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 𝛾𝑓(𝑧)𝑑𝑧= 𝛾1𝑓(𝑧)𝑑𝑧+ 𝛾2𝑓(𝑧)𝑑𝑧+⋯+ 𝛾𝑛𝑓(𝑧)𝑑𝑧

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TheoremLet 𝑓:[𝑎,𝑏]→ℝ be continuous, and define 𝐹(𝑥)=𝑥𝑎𝑓(𝑡)𝑑𝑡. Then 𝐹 is differentiable and 𝐹′(𝑥)=𝑓(𝑥) for 𝑥∈[𝑎,𝑏].

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DefinitionLet 𝛾 be a smooth curve, and let 𝑓 be a complex-valued and continuous function on 𝛾. Then  𝛾𝑓(𝑧)|𝑑𝑧|= 𝑏𝑎𝑓(𝛾(𝑡))|𝛾′(𝑡)|𝑑𝑡 is the integral of 𝑓 over 𝛾 with respect to arc length

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One way to prove this theorem is as follows First, show Morera's Theorem: If 𝑓 is continuous on a simply connected domain 𝐷, and if  𝛾𝑓(𝑧)𝑑𝑧=0 for any triangular curve 𝛾 in 𝐷, then 𝑓 has a primitive in 𝐷. Next, show the Cauchy Theorem for Triangles: For any triangle 𝑇 that fits into 𝐷 (including its boundary),  ∂𝑇𝑓(𝑧)𝑑𝑧=0.

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ID: 210800016 Last Updated: 8/16/2021 Revision: 0 Ref:

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References

  1. http://www.w3.org/TR/1999/REC-html401-1999, 1999, HTML 4.01 Specification: W3C Recommendation, updated 24 December 1999
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