Talk:Confluent hypergeometric function

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Whattaker function

Is U(a,b,z) the Whittaker function? (anon, Oct 2006)

I don't know, that's not what A&S calls them.linas 00:43, 11 December 2006 (UTC)[reply]

I am certainly not an expert, but I now know a bit about Kummer/Whittaker functions. Enough to find severe discrepancies between A&S and maple. Anybody have an opinion about whether I should tack some things up on the main page?

Kummer's function

I am interested in the real part of Kummmer's function in the case a=2n+1, b=a+1 (real part of incomplete gamma). From a numerical point of view, which is cheaper to approximate, what is the convergence like for each and what methods are used? (anon, Nov 2006)

continuous fraction for $e z$

The original text used to say

by setting $b = 0$ and $c = 1$

It is hard to tell what it meant because there was no $c$ around.

$.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}M(1, 2, z)⁄M(0, 1, z) = 1/ 1 − 1⁄2 z/ 1 + 1⁄6 z/ 1 − 2⁄12 z/ 1 + 2⁄20 z/ … 1 − k⁄(2 k − 1) (2 k) z/ 1 + k⁄(2 k) (2 k + 1) z/ … = 1 + 1/ 1 − 1⁄2 z/ 1 + 1⁄6 z/ 1 − 1⁄6 z/ 1 + 1⁄10 z/ … 1 − 1⁄2 (2 k − 1) z/ 1 + 1⁄2 (2 k + 1) z/ …$

Transforming this fraction with the sequence $(1, 2, 3, 2, \dots, 2 k + 1, 2, \dots)$ gives

$1/ 1 - z / 2 + z / 3 - z / 2 + z / \dots (2 k - 1) - z / 2 + z / \dots = (e z - 1) ⁄ z$

which is not quite what was postulated.

--Yecril (talk) 13:47, 3 October 2008 (UTC)[reply]

Whattaker function

Kummer's function

continuous fraction for ez

continuous fraction for $e z$