From the course: Complete Guide to Calculus Foundations for Data Science
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Cauchy's theorem
From the course: Complete Guide to Calculus Foundations for Data Science
Cauchy's theorem
- [Instructor] In the previous video, you learned that there are two main methods when solving contour integrals with closed curves. In this video, you'll focus on using Cauchy's Theorem. Let's first review what this theorem is. This is what Cauchy's Theorem states. Let f of z be a complex function that is holomorphic, meaning it is complex differentiable and it has no singularities in a simple connected domain, capital D. If capital C is a simple closed contour lying entirely within D, then you have the complex function integral of f of z dz equals zero. This means the contour integral of f of z around any closed curve, capital C in a simply connected domain is equal to zero. So how does this theorem work? A simply connected domain is where any close curve can be continuously deformed to a point without leaving the domain. If a function is holomorphic in a domain, then any closed loop integral within that domain must vanish. This is similar to the fundamental theorem of calculus…
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