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 2.2.2
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,
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