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## Computational definitions for interesting complex functions [closed]

I'm after a reading source for a set of 'interesting' functions $f:\mathbb{C}^m \rightarrow \mathbb{C}$, complete with definitions that can be used to compute them numerically.

I'm looking for functions bearing graphs with interesting (read: varied) behaviour, or that can be composed in order to generate more diverse ones.

An example building block is the exp function which we can define as

$e^{a \ + \ ib} = e^{a} \ (\cos a + i\sin b)$

which can be immediately implemented numerically.

Functions

• like exp, that can be defined in terms of real ones;
• that have some known algorithm for computing the complex components;
• that are defined in terms of others in the list,

are good.

Does anyone know of a 'list' of such common/interesting computable functions, from books or web (preferably)? Making it up on the spot is good too.

Apologies for the lack of formality :) Thanks!

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Even though this looks like a fishing expedition, I will mention hypergeometric functions. Voting to close. – Steve Huntsman Jul 7 2010 at 22:04
Interesting thanks! Let's not close it before we have a few fish in the bucket :-) – Mau Jul 7 2010 at 22:08
I'm not convinced that "varied behavior", even if made precise, is a good criterion for a function being interesting. In any case, I think perusing dlmf.nist.gov will answer this question. – jc Jul 7 2010 at 22:25
I can't see how this is a mathematical question; it looks like an aesthetic one to me. Voting to close. – Qiaochu Yuan Jul 8 2010 at 0:03
I don't think this question is so bad. All of the standard "closed-form" functions are built from sum, negation, product, and exp. It's natural to ask for some other, more general functions that have some of the same nice properties as the elementary functions, and certainly one of these properties is that they are not too hard to compute. – Charles Staats Jul 8 2010 at 2:05