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Contour integration

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This page is a ruby.

In mathematics, contour integration, invented by Augustin-Lous Cauchy, is a method in complex analysis that can :

  • evaluate definite integrals (residue theorem)
  • find location of poles and zeroes of a function (argument principle )
  • give upper and lower bounds for integrals (ML inequality)

It is a mandatory course in TND school.

note : I half assed this page and it may contain some errors


Consider the following contour in the complex plane :

where R is a limit going to infinity...or something

Let's find

the first half of the contour is a straight line that goes from -R to R

hence, this can be represented as, (H is the entire contour, because i said so)

the other half of the contour, , can be represented as the equation : where v is a variable that goes from to 0

differentating, we have

plugging it back into our original integrals

if we set :

Now take the absolute value

from the triangle inequality for integrals :

It's known that if z is real, then

thus :

Using the triangle inequality :

Since there is no more v, we can move the thing out of the integral, or so i think

If we set a limit with R go to infinity :

letting the limit go, we get

Since the absolute value of something is always greater than 0 unless its 0, then P=0, this gives us

We can let R go to infinity

inside 1/(x^2+1), there is only 2 poles, where x=-i, and where x=i, inside our contour, there is only one pole, x=i, hence, using the residue theorem:

x^2+1=(x-i)(x+i), hence :

Setting the limit

mathGODS won