Wolfram Function Repository
Instantuse addon functions for the Wolfram Language
Function Repository Resource:
Find a numerical approximation of a residue of an expression around the specified point
ResourceFunction["NResidue"][expr,{z,z_{0}}] numerically finds the residue of expr near the point z=z_{0}. 
Residue of the function about the origin:
In[1]:= 

Out[1]= 

NResidue can find residues of functions with essential singularities:
In[2]:= 

Out[2]= 

Since Series is unable to handle essential singularities, Residue returns unevaluated:
In[3]:= 

Out[3]= 

NResidue allows for some error in the location of the pole:
In[4]:= 

Out[4]= 

Due to machineprecision arithmetic, z → 1. is not a pole:
In[5]:= 

Out[5]= 

With Residue, the error in the location of the pole yields a result of zero:
In[6]:= 

Out[6]= 

NResidue threads elementwise over lists:
In[7]:= 

Out[7]= 

Use "Radius" to shrink the radius of the contour of integration to isolate a single pole:
In[8]:= 

Out[8]= 

In[9]:= 

Out[9]= 

Increase the radius to improve convergence of the integration if no other poles are nearby:
In[10]:= 

Out[10]= 

In[11]:= 

Out[11]= 

NResidue accepts options for NIntegrate, which are sometimes necessary to get an accurate result:
In[12]:= 

Out[12]= 

Use NResidue to evaluate derivatives of a function evaluated at a point:
In[13]:= 

Out[13]= 

In[14]:= 

Out[14]= 

Residues of numerical functions:
In[15]:= 

In[16]:= 

Out[16]= 

NSeries can also compute residues of numerical functions:
In[17]:= 

In[18]:= 

Out[18]= 

Using NSeries:
In[19]:= 

Out[19]= 

NResidue will return an incorrect result when the integration contour contains branch cuts:
In[20]:= 

Out[5]= 

In[21]:= 

Out[21]= 

Wolfram Language 11.3 (March 2018) or above
This work is licensed under a Creative Commons Attribution 4.0 International License