Wolfram Koepf - Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities

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Wolfram Koepf Universitext Hypergeometric Summation 2nd ed. 2014 An Algorithmic Approach to Summation and Special Function Identities 10.1007/978-1-4471-6464-7_1

Springer-Verlag London 2014

1. The Gamma Function

Wolfram Koepf 1

(1)

Fachbereich 10 Mathematik und Naturwissenschaften, Universitt Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germany

Wolfram Koepf

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Abstract

Apart from the elementary transcendental functions such as the exponential and trigonometric functions and their inverses, the Gamma function is probably the most important transcendental function. It was defined by Euler to interpolate the factorials at noninteger arguments.

Apart from the elementary transcendental functions such as the exponential and trigonometric functions and their inverses, the Gamma function is probably the most important transcendental function. It was defined by Euler to interpolate the factorials at noninteger arguments.

Following Euler, we define

and call it the Gamma function .

This improper integral exists for complex with (or, if you prefer only to think of real variables, for real ). Using integration by parts, we get the fundamental functional equation

(1.1)

From the initial value

it follows further by induction that

(1.2)

for . Therefore the -function interpolates the factorial function continuously, and we may define the factorial function by (

For points with nonpositive real part one reads the fundamental functional equation () from right to left to obtain

and defines the -function by a recursive application of this rule for with nonpositive real part (in particular for with ) (Figs. ).

Fig. 1.1

The Gamma function on the real axis

The resulting function is differentiable in the whole complex plane (proved by standard differentiation under the integral sign) except at the nonpositive integers where it has poles of order 1. By continuity, we may set

(1.3)

and by our general interpretation this reads as for . In function-theoretic terms this means that the function is an entire function, i.e., it is analytic in the entire complex plane with zeros exactly at the negative integers and the origin, the poles of .

By induction, we get from () for

(1.4)

The shifted factorial

(1.5)

which occurs in (). In this book, however, we will use the Pochhammer symbol only for integer values of .

From the fundamental identities (), we get the following limit relation at the poles of the -function

(1.6)

This computation may be interpreted as the residue computation

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