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)

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Then:   ò _C  f(z)  dz =      2 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:

ò_C  f(z)  dz =   2 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 =       ò_C1  f(z)  dz   + ò _C2  f(z)  dz  + ò _C3  f(z)  dz   +    ..........


Where:

 ò _C1  f(z)  dz    =   2 pi   a _{- 1}       

  ò _C2  f(z)  dz   =  2 p  b _{- 1}       

  ò _C3  f(z)  dz  =   2 p   c _{- 1}       

So that:

 ò _C  f(z)  dz=   2 pi   [a _{- 1}  +  b _{- 1}  +  c _{- 1}   + ..] = 2 p   (sum of residues)

Example 1:

Evaluate the integral:  ò _C   cot (z)  dz

f(z) = cot (z)

For which: ò _C  f(z)  dz   =   2 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 = + n p}     =    1/ (1/ cos² z)

= cos² z÷ _{z = + n p}     =    cos² (np)  

and:  cos² (np)    = 1   at z =  (2n + 1) p)/ 2

Therefore:    c _{- 1}  =  1, and

  ò _C  cot (z)  dz  =    2 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)