I want to prove the inversion theorem on $R^{k}$.Then I need to compute the integral :
$\int_{R^{1}}\exp^{-\sqrt{x^{2}+a^{2}}+itx}dx$
I have no ideal how to deal with it.I will appreciate your help.
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1What is $R^1$?${}$ – Pratyush Sarkar Feb 21 '14 at 03:53
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$a,t$ is real parameter. – gilliatt Feb 21 '14 at 03:55
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What are the integration limits ? $(0,\infty)$ or $(-\infty,\infty)$ ? – Felix Marin Feb 21 '14 at 04:02
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$(-\infty,\infty)$ – gilliatt Feb 21 '14 at 04:08
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Where did this problem come from? – Mhenni Benghorbal Feb 21 '14 at 06:07
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I want to prove the inversion theorem of Fourier Transform on $L^{1}(R^{k})$ so I imitate the proof of inversion theorem on $L^{1}(R^{1})$ .In that proof,it chooses auxiliary functions $\exp^{-|t|}$. – gilliatt Feb 21 '14 at 07:57