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Unformatted text preview: Fourier transforms We can imagine our periodic function having periodicity taken to the limits In this case, the function f ( x ) is not necessarily periodic, but we can still use Fourier transforms (related to Fourier series) Consider the complex Fourier series, periodic with periodicity 2 l f ( x ) = X n = c n e in x l Write same thing in an equivalent form, using n = 1, f ( x ) = l X n = c n e in x l n l Fourier transforms continued Next we take the limit l , and the summation becomes an integral f ( x ) = Z  g ( k ) e ikx dx Here k = n l , dk = n / l , and g ( k ) = c n l / We say that f ( x ) is the inverse Fourier transform of g ( k ) To get g ( k ) given f ( x )( wesay g(k) istheFouriertransformof f(x)), we again start with the version for periodic functions, c n = 1 2 l Z l l f ( x ) e in x l dx Again use g ( k ) = c n l / , k = n / l , and take the limits of integration from to g ( k ) = 1 2 Z  f ( x ) e ikx dx Example: Problem 21 Find the Fourier transform of f ( x ) = e x 2 / 2 2 g ( k ) = 1 2 Z  e x 2 / 2 2 e ikx dx Notice that for the exponent, we can write x 2 / 2 2 ikx = x 2 1 / 2 + i k 2 1 / 2 2 2 k 2 2 Then the integral can be written g ( k ) = 1 2 e 2 k 2 2 Z  e h 1 2 2 ( x + i 2 k ) 2 i dx Change variables to y = x + i 2 k , then dx = dy , and we get g ( k ) = 1 2 e 2 k 2 2 Z  e y 2 2 2 dx = 2 e 2 k 2 2 Fourier sine and cosine transforms Less commonly used are the Fourier sine/cosine transforms f s ( x ) = r 2 Z g s ( k )sin kxdk g s ( k ) = r 2 Z f s ( x )sin kxdx f c ( x ) = r 2 Z g c ( k )cos kxdk g c ( k ) = r 2 Z f c ( x )cos kxdx Here, it is assumed that f s ( x ) and g s ( k ) are odd functions of x and k respectively, and f c ( x ) and g c ( k ) are even functions of x and k respectively Chapter 8: Ordinary differential equations: Introduction and goals By the end of the chapter you should be able to I Solve separable differential equations Note that we will skip Laplace transforms in this chapter, only because of our time limit Chapter 8: Ordinary differential equations: Introduction...
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 Spring '03
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