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A power series Σan has radius of convergence R > 0 and defines a function f(z) on {z C || < R}. Write down the power series for the functions f'(z) and f(22), and state without proof their radii of convergence. Show that there is an entire function ƒ : C→ C, expressible as the sum of a power series, such that f(0) = 0, f'(0) = 0, and f"(z) = exp(22) for all z Є C.

A power series Σan has radius of convergence R > 0 and defines a function f(z) on {z C || < R}. Write down the power series for the functions f'(z) and f(22), and state without proof their radii of convergence. Show that there is an entire function ƒ : C→ C, expressible as the sum of a power series, such that f(0) = 0, f'(0) = 0, and f"(z) = exp(22) for all z Є C.

Advanced Engineering Mathematics
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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A power series Σan has radius of convergence R > 0 and defines a function f(z) on {z C || < R}. Write down the power series for the functions f'(z) and f(22), and state without proof their radii of convergence. Show that there is an entire function ƒ : C→ C, expressible as the sum of a power series, such that f(0) = 0, f'(0) = 0, and f"(z) = exp(22) for all z Є C.
Transcribed Image Text:A power series Σan has radius of convergence R > 0 and defines a function f(z) on {z C || < R}. Write down the power series for the functions f'(z) and f(22), and state without proof their radii of convergence. Show that there is an entire function ƒ : C→ C, expressible as the sum of a power series, such that f(0) = 0, f'(0) = 0, and f"(z) = exp(22) for all z Є C.
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