Complex analysis review 6
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Laurent Series
The special properties of a complex function is much more determined by its singularity, to study the singularity of a function, we first give a useful theorem that does not hold for real functions.
Theorem 1 (Weierstrass)
Suppose that
For any closed rectifiable simple curve
Then by Morera theorem,
If
Then we can get a open covering and use Heine-Borel theorem, the conclusion is true for any closed bounded subset of
Now we can define a Laurent series at
Theorem 2
If
where
Isolated Singular Point
If
From theorem 2, there is a Laurent series of
There are three case to be considered.
Removable singular point
Poles of order m
Since
Without lost of generality, we assume that
Essential singular point
In this case,
Theorem 3 (Weierstrass)
If
Which means that the values of
This can be showed easily by prove the converse.
Residual Theorem
Define residual of
Use Laurent series we can deduce that
If
If
where
So
Theorem 4
If
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