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Residue Theorem and friends

A compendium of difficult calculus problems, some of which lean on the residue theorem

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Residue Theorem and friends | residuetheorem.com Reviews
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A compendium of difficult calculus problems, some of which lean on the residue theorem
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3 moderate nerd
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5 cauchy principal value
6 classical electrodynamics
7 then
8 by ron
9 posted in uncategorized
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Residue Theorem and friends | residuetheorem.com Reviews

https://residuetheorem.com

A compendium of difficult calculus problems, some of which lean on the residue theorem

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1

An integral that illustrates the beauty of contour integration methods – Residue Theorem and friends

http://residuetheorem.com/2014/01/27/an-integral-that-illustrates-the-beauty-of-contour-integration-methods

Residue Theorem and friends. A compendium of difficult calculus problems, some of which lean on the residue theorem. An integral that illustrates the beauty of contour integration methods. I attack many of the integrals in this blog using contour integration methods, some of which are obvious and some of which take a little more imagination. Here is one that illustrates the beauty an power of such methods as much as any other integral here. Is to derive the integral representation. For $ a le pi/2$.

2

An integral involving a quadratic phase – Residue Theorem and friends

http://residuetheorem.com/2014/02/08/an-integral-involving-a-quadratic-phase

Residue Theorem and friends. A compendium of difficult calculus problems, some of which lean on the residue theorem. An integral involving a quadratic phase. This one was first posted on the site Integrals and Series. And was brought to my attention on M.SE. By Cody. My solution involves a contour integration, although the approach is far from trivial. Yet again, the solution boils down to finding a contour and a function to integrate over the contour. The problem involves evaluating. Oint C dz ;f(z) $.

3

Inverse Laplace Transform with a coinciding pole and branch point – Residue Theorem and friends

http://residuetheorem.com/2015/03/07/inverse-laplace-transform-with-a-coinciding-pole-and-branch-point

Residue Theorem and friends. A compendium of difficult calculus problems, some of which lean on the residue theorem. Inverse Laplace Transform with a coinciding pole and branch point. Recently, the following Laplace transform. Was asked to be inverted:. F(s) = s {-a-1} e {-s a} $. Because of the branch point, the way to attack a problem like this is via Cauchy’s theorem on a properly distorted Bromwich contour. Here, we want our contour to avoid the branch point at $z=0$. This, we consider. C 5$ is the c...

4

Uncategorized – Residue Theorem and friends

http://residuetheorem.com/category/uncategorized

Residue Theorem and friends. A compendium of difficult calculus problems, some of which lean on the residue theorem. A method of evaluating a double integral that nobody taught you in school. May 4, 2016 – 3:19 am. The problem is to evaluate $ prod {n=2} { infty} left (1 frac{(-1) {n-1} {a n} right ) $ where $ a n = n! Sum {k=1} {n-1} frac{(-1) {k-1} {k! This is one of those cases where trying out a bunch of numbers really isn’t going to help all that much. The $a n$ look kind of like $n! I stumbled onto...

5

Algebraically difficult integral – Residue Theorem and friends

http://residuetheorem.com/2014/02/18/algebraically-difficult-integral

Residue Theorem and friends. A compendium of difficult calculus problems, some of which lean on the residue theorem. The integral to evaluate is. Int {2 pi} 0 d theta dfrac{ cos 2 theta}{ 2e {i theta}-z 2} , qquad mbox {when} , z neq 2.$. Write $ zeta=e {i theta}$ as usual to get, for the integral,. I frac14 oint { zeta =1} frac{d zeta}{ zeta} frac{( zeta zeta {-1}) 2}{ 2 zeta-z 2} = -i frac14 oint { zeta =1} frac{d zeta}{ zeta 2} frac{( zeta 2 1) 2}{(2 zeta-z)(2 - bar{z} zeta)}$. Int 0 {2 pi} d theta fr...

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