Special Functions & Their Applications

by N. N. Lebedev, Richard A. Silverman (Translator)

Overview

Richard Silverman's new translation makes available to English readers the work of the famous contemporary Russian mathematician N. N. Lebedev. Though extensive treatises on special functions are available, these do not serve the student or the applied mathematician as well as Lebedev's introductory and practically oriented approach. His systematic treatment of the basic theory of the more important special functions and the applications of this theory to specific problems of physics and engineering results in a practical course in the use of special functions for the student and for those concerned with actual mathematical applications or uses. In consideration of the practical nature of the coverage, most space has been devoted to the application of cylinder functions and particularly of spherical harmonics. Lebedev, however, also treats in some detail: the gamma function, the probability integral and related functions, the exponential integral and related functions, orthogonal polynomials with consideration of Legendre, Hermite and Laguerre polynomials (with exceptional treatment of the technique of expanding functions in series of Hermite and Laguerre polynomials), the Airy functions, the hypergeometric functions (making this often slighted area accessible to the theoretical physicist), and parabolic cylinder functions. The arrangement of the material in the separate chapters, to a certain degree, makes the different parts of the book independent of each other. Although a familiarity with complex variable theory is needed, a serious attempt has been made to keep to a minimum the required background in this area. Various useful properties of the special functions which do not appear in the text proper will be found in the problems at the end of the appropriate chapters. This edition closely adheres to the revised Russian edition (Moscow, 1965). Richard Silverman, however, has made the book even more useful to the English reader. The bibliography and references have been slanted toward books available in English or the West European languages, and a number of additional problems have been added to this edition.

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ISBN-13:	9780486139890
Publisher:	Dover Publications
Publication date:	04/02/2012
Series:	Dover Books on Mathematics
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CHAPTER 1

THE GAMMA FUNCTION

1.1. Definition of the Gamma Function

One of the simplest and most important special functions is the gamma function, knowledge of whose properties is a prerequisite for the study of many other special functions, notably the cylinder functions and the hypergeometric function. Since the gamma function is usually studied in courses on complex variable theory, and even in advanced calculus, the treatment given here will be deliberately brief.

The gamma function is defined by the formula

[MATHEMATICAL EXPRESSION OMITTED] (1.1.1)

whenever the complex variable z has a positive real part Re z. We can write (1.1.1) as a sum of two integrals, i.e.,

[MATHEMATICAL EXPRESSION OMITTED] (1.1.2)

where it can easily be shown that the first integral defines a function P(z) which is analytic in the half-plane Re z > 0, while the second integral defines an entire function. It follows that the function Γ(z) = P(z) + Q(z) is analytic in the half-plane Re z > 0.

The values of Γ(z) in the rest of the complex plane can be found by analytic continuation of the function defined by (1.1.1). First we replace the exponential in the integral for P(z) by its power series expansion, and then we integrate term by term, obtaining

[MATHEMATICAL EXPRESSION OMITTED] (1.1.3)

where it is permissible to reverse the order of integration and summation since

[MATHEMATICAL EXPRESSION OMITTED]

(the last integral converges for x = Re z > 0). The terms of the series (1.1.3) are analytic functions of z, if z ≠ 0, -1, -2, ... Moreover, in the region

[MATHEMATICAL EXPRESSION OMITTED]

(1.1.3) is majorized by the convergent series

[MATHEMATICAL EXPRESSION OMITTED]

and hence is uniformly convergent in this region. Using Weierstrass' theorem and the arbitrariness of d, we conclude that the sum of the series (1.1.3) is a meromorphic function with simple poles at the points z = 0, — 1, — 2, ... For Re z > 0 this function coincides with the integral P(z), and hence is the analytic continuation of P(z).

The function Γ(z) differs from P(z) by the term Q(z), which, as just shown, is an entire function. Therefore Γ(z) is a meromorphic function of the complex variable z, with simple poles at the points z = 0, — 1, — 2, ... An analytic expression for Γ(z), suitable for defining Γ(z) in the whole complex plane, is given by

[MATHEMATICAL EXPRESSION OMITTED] (1.1.4)

It follows from (1.1.4) that Γ(z) has the representation

[MATHEMATICAL EXPRESSION OMITTED] (1.1.5)

in a neighborhood of the pole z = – n (n = 0, 1, 2, ...), with regular part Ω(z + n).

