Hypergeometric Function
engineering fundamentals Hypergeometric Functions
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Definition

The Gaussian Hypergeometric Differential Equation is:

HyperGeoDE

where a, b, and c are constants. The indicial equation of the hypergeometric differential equation is:

which has the roots r1=0 and r2=1-c. Using the Frobenius method, the series solution for r1=0 can be express as:


where c!=0,-1,-2,-3,... and the series converges for -1<x<1. This series is called Hypergeometric Series. The sum of the hypergeometric series denoted by F(a,b;c;x) is called Hypergeometric Function, which is:

Important Properties

General Solution: If c, a-b, and c-a-b are all non-integers, the general solution for the hypergeometric differential equation is:

which is valid for -1<x<1.

Gamma Function: A hypergeometric function can be expressed in terms of gamma functions.


For x=1,


Other Formulas:

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