# How do I compute this integration? (the characteristic function of Levy distribution)

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?

• 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 .... Jul 18 '19 at 17:39
• @bob.sacamento How do I adjust the font size..? Jul 18 '19 at 17:40
• seems to be aduplicate math.stackexchange.com/questions/2566188/… Jul 18 '19 at 18:10
• @MikeHawk The answer in that post must be justified. The step using the change of variable is informal, since it is complex-valued. Jul 18 '19 at 18:43
• @MikeHawk See for instance this: math.stackexchange.com/q/2949892/44669 Jul 18 '19 at 18:45