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Modern engineering and physical science applications demand a thorough knowledge of applied mathematics, particularly special functions. These typically arise in applications such as communication systems, electro-optics, nonlinear wave propagation, electromagnetic theory, electric circuit theory, and quantum mechanics. This text systematically introduces special functions and explores their properties and applications in engineering and science.

Copublished with Oxford University Press.

Softcover version of PM49.

This front matter contains the table of contents, preface to the 1st and 2nd editions, and notation for special functions.

This back matter contains the index and author biography.

1.1 Introduction Because of the close relation of infinite series and improper integrals to the special functions, it can be useful to first review some basic concepts of series and integrals. Infinite products, which are generally less well known, are introduced here mostly for the sake of completeness, but in some instances they are also useful. Infinite series are important in almost all areas of pure and applied mathematics. In addition to numerous other uses, they are used to define certain functions and to calculate accurate numerical estimates of the values of these functions. In calculus the primary problem is deciding whether a given series converges or diverges. In practice, however, the more crucial problem may actually be summing the series. If a convergent series converges too slowly, the series may be worthless for computational purposes. On the other hand, the first few terms of a divergent series in some instances may give excellent results. Improper integrals and infinite products are used in much the same fashion as infinite series, and, in fact, their basic theory closely parallels that of infinite series. In the application of mathematics frequently two or more limiting processes have to be performed successively. For example, we often find the derivative (or integral) of an infinite sum of functions by taking the sum of derivatives (or integrals) of the individual terms of the series. However, in many cases of interest, performing two limit operations in one order may yield an answer different from that obtained using the other order. That is, the order in which the limiting processes are carried out may be critical.

2.1 Introduction In the eighteenth century, L. Euler (1707â1783) concerned himself with the problem of interpolating between the numbers n!=â« â 0 e ât t n dtn=0,1,2,â¦ with nonintegral values of n. This problem led Euler in 1729 to the now famous gamma function, a generalization of the factorial function that gives meaning to x! when x is any positive number. His result can be extended to certain negative numbers and even to complex numbers. The notation Î(x) that is now widely accepted for the gamma function is not due to Euler, however, but was introduced in 1809 by A. Legendre (1752â1833), who was also responsible for the duplication formula for the gamma function. Nearly 150 years after Euler's discovery of it, the theory concerning the gamma function was greatly expanded by means of the theory of entire functions developed by K. Weierstrass (1815â1897). Because it is a generalization of n!, the gamma function has been examined over the years as a means of generalizing certain functions, operations, etc., that are commonly defined in terms of factorials. In addition, the gamma function is useful in the evaluation of many nonelementary integrals and in the definition of other special functions. Another function useful in various applications is the related beta function, often called the eulerian integral of the first kind.

3.1 Introduction In addition to the gamma function, there are numerous other special functions whose primary definition involves an integral. Some of these functions were introduced in Chap. 2 along with the gamma function, and in this chapter we consider several others. The error function derives its name from its importance in the theory of errors, but it also occurs in probability theory and in certain heat conduction problems on infinite domains. The closely related Fresnel integrals, which are fundamental in the theory of optics, can be derived directly from the error function. A special case of the incomplete gamma function (Sec. 2.5) leads to the exponential integral and related functionsâthe logarithmic integral, which is important in analysis and number theory, and the sine and cosine integrals, which arise in Fourier transform theory. Elliptic integrals first arose in the problems associated with computing the arclength of an ellipse and a lemniscate (a curve in the shape of a figure eight). Some early results concerning elliptic integrals were discovered by L. Euler and J. Landen, but virtually the whole theory of these integrals was developed by Legendre over a period spanning 40 years. The inverses of the elliptic integrals, called elliptic functions, were independently introduced in 1827 by C. G. J. Jacobi (1802â1859) and N. H. Abel (1802â1829). Many of the properties of elliptic functions, however, had already been developed as early as 1809 by Gauss. Elliptic functions have the distinction of being doubly periodic, with one real period and one imaginary period. Among other areas of application, the elliptic functions are important in solving the pendulum problem (Sec. 3.5.2).

