Special Functions

 

Special Functions

A number of special functions have become important in physics because they arise in frequently encountered situations. Identifying a one-dimensional (1-D) integral as one yielding a special function is almost as good as a straight-out evaluation, in part because it prevents the waste of time that otherwise might be spent trying to carry out the integration. But of perhaps more importance, it connects the integral to the full body of knowledge regarding its properties and evaluation. It is not necessary for every physicist to know everything about all known special functions, but it is desirable to have an overview permitting the recognition of special functions which can then be studied in more detail if necessary.

It is common for a special function to be defined in terms of an integral over the range for which that integral converges, but to have its definition extended to a larger domain by analytic continuation in the complex plane (cf. Chapter 11) or by the establishment of suitable functional relations. We present in Table 1.2 only the most useful integral representations of a few functions of frequent occurrence. More detail is provided by a variety of on-line sources and in material listed under Additional Readings at the end of this chapter, particularly the compilations by Abramowitz and Stegun and by Gradshteyn and Ryzhik.

THE EULERIAN FUNCTIONS:

 1 1. THE GAMMA FUNCTION : 

Γ(z) The Gamma function is the most widely used of all the special functions: it is usually discussed first since it appears in almost every integral or series representation of the other advanced mathematical functions. This function, denoted by Γ(z), can be defined by Euler’s Integral Representation (1)         Γ(x) = 

Integral on the interval [0, ∞] of∫ 0∞

t ^x −1 e^−t dt.  , Re(z) > 0 .

properties:

Γ( 1) = 1

Γ(z + 1) = z Γ(z),

Γ(n + 1) = n! , n = 0, 1, 2, . . . .

Γ  1 /2  = Z +∞ −∞ e −v 2 dv = √ π

Complementary Formula:Γ(z) Γ(1 − z) = π/ sin πz .

THE BETA FUNCTION : B(p, q) 

Euler’s Integral Representation :

B(p, q) = Z 1 0 u p−1 (1 − u) q−1 du , Re(p) > 0 , Re(q) > 0 .

Symmetry property of beta function :B(p, q) = B(q, p).

Relation with the Gamma Function :B(p, q) = Γ(p) Γ(q)/ Γ(p + q) .