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let be the function

$$ e^{-a|x|^{b}} $$

with $ a,b $ positive numbers bigger than zero

then how could i evaluate this 2 integrals ?

$$ \int_{-\infty}^{\infty}dxe^{-a|x|^{b}}e^{cx}$$

here 'c' can be either positive or negative or even pure complex (Fourier transform)

also how i would evaluate the Fourier cosine trasnform

$$ \int_{0}^{\infty}dxe^{-a|x|^{b}}cos(cx)$$

thanks in advance if possible give a hinto of course i know tht i could expand the function in powers of $ |x| $ but if possible i would like a closed answer thanks.

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1 Answer

up vote 1 down vote accepted

If you really interested in finding a closed form formula, you can have it in terms of the Fox $H$-function. However, I am adopting the following convention (see section 3 & 5) of the $H$-function which follows from the Mellin-Barnes integrals.

$$\int_{-\infty}^{\infty}e^{-a|x|^{b}}e^{cx} dx= \frac{1}{ac}H^{1,1}_{1,1} \left[ \frac{c}{a^b} \left| \begin{matrix} ( 1 , \frac{1}{b} ) \\ ( 1 , 1 ) \end{matrix} \right. \right] -\frac{1}{ac}H^{1,1}_{1,1} \left[ \frac{-c}{a^b} \left| \begin{matrix} ( 1 , \frac{1}{b} ) \\ ( 1 , 1 ) \end{matrix} \right. \right] \,.$$

Offcourse there are existence conditions for the above formula. The above formula can be simplified in terms of less general functions depending on $b$. Just try it for some special values of $b$.

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ok thank you D ::D :D :D ,also what would happen if we take the limit $ \epsilon \to 0 $ with $ a= \epsilon $ and $ b= 1+\epsilon $ –  Jose Garcia Oct 31 '12 at 18:12
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