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Chapter 3:
Other Functions Defined by Integrals
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 functions—the 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 (1802–1859) and N. H. Abel (1802–1829). 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).
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