Hypergeometric Functions - The Hypergeometric Series

The Hypergeometric Series

The hypergeometric function is defined for |z| < 1 by the power series

provided that c does not equal 0, −1, −2, ... . Here (q)n is the Pochhammer symbol, which is defined by:

(q)_n = \left\{ \begin{array}{ll} 1 & \mbox{if } n = 0 \\ q(q+1) \cdots (q+n-1) & \mbox{if } n > 0 \end{array} \right .

Notice that the series terminates if either a or b is a nonpositive integer. For complex arguments z with |z| ≥ 1 it can be analytically continued along any path in the complex plane that avoids the branch points 0 and 1.

Read more about this topic:  Hypergeometric Functions

Famous quotes containing the word series:

    “Depression moods lead, almost invariably, to accidents. But, when they occur, our mood changes again, since the accident shows we can draw the world in our wake, and that we still retain some degree of power even when our spirits are low. A series of accidents creates a positively light-hearted state, out of consideration for this strange power.”
    —Jean Baudrillard (b. 1929)