Consider differentiable functions a(t), B(1), a, B: [lo, l₁] → R, and set µ(t) = a(t)+iß(t). Explain why rt₁ [{"^ p² (1) dt = µ(tr) — µ(lo). to [You may use the Fundamental Theorem of Calculus for real-valued functions in a real variable. The question is asking you to deduce a version for complex functions.] Now let f: U → C be a holomorphic function and y: [lo, t₁] → U a differentiable path from a € U to be U. Suppose that f(z) = g'(z) where g: U →→ C is another holomorphic function: that is, f has an antiderivative g on U. Show the Complex Fundamental Theorem of Calculus, [ f(z) dz = f*^* f(y(1))7′(1) dl = g(b) – g(a). [You can assume the first equality holds; the question is about the second.] [Hint: set u(t)=h(y(1)). What is '(t)? What does (a) give?]

Consider differentiable functions a(t), B(1), a, B: [lo, l₁] → R, and set µ(t) = a(t)+iß(t). Explain why rt₁ [{"^ p² (1) dt = µ(tr) — µ(lo). to [You may use the Fundamental Theorem of Calculus for real-valued functions in a real variable. The question is asking you to deduce a version for complex functions.] Now let f: U → C be a holomorphic function and y: [lo, t₁] → U a differentiable path from a € U to be U. Suppose that f(z) = g'(z) where g: U →→ C is another holomorphic function: that is, f has an antiderivative g on U. Show the Complex Fundamental Theorem of Calculus, [ f(z) dz = f^ f(y(1))7′(1) dl = g(b) – g(a). [You can assume the first equality holds; the question is about the second.] [Hint: set u(t)=h(y(1)). What is '(t)? What does (a) give?]

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter4: Calculating The Derivative

Section4.2: Derivatives Of Products And Quotients

Problem 36E

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