not recommended.

*815 This notation is the usual one in English-speaking countries, although it tends to be misleading (see also Sec. 21.2-4). An alternative notation is ar sinh z, ar cosh z , ar tanh z , or Ar Sin z, Ar Cos z,Ar Tan z.