1.2. Some Relations Satisfied by the Gamma Function

We now prove three basic relations satisfied by the gamma function :

[MATHEMATICAL EXPRESSION OMITTED]

These formulas play an important role in various transformations and calculations involving Γ(z).

To prove (1.2.1), we assume that Re z > 0 and use the integral representation (1.1.1). An integration by parts gives

[MATHEMATICAL EXPRESSION OMITTED]

The validity of this result for arbitrary complex z ≠ 0, -1, -2, ... is an immediate consequence of the principle of analytic continuation, since both sides of the formula are analytic everywhere except at the points z = 0, -1, -2, ...

To derive (1.2.2), we temporarily assume that 0 < Rez < 1 and again use (1.1.1), obtaining

[MATHEMATICAL EXPRESSION OMITTED]

Introducing the new variables

[MATHEMATICAL EXPRESSION OMITTED]

we find that

[MATHEMATICAL EXPRESSION OMITTED]

Using the principle of analytic continuation, we see that this formula remains valid everywhere in the complex plane except at the points z = 0, ±1, ±2, ...

To prove (1.2.3), known as the duplication formula, we assume that Re z > 0 and then use (1.1.1) again, obtaining

[MATHEMATICAL EXPRESSION OMITTED]

where we have introduced new variables [MATHEMATICAL EXPRESSION OMITTED]. To this formula we add the similar formula obtained by permuting α and ß. This gives the more symmetric representation

[MATHEMATICAL EXPRESSION OMITTED]

where the last integral is over the sector σ: 0 ≤ α < ∞, 0 ≤ ß ≤ α. Introducing new variables

u = α2 + ß2, v = 2αß,

we find that

[MATHEMATICAL EXPRESSION OMITTED]

As before, this result can be extended to arbitrary complex values z ≠ 0, -1/2 -1; -3/2, ..., by using the principle of analytic continuation.

We now use formula (1.2.1) to calculate Γ(z) for some special values of the variable z. Applying (1.2.1) and noting that Γ(1) = 1, we find by mathematical induction that

[MATHEMATICAL EXPRESSION OMITTED]

Moreover, setting z = 1/2 in (1.1.1), we obtain

[MATHEMATICAL EXPRESSION OMITTED] (1.2.5)

and then (1.2.1) implies

[MATHEMATICAL EXPRESSION OMITTED] (1.2.6)

Finally we use (1.2.2) to prove that the Function Γ(z) has no zeros in the complex plane. First we note that the points z = n (n = 0, ±1, ±2, ... ) cannot be zeros of Γ(z), since Γ(n) = (n -1)! if n = 1, 2, ... while Γ(n) = 8 if n = 0, -1, -2, ... The fact that no other value of z can be a zero of Γ(z) is an immediate consequence of (1.2.2), since if a nonintegral value of z were a zero of Γ(z) it would have to be a pole of Γ(1 - z), which is impossible. It follows at once that [Γ(z)]-1is an entire function.

1.3. The Logarithmic Derivative of the Gamma Function

The theory of the gamma function is intimately related to the theory of another special function, i.e., the logarithmic derivative of Γ(z):

[MATHEMATICAL EXPRESSION OMITTED]

Since Γ(z) is a meromorphic function with no zeros, ψ(z) can have no singular points other than the poles Γz = – n (n = 0, 1, 2, ...) of Γ(z). It follows from (1.1.5) that (z) has the representation

[MATHEMATICAL EXPRESSION OMITTED] (1.3.2)

in a neighborhood of the point z = – n, and hence ψ (z), like Γ(z), is a meromorphic function with simple poles at the points z = 0, -1, -2, ...

