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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 π x l • Write same thing in an equivalent form, using Δ n = 1, f ( x ) = l π ∞ X n =∞ c n e in π x l Δ n π 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 π l , dk = Δ n π/ l , and g ( k ) = c n l /π • 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 π x l dx • Again use g ( k ) = c n l /π , k = n π/ l , and take the limits of integration from∞ to ∞ g ( k ) = 1 2 π Z ∞∞ f ( x ) e ikx dx Example: Problem 21 • Find the Fourier transform of f ( x ) = e x 2 / 2 σ 2 g ( k ) = 1 2 π Z ∞∞ e x 2 / 2 σ 2 e ikx dx • Notice that for the exponent, we can write x 2 / 2 σ 2 ikx = x 2 1 / 2 σ + i σ k 2 1 / 2 2 σ 2 k 2 2 • Then the integral can be written g ( k ) = 1 2 π e σ 2 k 2 2 Z ∞∞ e h 1 2 σ 2 ( x + i σ 2 k ) 2 i dx • Change variables to y = x + i σ 2 k , then dx = dy , and we get g ( k ) = 1 2 π e σ 2 k 2 2 Z ∞∞ e y 2 2 σ 2 dx = σ √ 2 π e σ 2 k 2 2 Fourier sine and cosine transforms • Less commonly used are the Fourier sine/cosine transforms f s ( x ) = r 2 π Z ∞ g s ( k )sin kxdk g s ( k ) = r 2 π Z ∞ f s ( x )sin kxdx f c ( x ) = r 2 π Z ∞ g c ( k )cos kxdk g c ( k ) = r 2 π 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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 Differential Equations, Fourier Series, Dirac delta function

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