Help me fast......I will give Upvote....... Compute the complex exponenti al foomOf the Fourier series for f(x) =e[-^, ^]on

Answered: Compute the complex exponenti al form…

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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Help me fast......I will give Upvote.......

Compute the complex exponenti al foom
Of the Fourier series for f(x) =e
[-^, ^]
on

Expert Solution

Step 1

Given function,

$f (x) = e^{x}$

Find the Fourier series of $f (x)$ in the interval $[- π, π]$ .

Step 2

Fourier series representation of $f (x)$ in the interval $[- π, π]$ is given by,

$f (x) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} (a_{n} \cos (n x) + b_{n} \sin (n x)) where a_{0} = \frac{1}{π} \int_{- π}^{π} f (x) d x a_{n} = \frac{1}{π} \int_{- π}^{π} f (x) \cos (n x) d x b_{n} = \frac{1}{π} \int_{- π}^{π} f (x) \sin (n x) d x$

Step 3

Now

$a_{0} = \frac{1}{π} \int_{- π}^{π} e^{x} d x = \frac{1}{π} {[e^{x}]}_{- π}^{π} = \frac{1}{π} \times (e^{π} - e^{- π}) = \frac{e^{π} - e^{- π}}{π}$

$a_{n} = \frac{1}{π} \int_{- π}^{π} e^{x} \cos (n x) d x = \frac{1}{π} {[\frac{e^{x} (n \sin (n x) + \cos (n x))}{n^{2} + 1}]}_{- π}^{π} = \frac{1}{π} [\frac{2 \{\cos (n π) \sinh (π) + n \sin (n π) \cosh (π)\}}{n^{2} + 1}] = \frac{2 \{\cos (n π) \sinh (π) + n \sin (n π) \cosh (π)\}}{(n^{2} + 1) π}$

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