Page 312 - Schaum's Outline of Differential Equations

CHAPTER 30

Gamma and Bessel

Functions

GAMMA FUNCTION
The gamma function, T(p), is defined for any positive real number/? by

Consequently, F(l) = 1 and for any positive real number/?,

Furthermore, when p = n, a positive integer,

Thus, the gamma function (which is defined on all positive real numbers) is an extension of the factorial function
(which is defined only on the nonnegative integers).
Equation (30.2) may be rewritten as

which defines the gamma function iteratively for all nonintegral negative values of p. F(0) remains undefined,
because

It then follows from Eq. (30.4) that F(/?) is undefined for negative integer values of p.
Table 30-1 lists values of the gamma function in the interval 1 <p<2. These tabular values are used with
Eqs. (30.2) and (30.4) to generate values of T(p) in other intervals.

BESSEL FUNCTIONS
Let/? represent any real number. The Bessel function of the first kind of order p, J p(x), is

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