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Integration by substitution

  1. Apr 29, 2014 #1
    integral between -infinity and +infinity of

    e^-iwt / (a - iw) dw



    u = a - iw

    du/dw = -i

    dw = du / -i

    e^-iwt = e^(u+a)t / -iu du


    its a homework question, so just a pointer as to which method to use would be appreciated, it says substitution, but ive tried the above and doesnt get me anywhere, maybe im not choosing the correct substitution?

    thanks for any help
     
  2. jcsd
  3. Apr 29, 2014 #2

    Simon Bridge

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    Have you been instructed to use substitution?
    Have you tried ratonalizing the denominator?
     
  4. Apr 29, 2014 #3

    Ray Vickson

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    Your exp(at) part has the wrong sign in the exponent. You should find that your integral, J, is
    [tex] J \equiv \int \frac{e^{-iwt}}{a-iw} \, dw = i e^{-at} \int \frac{e^{ut}}{u} \, du [/tex]
    This last form reveals that the integral is "non-elementary", because ##\exp(tu)/u## cannot be integrated found as a finite expression involving elementary functions; it involves the so-called exponential integral; see http://en.wikipedia.org/wiki/Exponential_integral .
     
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