Contour integration: Difference between revisions
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hence, this can be represented as, (H is the entire contour, because i said so) | hence, this can be represented as, (H is the entire contour, because i said so) | ||
<math> \oint\limits_{H}\frac{1}{x^2+1} \mathrm{d} x=\int\limits_{\gamma_R}\frac{1}{x^2+1} \mathrm{d} x+\int\limits_{R}^{R}\frac{1}{x^2+1} \mathrm{d} x</math></blockquote> | <math> \oint\limits_{H}\frac{1}{x^2+1} \mathrm{d} x=\int\limits_{\gamma_R}\frac{1}{x^2+1} \mathrm{d} x+\int\limits_{R}^{R}\frac{1}{x^2+1} \mathrm{d} x</math></blockquote> | ||
the other half of the contour, <math>\gamma_R</math></blockquote>, can be represented as the equation : | the other half of the contour, <math>\gamma_R</math></blockquote>, can be represented as the equation : |
Revision as of 14:42, 21 July 2023
The residue theorem, invented by Cauchy, is a method of evaluating definite integrals.
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