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What is Residue Theorem?

Welcome back MechanicaLEi, did you know that the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals as well.? This makes us wonder, what is Residue theorem? Before we jump in check out the previous part of this series to learn about what Taylor’s and Laurent’s series are? First, we need to learn what residue is; Suppose f of z is a function which is analytic inside and on a closed contour C, except for a pole of order m at z equals to z not, which lies inside C. To evaluate closed integral of f of z dz around c we can expand f of z in a Laurent series in powers of z minus z not. We know that the integral of each of the positive and negative powers of z minus z not is zero with the exception of b one upon z minus z not and this has the value 2 into pi into b one. Since it is the only coefficient remaining after the integration, it is called residue of f of z at z equals to z not and is given by b one equals to one upon 2 pi i into closed integral of f of z d z about c. Now, the residue theorem states that for a complex variable f of z in the region C, closed integral of f of z d z over c is given by 2 pi i into sum of the residues at the poles inside c. Hence, we first saw what residue is and then went on see what Residue theorem is? In the next episode of MechanicaLEi find out what orthogonal and orthonormal functions are? Attributions: Doh De Oh by Kevin MacLeod is licensed under a Creative Commons Attribution licence (https://creativecommons.org/licenses/...) Source: http://incompetech.com/music/royalty-... Artist: http://incompetech.com/ Subtle Library by Fabian Measures (http://freemusicarchive.org/music/Fab...) is licensed under a Creative Commons Attribution licence ( https://creativecommons.org/licenses/...) Source: http://freemusicarchive.org/music/Fab...