Let
be a domain,
, and let
be holomorphic except for isolated singularities
, i.e.,
is holomorphic. If
is an Isolated singularity of
with
, the residue is defined as:
.
If
is represented around an isolated singularity
as a Laurent series, the residue can be computed as follows.
With
as the Laurent Series expansion of
around
, it holds that:
.
It must be taken into account that the closed disk
contains only the singularity
, i.e
.
Thus, one can read off the 'residue'
from the Laurent expansion of around at the
um
an -1-ten coefficient of .
The closed disk
must contain only the singularity
, meaning
.
Thus, the residue
can be read off directly as the coefficient of
in the Laurent series expansion of
around
.
The term "residue" (from Latin residuere – to remain) is used because in integration along the path
with
around the circle centered at
, the following holds:
Thus, the residue is what remains after integrating.
If
is a pole of order
of
, the Laurent Series expansion of
around
has the form:

with
.
By multiplying with
, we get:

The residue
is then the coefficient of
in the power series of
.
By differentiating
times, the first
terms from
to
vanish. The residue is then found as the coefficient of
in:

Proof 3: Limit Process to Compute Coefficient of 
[edit | edit source]
By shifting the index:

Taking the limit
, all terms with
vanish, leaving:

Thus, the residue can be computed
using:

- Explain why, in the Laurent series expansion, all terms from the principal and outer parts, i.e.,
with
, yield integrals that evaluate to zero:

- Why can the order of integration and series expansion be interchanged?

- Given the function
with
, calculate the residue
at
.
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https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Residuum