2. Suppose g: [0, 1] → C is a continuous function g(t) in one real variable. Define a complex function J: U → C where U = C\ [0, 1], by (a) For a, z EU, show that S(z) = f' g(t) Z dt. g(t) √(z) - ƒ(a) = f'(z-a). (t-z)(t-a) (b) Show that f is holomorphic on U and find an expression for the derivative f'(a). dt.
2. Suppose g: [0, 1] → C is a continuous function g(t) in one real variable. Define a complex function J: U → C where U = C\ [0, 1], by (a) For a, z EU, show that S(z) = f' g(t) Z dt. g(t) √(z) - ƒ(a) = f'(z-a). (t-z)(t-a) (b) Show that f is holomorphic on U and find an expression for the derivative f'(a). dt.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 2**: Consider a continuous function \( g : [0, 1] \to \mathbb{C} \) in one real variable. We define a complex function \( f : U \to \mathbb{C} \) where \( U = \mathbb{C} \backslash [0, 1] \), by
\[ f(z) = \int_0^1 \frac{g(t)}{t - z} \, dt. \]
**(a)** For \( a, z \in U \), show that
\[ f(z) - f(a) = \int_0^1 \frac{(z - a) \, g(t)}{(t - z)(t - a)} \, dt. \]
**(b)** Show that \( f \) is holomorphic on \( U \) and find an expression for the derivative \( f'(a) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27adff13-7954-4bd1-a3e1-459b28324214%2F5b0cefce-a27e-4aef-89d3-2f6d85fa4490%2Fcbxvyr9_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 2**: Consider a continuous function \( g : [0, 1] \to \mathbb{C} \) in one real variable. We define a complex function \( f : U \to \mathbb{C} \) where \( U = \mathbb{C} \backslash [0, 1] \), by
\[ f(z) = \int_0^1 \frac{g(t)}{t - z} \, dt. \]
**(a)** For \( a, z \in U \), show that
\[ f(z) - f(a) = \int_0^1 \frac{(z - a) \, g(t)}{(t - z)(t - a)} \, dt. \]
**(b)** Show that \( f \) is holomorphic on \( U \) and find an expression for the derivative \( f'(a) \).
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