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Homework Help: Fourier cosine problem

  1. Oct 13, 2008 #1
    1. The problem statement, all variables and given/known data

    given 2 functions f and g related by a cosine transform

    [tex] g( \alpha ) = \int_{0}^{\infty}dx f(x)Cos( \alpha x) [/tex]

    then if the integral

    [tex] \int_{0}^{\infty}dx f(x)exp(cx) [/tex]

    exists for every positive or negative 'c' then should it be equal to

    [tex] \int_{0}^{\infty}dx f(x)exp(cx)= \frac{g(ic)+g(-ic)}{2} [/tex] ??


    2. Relevant equations

    [tex] g( \alpha ) = \int_{0}^{\infty}dx f(x)Cos( \alpha x) [/tex]


    3. The attempt at a solution

    where i have used the Euler identity to express the cosine as a linear combination of complex

    exponentials.
     
  2. jcsd
  3. Oct 13, 2008 #2

    HallsofIvy

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    Science Advisor

    Yes, that should work. Unfortunately, since you chose not to show us what you did, I can't say where you might have made a mistake.
     
  4. Oct 14, 2008 #3
    thanks Hallsoftivy.. i think this would be the result since

    [tex] \int_{0}^{\infty}dx f(x)exp(cx) [/tex] should be real

    then i used Euler's formula so [tex] 2exp(cx)Cos(ax)=exp(iax+cx)+exp(-iax+cx) [/tex]

    then somehow (of course this all is completely nonrigorous) expanding the exponential into a real and complex part, the contribution to the integral would come from

    [tex] Cos(ax+icx) [/tex] and [tex] cos(ax-icx) [/tex] this kernel is precisely the Kernel of a Fourier cosine transform with complex argument.
     
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