# HW8 - Math 401(Sec 502 Homework 8(due March 26 Spring 2009...

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Math 401 (Sec. 502) Spring, 2009 Homework 8 (due March 26) Q1. (a) Show that any function f ( x ) deﬁned on [ L, L ] can be written as the sum of an even function and an odd function. [ Hint. Suppose f ( x ) = p ( x ) + q ( x ), where p ( x ) is even and q ( x ) is odd. Then solve the equations, f ( x ) = p ( x ) + q ( x ) and f ( x ) = p ( x ) q ( x ), simultaneously for p ( x ) and q ( x ) in terms of f ( x ) and f ( x ).] (b) Assuming that f ( x ) is deﬁned on [ L, L ], prove that, f ( x ) is even when f ( x ) is odd and f ( x ) is odd when f ( x ) is even. Q2. If f ( x ) is continuous on the interval 0 < x < a , is its even periodic extension continuous? What about the odd periodic extension? [ Hint. check at x = 0 and ± a .] Q3. Sketch both the even and odd extensions of the functions (a) f ( x ) = 1 , 0 < x < 2 (b) f ( x ) = x, 0 < x < 1 (c) f ( x ) = cos x, 0 < x < π . Q4.

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## This note was uploaded on 04/27/2009 for the course MATH 401 taught by Professor Sarkar during the Spring '09 term at Texas A&M.

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HW8 - Math 401(Sec 502 Homework 8(due March 26 Spring 2009...

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