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 ∫𝛾𝑓(𝑧)𝑑𝑧= ∫𝛾₁𝑓(𝑧)𝑑𝑧+ ∫𝛾₂𝑓(𝑧)𝑑𝑧+⋯+ ∫𝛾_𝑛𝑓(𝑧)𝑑𝑧

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TheoremLet 𝑓:[𝑎,𝑏]→ℝ be continuous, and define 𝐹(𝑥)=𝑥∫𝑎𝑓(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|=
        𝑏∫𝑎𝑓(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: