Monday, March 10, 2025

Re-Exploring Features of Integration in the Calculus of Residues

 Let f(z) be analytic on and inside a closed contour C, except for a finite number of isolated singularities: z= a1, a2, etc. which are enclosed by C (e.g. diagram below)

.


Then:   ò C  f(z)  dz =      pi    ån k = 1    Res f (a k) 

 We now want to elaborate this a bit more by reference to the diagram shown. In this case we consider the function f(z) is analytic inside and on the simple closed curve C except at a finite number of specified points: a, b, c, etc.  at which there exist residues:   a - 1  ,        b - 1 ,  c - 1      , etc.


In which case we can write:

òC  f(z)  dz =   pi   [a - 1        +  b - 1          +  c - 1        + …………………….]
That is, 2 pi    times the sum of the residues at all the singularities enclosed by C. To ensure this, one would respectively construct circles C1, C2, C3 etc. as I have done with respective centers at a, b, c etc. If we take care to do this properly then we can write:


 ò C  f(z)  dz =       òC1  f(z)  dz   + ò C2  f(z)  dz  + ò C3  f(z)  dz   +    ..........


Where:
 ò C1  f(z)  dz    =   pi   a - 1       

  ò C2  f(z)  dz   =  p  b - 1       

  ò C3  f(z)  dz  =   p   c - 1       

So that:

 ò C  f(z)  dz=   pi   [a - 1  +  b - 1  +  c - 1   + ..] = p   (sum of residues)

Example 1:

Evaluate the integral:  ò   cot (z)  dz

f(z) = cot (z)

For which: ò C  f(z)  dz   =   pi   c - 1      
 
Re-write: f(z) = cot (z) = 1/ tan z

For which singularities occur at tan z = 0

Or: o, + p+ 2p,+  3p  etc.

Then Res f(z) =   1/ sec2 z ÷ z = + p     =    1/ (1/ cos2 z)
= cos2 z÷ z = + p     =    cos2 (np)  
and:  cos2 (np)    = 1   at z =  (2n + 1) p)/ 2

Therefore:    c - 1  =  1, and

  ò C  cot (z)  dz  =    pi   (1) = 2 pi  

Model residue solutions from Mathcad:




    The last can be solved to obtain:

                         I=    2 p/ Ö 3

Suggested Problem:

Integrate:

  ò C (z + 1)   dz / (2z +  i)



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