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Proving inverse Fourier transform of 1/(1+x^2) = 1/(1+x^2)

  1. Apr 11, 2017 #1
    1. The problem statement, all variables and given/known data
    F(t) = sqrt(π/2)e-t for t>0
    F(t) = sqrt(π/2)et for t<0
    In other words the question asks to solve this integral: 1/sqrt(2π) ∫F(t)eitxdt and show that it equals 1/(1+x2)
    2. Relevant equations
    F(t) = sqrt(π/2)e-t for t>0
    F(t) = sqrt(π/2)et for t<0
    1/sqrt(2π) ∫F(t)eitxdt
    3. The attempt at a solution
    I want to use complex numbers but these functions are analytic and so I can not do this using residues. A naive approach to integrating with respect to t is straightforward but the limits cause the result to be infinity. How should I approach this properly?
     
  2. jcsd
  3. Apr 11, 2017 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    Break up the real line into ##-(\infty, 0)## and ##(0,\infty)##, then do two separate integrals. Each integral is really easy.
     
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