4.1 Introduction The Legendre polynomials are closely associated with physical phenomena for which spherical geometry is important. In particular, these polynomials first arose in the problem of expressing the newtonian potential of a conservative force field in an infinite series involving the distance variables of two points and their included central angle (see Sec. 4.2). Other similar problems dealing with either gravitational potentials or electrostatic potentials also lead to Legendre polynomials, as do certain steady-state heat conduction problems in spherical solids, and so forth. There exist a whole class of polynomial sets which have many properties in common and for which the Legendre polynomials represent the simplest example. Each polynomial set satisfies several recurrence formulas, is involved in numerous integral relationships, and forms the basis for series expansions resembling Fourier trigonometric series, where the sines and cosines are replaced by members of the polynomial set. Because of all the similarities in these polynomial sets and because the Legendre polynomials are the simplest such set, our development of the properties associated with the Legendre polynomials will be more extensive than similar developments in Chap. 5, where we introduce other polynomial sets. In addition to the Legendre polynomials, we present a brief discussion of the Legendre functions of the second kind and associated Legendre functions. The Legendre functions of the second kind arise as a second solution set of Legendre's equation (independent of the Legendre polynomials), and the associated functions are related to derivatives of the Legendre polynomials.

5.1 Introduction A set of functions {Ï n (x)},n=0,1,2,â¦, is said to be orthogonal on the interval a<x<b , with respect to a weight function r(x)<0 , if â« b a r(x)Ï n (x)Ï k (x)dx=0kâ n Sets of orthogonal functions play an extremely important role in analysis, primarily because functions belonging to a very general class can be represented by series of orthogonal functions, called generalized Fourier series. A special class of orthogonal functions consists of the sets of orthogonal polynomials{p n (x)} , where n denotes the degree of the polynomial p n (x) . The Legendre polynomials discussed in Chap. 4 are probably the simplest set of polynomials belonging to this class. Other polynomial sets which commonly occur in applications are the Hermite, Laguerre, and Chebyshev polynomials. More general polynomial sets are defined by the Gegenbauer and Jacobi polynomials, which include the others as special cases. The study of general polynomial sets like the Jacobi polynomials facilitates the study of each polynomial set by focusing on those properties that are characteristic of all the individual sets.

The German astronomer F. W. Bessel (1784â1846) first achieved fame by computing the orbit of Halley's comet. In addition to many other accomplishments in connection with his studies of planetary motion, he is credited with deriving the differential equation bearing his name and carrying out the first systematic study of the general properties of its solutions (now called Bessel functions) in his famous 1824 memoir. Nonetheless, Bessel functions were first discovered in 1732 by D. Bernoulli (1700â1782), who provided a series solution (representing a Bessel function) for the oscillatory displacements of a heavy hanging chain (see Sec. 6.7.1). Euler later developed a series similar to that of Bernoulli, which was also a Bessel function, and Bessel's equation appeared in a 1764 article by Euler dealing with the vibrations of a circular drumhead. J. Fourier (1768â1836) also used Bessel functions in his classical treatise on heat in 1822, but it was Bessel who first recognized their special properties. Bessel functions are closely associated with problems possessing circular or cylindrical symmetry. For example, they arise in the study of free vibrations of a circular membrane and in finding the temperature distribution in a circular cylinder. They also occur in electromagnetic theory and numerous other areas of physics and engineering. In fact, Bessel functions occur so frequently in practice that they are undoubtedly the most important functions beyond the elementary ones. Because of their close association with cylindrical domains, the solutions of Bessel's equation are also called cylinder functions.

7.1 Introduction The Bessel functions of the first and second kinds studied in Chap. 6 are often referred to as the standard Bessel or cylinder functions. In addition to these, there are a host of related functions also belonging to the general family of cylinder functions, the most notable of which are the modified Bessel functions of the first and second kinds. Although similar in definition to the standard Bessel functions, the modified Bessel functions are most clearly distinguished by their nonoscillatory behavior. For this reason, they often appear in applications that are different in nature from those for the standard functions. The general family of cylinder functions also include spherical Bessel functions, Hankel functions, Kelvin's functions, Lommel functions, Struve functions, Airy functions, and Anger and Weber functions. Of these, Hankel functions have special significance in that they enable us to obtain asymptotic formulas for large arguments for all the other types of Bessel functions.

