Residue

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`Residue`

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Residue[expr,{z,z₀}]

finds the residue of expr at the point z=z₀.

Details and Options

The residue is defined as the coefficient of (z-z₀)^-1 in the Laurent expansion of expr.
The Wolfram Language can usually find residues at a point only when it can evaluate power series at that point.

Examples

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Basic Examples (2)Summary of the most common use cases

In[1]:=1

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https://wolfram.com/xid/0c129jq-qr2

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0c129jq-v0u

Out[2]=2

In[1]:=1

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https://wolfram.com/xid/0c129jq-6a

Out[1]=1

Scope (3)Survey of the scope of standard use cases

In[1]:=1

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https://wolfram.com/xid/0c129jq-eje

Out[1]=1

In[1]:=1

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https://wolfram.com/xid/0c129jq-rbb

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0c129jq-sbm

Out[2]=2

In[1]:=1

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https://wolfram.com/xid/0c129jq-psu

Out[1]=1

Applications (1)Sample problems that can be solved with this function

Illustrate Cauchy's theorem for the integral of a complex function:

In[1]:=1

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https://wolfram.com/xid/0c129jq-hrv

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0c129jq-db3

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0c129jq-gz

Out[3]=3

Properties & Relations (1)Properties of the function, and connections to other functions

Use FunctionPoles to find the poles of a function:

In[1]:=1

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https://wolfram.com/xid/0c129jq-dawhi

Out[1]=1

Use Residue to find the residues at the poles:

In[2]:=2

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https://wolfram.com/xid/0c129jq-dk8axc

Out[2]=2

ResidueSum gives the sum of the residues at all poles:

In[3]:=3

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https://wolfram.com/xid/0c129jq-lr5yac

Out[3]=3

Possible Issues (1)Common pitfalls and unexpected behavior

Residues are not defined at branch points:

In[1]:=1

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https://wolfram.com/xid/0c129jq-gbd

Out[1]=1

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