# Algebraic form for a function of a complex variable.

Let be a function of a complex variable, defined over a domain in . We write , where and are real numbers, i.e. is written in algebraic form. We can write in algebraic form too, i.e.

 (2.1)

Example 2.2.1   Let , for . With , where and , as above, we have:

Example 2.2.2   Let , for . With , where and , as above, we have:

Thus

 and

Conversely, if we have a function given in algebraic form (v.s. 3)

we can compute a closed'' form for , using the following remark:

 (2.2)

Example 2.2.3   Let . Using Euler formulas, i.e. Equation (4), we have:

Of course, the converse process is possible, i.e. for a function given by a formula like , Euler formulas can be used to give an expression in and for .

Example 2.2.4   Let . We use Euler formulas:

Thus,

Noah Dana-Picard 2007-12-24