From the course: Complete Guide to Calculus Foundations for Data Science
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Residue theorem
From the course: Complete Guide to Calculus Foundations for Data Science
Residue theorem
- [Instructor] In the previous video, you learned how to use Cauchy's theorem to solve contour integrals with closed curves. But what happens if your complex function contains singularities? This is where the residue theorem shines. Let's first review what this theorem is. This is what the residue theorem states. Let f of z be a function that is meromorphic, meaning it is holomorphic except for isolated singularities, and this is going to be on a simple closed contour C in the complex plane. Suppose f of z has a finite number of isolated singularities notated by z1, z2, all the way to zn, and this will be inside of C. Then the contour integral is given by the contour integral of f of z over z minus z nought dz is equal to two multiplied by pi, multiplied by i, multiplied by the sum of k equals one to n of the residues of (f,zk). Let's review the contour integral for f of z with the residue theorem. So looking at your formula that you just reviewed, you first have the singularities of…
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