Mathematical Methods in Science and Engineering, 2nd Edition

Chapter 6Hypergeometric Functions

Hypergeometric function, is a special function defined by the hypergeometric series. It is the solution of a linear second-order ordinary differential equation called the hypergeometric equation:

6.1

Majority of the second-order ordinary linear differential equations encountered in science and engineering can be expressed in terms of the three parameters of the hypergeometric equation and its transformations.

6.1 Hypergeometric Series

The hypergeometric equation has three regular singular points at and ; hence we can find a series solution about the origin using the Frobenius method. Substituting the series

6.2

into Eq. (6.1) gives

6.3

which we write as

6.4

After rearranging, we obtain

6.5

Writing the first term explicitly:

6.6

and setting the ...