# Characteristic function vs Moment generating function

Mathematics Asked by nessy on November 9, 2020

Normal distribution $$N(mu, sigma^2$$) has the moment generating function $$m_X(t) = exp (mu t+frac{sigma^2t^2}{2})$$ and the characteristic function $$phi_X(t) = exp (i mu t-frac{sigma^2t^2}{2})$$ which looks almost the same. In fact, it satisfies the equation $$m_X(it) = phi_X(t)$$ for all $$tin mathbb{R}$$.

My question : Is there a criterion for a distribution to satisfy $$m_X(it) = phi_X(t)$$ ? I’m especially interested in continuous distributions.

I had a course on measure theory, and I’m new to probability theory. I know that moment generating function can be failed to be defined for all $$tin mathbb{R}$$. I saw some examples of moment generating functions and characteristic functions, and all of them satisfy the equation above.

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