#Estimation lemma for contour integration

31 messages · Page 1 of 1 (latest)

visual sail
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How do I actually use Estimation lemma to prove that a curve (Suppose gamma) equals 0?

cosmic troutBOT
visual sail
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The integral im trying to evaluating is the integral from negative infinity to positive infinity of cosx/x^2+1 in dx

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Already did all the stuff, like defining the entire contour's value (Which if I'm not wrong is pi/e) and I just need to prove what I wrote in the title

upper breach
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Gamma usually refers to the parametrization of the contour

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The contour integral of a holomorphic function is zero

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If we apply the Cauchy integral theorem

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But in this case, you must take a path that encloses a residue of the function

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Which function did you use

visual sail
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i said i was integrating cosx/x^2+1

upper breach
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Yes but

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When you apply the residue theorem

visual sail
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as there was just a pole in i (and another in -i, which i didnt take because i was considering only the upper bound. i had so the contour integral of cosz/z^2 in the contour c (which is all the semicircle, that give with the residue theorem pi/6) equal to the integral of my f(z) from -R to R and the integral of the curve i choose to call gamma

upper breach
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Which holomorphic function did you choose

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The f(z) to use should be

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Exp(iz)

visual sail
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cosz/z^2+1

upper breach
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/(z^2+1)

visual sail
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but i wasnt struggling with that

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i was struggling with the estimation lemma

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to prove that the integral of the curve (in my case gamma) of my f(z) was 0

upper breach
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It's not zero ... It's why we compute the residue of f at z = i

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The estimation lemma is used to make a part of the integral "vanish" as R goes to infinity

noble bone
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@visual sail as Daddy_314 says. Here, you have to switch to exp(iz) otherwise your integral does not tend to 0.

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Here is an exemple with your integral

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hope it will help you

visual sail
# noble bone

yeah,as i said in my paper i did that (i didnt said the part where i changed from trig to exp because it was obvious). im struggling only to prove with estiamtion lemma that the integral of the curve of the contour is 0

noble bone
visual sail
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yeah i saw it