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Define

$$f(x) := \begin{cases} \displaystyle\frac{1}{\sqrt{2\pi}}\frac{e^{-1/2x}}{x^{3/2}}, & \text{for $x>0$} \\[4ex] 0 & \text{otherwise} \end{cases}$$

Then, how do I prove that $$\displaystyle\int_{-\infty}^{\infty} e^{itx}f(x) dx = e^{-\sqrt{-2it}}$$ for all $t\in \mathbb{R}$?

Since $f$ does not have a moment generating function, we cannot apply analytic continuation technique here to prove the above equality. Hence, I think we must directly prove this; but how?

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  • $\begingroup$ Just a suggestion: You might want to make that first equation larger with "\displaystyle". I had trouble reading the 3/2 exponent. Now, let me see if I can figure this out .... $\endgroup$ Jul 18 '19 at 17:39
  • $\begingroup$ @bob.sacamento How do I adjust the font size..? $\endgroup$
    – user11
    Jul 18 '19 at 17:40
  • $\begingroup$ seems to be aduplicate math.stackexchange.com/questions/2566188/… $\endgroup$
    – Mike Hawk
    Jul 18 '19 at 18:10
  • $\begingroup$ @MikeHawk The answer in that post must be justified. The step using the change of variable is informal, since it is complex-valued. $\endgroup$
    – Rubertos
    Jul 18 '19 at 18:43
  • $\begingroup$ @MikeHawk See for instance this: math.stackexchange.com/q/2949892/44669 $\endgroup$
    – Rubertos
    Jul 18 '19 at 18:45

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