 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 and 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 Lorentz 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 counter c, except for a pole of order m, at z equals to z0, which lies inside c. To evaluate closed integral of f of z dz around c, we can expand f of z in a Lorentz series in powers of z minus z0. We know that the integral of each of the positive and negative powers of z minus z0 is 0 with the exception of b1 upon z minus z0 and this has a value 2 into pi into b1. Since it is the only coefficient remaining after integration, it is called residue of f of z at z equals to z0 and is given by b1 equals to 1 upon 2 pi i into closed integral of f of z dz about c. Now the residue theorem states that for a complex variable f of z in this region c, closed integral of f of z dz over c is given by 2 pi i into some of the residues at the poles inside c. Hence we first saw what residue is and then went on to see what residue theorem is. In the next episode of Mechanical EI, find out what orthogonal and orthonormal functions are.