Question

Problem 4.9

(e) /(z) = and γ is parametrized by r(t), 0 z + t 1, and satisfies Imr(t)> 0, r(0) -4 + i, and γ(1) 6 + 2i (f) f(s) sin(z) and γ is some piecewise smooth path from 1 to π. 4.2 and the fact that the length of γ does not change under 4.9. Prove Proposi reparametrization. (Hint: Assume γ, σ, and τ are smooth. Start with the definition off, f, apply the chain rule to σ γ or, and then use the change of variables formula, Theorem A.6.) tion It is apparent but nevertheless noteworthy that the se integrals evaluate to different complex integral does not simply cesults; in particular--unlike in calculus -a depena the orher hand, the complex integral has some standard properties, most of d on the endpoints of the path of integration. w from their real siblings in a straightforward way. Our first observation is that the actual choice of parametrization of (r), a t b that σ is a reparametrization of γ if there is an increasin γ does not matter. More precisely, if d are parametrizations of a curve then we say g piecewise smooth map of and σ(t), c that takes γ to σ, in the sense that σ If r(), ass t γ τ. [c,d] onto [a, b] b is a piecewise smooth parametrization of a Proposition 4.2. curve and σ(t), c d is a reparametrization of γ then t f(o(t))ơ,@dt | fly(t))r'(t) dt. Example 4.3. To appreciate this statement, consider the two parametrizations

Answer 1

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