The function ψ(z) satisfies relations obtained from formulas (1.2.1-3) by taking logarithmic derivatives. In this way, we find that

[MATHEMATICAL EXPRESSION OMITTED] (1.3.3)

[MATHEMATICAL EXPRESSION OMITTED] (1.3.4)

[MATHEMATICAL EXPRESSION OMITTED] (1.3.5)

These formulas can be used to calculate ψ(z) for special values of z. For example, writing

[MATHEMATICAL EXPRESSION OMITTED] (1.3.6)

where γ = 0.57721566 ... is Euler's constant, and using (1.3.3), we obtain

[MATHEMATICAL EXPRESSION OMITTED] (1.3.7)

Moreover, substituting z = 1/2 into (1.3.5), we find that

[MATHEMATICAL EXPRESSION OMITTED] (1.3.8)

and then (1.3.3) gives

[MATHEMATICAL EXPRESSION OMITTED] (1.3.9)

The function ψ(z) has simple representations in the form of definite integrals involving the variable z as a parameter. To derive these representations, we first note that (1.1.1) implies

[MATHEMATICAL EXPRESSION OMITTED] (1.3.10)

If we replace the logarithm in the integrand by its expression in terms of the Frullani integral

[MATHEMATICAL EXPRESSION OMITTED] (1.3.11)

we find that

[MATHEMATICAL EXPRESSION OMITTED]

Introducing the new variable of integration u = t(x + 1), we find that the integral in brackets equals (x + l) ZΓ(z). This leads to the following integral representation of ψ(z):

[MATHEMATICAL EXPRESSION OMITTED] (1.3.12)

To obtain another integral representation of ψ(z), we write (1.3.12) in the form

[MATHEMATICAL EXPRESSION OMITTED]

and change the variable of integration in the second integral, by setting x + 1 = ey. This gives

[MATHEMATICAL EXPRESSION OMITTED]

and therefore, since the second integral approaches zero as δ -> 0,

[MATHEMATICAL EXPRESSION OMITTED] (1.3.13)

Setting z = 1 and subtracting the result from (1,3.13), we find that

[MATHEMATICAL EXPRESSION OMITTED] (1.3.14)

or

[MATHEMATICAL EXPRESSION OMITTED] (1.3.15)

where we have introduced the variable of integration x= e.-t

From formula (1.3.15) we can deduce an important representation of ψ(z) as an analytic expression valid for all z ≠ 0, -1, -2, ... , i.e., in the whole domain of definition of (z). To obtain this representation, we substitute the power series expansion

[MATHEMATICAL EXPRESSION OMITTED]

into (1.3.15) and integrate term by term (this operation is easily justified). The result is

[MATHEMATICAL EXPRESSION OMITTED] (1.3.16)

The series (1.3.16), whose terms are analytic functions for z ≠ 0, -1, -2, ... , is uniformly convergent in the region defined by the inequalities

[MATHEMATICAL EXPRESSION OMITTED]

since

[MATHEMATICAL EXPRESSION OMITTED]

for n ≥ N >a, and the series

[MATHEMATICAL EXPRESSION OMITTED]

converges. Therefore, since δ is arbitrarily small and a arbitrarily large, both sides of (1.3.16) are analytic functions except at the poles z = 0, -1, -2, ..., and hence, according to the principle of analytic continuation, the original restriction Re z > 0 used to prove this formula can be dropped. If we replace z by z+ 1 in (1.3.16), integrate the resulting series between the limits 0 and z, and then take exponentials of both sides, we find the following infinite product representation of the gamma function:

[MATHEMATICAL EXPRESSION OMITTED]

This formula can be made the starting point for the theory of the gamma function, instead of the integral representation (1.1.1).

Finally we derive some formulas for Euler's constant ?. Setting z = 1 in (1.3.12-13), we obtain

[MATHEMATICAL EXPRESSION OMITTED] (1.3.18)

Moreover, (1.3.10) implies

[MATHEMATICAL EXPRESSION OMITTED] (1.3.19)

which, when integrated by parts, gives

[MATHEMATICAL EXPRESSION OMITTED]

Replacing t by l/t in the last integral on the right, we find that

[MATHEMATICAL EXPRESSION OMITTED] (1.3.20)

1.4. Asymptotic Representation of the Gamma Function for Large |z| To describe the behavior of a given function f(z) as |z| ? 8 within a sector a arg z ß, it is in many cases sufficient to derive an expression of the form

[MATHEMATICAL EXPRESSION OMITTED] (1.4.1)

where f(z) is a function of a simpler structure than f(z), and r(z) converges uniformly to zero as |z| ? 8 within the given sector. Formulas of this type are called asymptotic representations of f(z) for large |z|. It follows from (1.4.1) that the ratio f(z)/f(z) converges to unity as |z| ? 8, i.e., the two functions f(z) and f(z) are "asymptotically equal," a fact we indicate by writing

[MATHEMATICAL EXPRESSION OMITTED] (1.4.2)

An estimate of |r(z)| gives the size of the error committed when f(z) is replaced by f(z) for large but finite |z|.

We now look for a description of the behavior of the function f(z) as |z| ? 8 which is more exact than that given by (1.4.1). Suppose we succeed in deriving the formula

[MATHEMATICAL EXPRESSION OMITTED] (1.4.3)

where zNrN(z) converges uniformly to zero as |z| ? 8, a arg z ß. [Note that (1.4.3) reduces to (1.4.1) for N= 0.] Then we write

[MATHEMATICAL EXPRESSION OMITTED] (1.4.4)

and the right-hand side is called an asymptotic series or asymptotic expansion of f(z) for large |z|. It should be noted that this definition does not stipulate that the given series converge in the ordinary sense, and on the contrary, the series will usually diverge. Nevertheless, asymptotic series are very useful, since, by taking a finite number of terms, we can obtain an arbitrarily good approximation to the function f(z) for sufficiently large |z|. In this book, the reader will find many examples of asymptotic representations and asymptotic series (see Secs. 1.4, 2.2, 3.2, 4.6, 4.14, 4.22, 5.11, etc.). For the general theory of asymptotic series, we refer to the references cited in the Bibliography on p. 300.

To obtain an asymptotic representation of the gamma function G(z), it is convenient to first derive an asymptotic representation of log G(z). To this end, let Re z > 0, and consider the integral representation (1.3.13), with z replaced by z + 1, i.e.,

[MATHEMATICAL EXPRESSION OMITTED]

or

[MATHEMATICAL EXPRESSION OMITTED]

where we have used (1.3.11). Integrating the last equation between the limits 1 and z, and bearing in mind that

log G(z + 1) = log G(z) + log z,

we find that

[MATHEMATICAL EXPRESSION OMITTED] (1.4.5)

where Re z > 0. It should be noted that the function

[MATHEMATICAL EXPRESSION OMITTED] (1.4.6)

appearing in the integrand in (1.4.5), is continuous for t 0, with f(0) = , as can easily be verified by expanding f(t) in a power series in a neighborhood of the point t = 0.

To simplify (1.4.5), we evaluate the integral

[MATHEMATICAL EXPRESSION OMITTED] (1.4.7)

This can be done by using the following trick: If

[MATHEMATICAL EXPRESSION OMITTED] (1.4.8)

then

[MATHEMATICAL EXPRESSION OMITTED]

It follows that

[MATHEMATICAL EXPRESSION OMITTED] (1.4.9)

On the other hand, substituting z = into (1.4.5), we find that

[MATHEMATICAL EXPRESSION OMITTED] (1.4.10)

and hence

[MATHEMATICAL EXPRESSION OMITTED] (1.4.11)

Using this result, we can write (1.4.5) in the form

[MATHEMATICAL EXPRESSION OMITTED] (1.4.12)

where

[MATHEMATICAL EXPRESSION OMITTED] (1.4.13)

Since f(t) decreases monotonically as t increases, the integral (1.4.13) also converges for Re z = 0, Im z [not equal] 0.

(Continues…)

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Table of Contents

1 THE GAMMA FUNCTION,
2 THE PROBABILITY INTEGRAL AND RELATED FUNCTIONS,
3 THE EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS,
4 ORTHOGONAL POLYNOMIALS,
5 CYLINDER FUNCTIONS: THEORY,
6 CYLINDER FUNCTIONS: APPLICATIONS,
7 SPHERICAL HARMONICS: THEORY,
8 SPHERICAL HARMONICS: APPLICATIONS,
9 HYPERGEOMETRIC FUNCTIONS,
10 PARABOLIC CYLINDER FUNCTIONS,
BIBLIOGRAPHY,
INDEX,