8.1 Introduction Bessel functions are prominent in a variety of applications, some of which were discussed in Chaps. 6 and 7. Now we wish to consider some additional examples typical of those occurring in more than one field of application. To provide some variety in our discussions, we have chosen examples from the fields of mechanics, wave propagation and scattering, fiber optics, heat conduction in solids, and vibration phenomena. (A working knowledge of each subject is generally sufficient to follow the discussion.) 8.2 Problems in Mechanics We begin this chapter on applications with some examples chosen from the fields of particle dynamics and the static displacements of beams and columns. Additional problems of a similar nature are taken up in the exercises.

9.1 Introduction Because of the many relations connecting the special functions to each other, and to the elementary functions, it is natural to inquire whether more general functions can be developed so that the special functions and elementary functions are merely specializations of these general functions. General functions of this nature have in fact been developed and are collectively referred to as functions of the hypergeometric type. There are several varieties of these functions, but the most common are the standard hypergeometric function (which we discuss in this chapter) and the confluent hypergeometric function (Chap. 10). Still, other generalizations exist, such as MacRobert's E function and Meijer's G function, for which even generalized hypergeometric functions are certain specializations (Chap. 11). The major development of the theory of the hypergeometric function was carried out by Gauss and published in his famous memoir of 1812, a memoir that is also noted as being the real beginning of rigor in mathematics. Some important results concerning the hypergeometric function had been developed earlier by Euler and others, but it was Gauss who made the first systematic study of the series that defines this function.

10.1 Introduction Whereas Gauss was largely responsible for the systematic study of the hypergeometric function, E. E. Kummer (1810â1893) is the person most associated with developing properties of the related confluent hypergeometric function. Kummer published his work on this function in 1836, and since that time it has been commonly referred to as Kummer's function. Like the hypergeometric function, the confluent hypergeometric function is related to a large number of other functions. Kummer's function satisfies a second-order linear differential equation called the confluent hypergeometric equation. A second solution of this DE leads to the definition of the confluent hypergeometric function of the second kind, which is also related to many other functions. At the beginning of the twentieth century (1904), Whittaker introduced another pair of confluent hypergeometric functions that now bears his name. The Whittaker functions arise as solutions of the confluent hypergeometric equation after a transformation to Liouville's standard form of the DE.

11.1 Introduction The special properties associated with the hypergeometric and confluent hypergeometric functions have spurred a number of investigations into developing functions even more general than these. Some of this work was done in the nineteenth century by Clausen, Appell, and Lauricella (among others), but much of it has occurred during the last 70 years. Even the most recent names are too numerous to mention, but MacRobert and Meijer are among the most famous. The importance of working with generalized functions of any kind stems from the fact that most special functions are simply special cases of them, and thus each recurrence formula or identity developed for the generalized function becomes a master formula from which a large number of relations for other functions can be deduced. New relations for some of the special functions have been discovered in just this way. Also the use of generalized functions often facilitates the analysis by permitting complex expressions to be represented more simply in terms of some generalized function. Operations such as differentiation and integration can sometimes be performed more readily on the resulting generalized functions than on the original complex expression, even though the two are equivalent. Finally, in many situations we resort to expressing our results in terms of these generalized functions because there are no simpler functions that we can call upon. Our treatment of generalized hypergeometric functions is brief.

12.1 Introduction In this final chapter we illustrate the use of the general family of hypergeometric functions in various applications. Although we have chosen specific examples from the fields of statistical communication theory, fluid mechanics, and random fields, the techniques we use are sufficiently general that they apply to a wider range of applications. As before, we assume only a working knowledge of the subjects in order to follow the exposition. 12.2 Statistical Communication Theory Communication systems may be broadly classified in terms of linear operations, such as amplification and filtering, and nonlinear operations, such as modulation and detection. Random noise, which appears at the input to any communications receiver, interferes with the reception of incoming radio and radar signals. When this noise is channeled through a passband linear filter whose bandwidth is narrow compared with the center frequency Ï 0 of the filter, the output is called narrowband noise and has the representation (recall Sec. 8.3.1) n(t)=x(t)cosÏ 0 tây(t)sinÏ 0 t where x(t) and y(t) are independent gaussian (or normal) random processes with zero means and equal mean-squared values N.

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