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Homework_5_Spring_2010

Homework_5_Spring_2010 - 6 4 1 3 1 2 1 1 1 2 2 2 2 2 5 Show...

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1 MAE 105: Introduction to Mathematical Physics Homework 5 Posted: May 4, 2010 , DUE: May 14, 2010 (Friday, 1 pm), 1. Write down the Fourier series representation for the following function, f(x) defined as follows: x x x x f 2 , 0 2 0 , 1 0 , 0 ) ( Is this an even or an odd function? Why? 2. Write down the Fourier series representation for the following function, f(x) defined as follows: x x x x f 0 , sin 0 , 0 ) ( (a) Why does the above function have both sine and cosine terms in the Fourier series? (b) Describe how you could represent the function f(x) = sin ( x ) , 0 < x < only with cosine terms? 3. Expand f(x) = x 2 , 0 < x < 2 in a Fourier series, if (a) the period is 2 Additionally, what is the value of f(x) at x= 0 and x = 2 ? (b) the period is not specified. 4. Using the results of Problem 3, prove that
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Unformatted text preview: 6 ... 4 1 3 1 2 1 1 1 2 2 2 2 2 5. Show, graphically, that a Fourier series can only be differentiated if f ( x ) is continuous and f (0) = f ( L ) = 0. Consequently, verify that the temperature distribution in a one-dimensional rod (that we previously derived in class) could be expanded in terms of a Fourier sine series, if and only if both ends of the rod have a temperature of zero. 6. Show (using the de Moivre relations) that an alternate representation of the function, f(x) expressed as a Fourier series over an interval (-L, L ), i.e., f(x) ~ ) sin( ) cos( 1 1 L x n b L x n a a n n n n o is: f(x) ~ ) ( L x n i n n e c where ) ( 2 1 n n n b i + a c...
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