$F(x)= \int_0^{\infty} t^{x^2-1} e^{-2t} \ dt$ studying for a measure theory exam I found the following exercise related to certain parametric integral:
Given $F(x)= \int_0^{\infty} t^{x^2-1} e^{-2t} dt$, study the points of $\mathbb{R}$ in which F is well defined. Can it be expressed as a power series centered at $x=0$? Finally, prove that F is analytic in it's whole domain.
I'm quite lost at answering this questions. As $F(x)$ is certainly similar to the Gamma Function, I tried to find a way of expressing $F(x)$ in terms of the Gamma Function and, from it, find quicker the results:
$$ \int_0^{\infty} t^{x^2-1} e^{-2t} dt = \int_0^{\infty} \frac{(2t)^{x^2-1}}{2^{x^2-1}} e^{-2t} dt = \frac{1}{2^{x^2-1}} \int_0^{\infty} (2t)^{x^2-1} e^{-2t} dt =$$
$$= \frac{1}{2^{x^2}} \int_0^{\infty} u^{x^2-1} e^{-u} du = \frac{1}{2^{x^2}} \Gamma (x^2)$$
From where I deduce that F(x) is defined in every point except 0. It's the reasoning correct. How will I proceed for the next questions? Any hint or solution would be appreciated.
 A: I agree that $F(x) = \frac 1 {2^{x^2}} \Gamma(x^2)$ for $x\neq 0$ (note you have a small typo in your final equality).

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*You can calculate the domain of $F$ is $(0,\infty)$ without the Gamma function. Indeed, $$ t \mapsto t^{x^2-1} e^{-2t}$$ is integrable at $\infty$ since exponentials decay much quicker than polynomials grow. Moreover, $ t^{x^2-1} e^{-2t} \sim t^{x^2-1} $ as $t\to 0^+$ which is integrable if and only if $x^2-1>-1$ i.e. $x\neq 0 $.


*$F$ cannot be represented as a power series centred at zero. Indeed, by Fatou's lemma $$ \liminf_{x\to 0^+} F(x) \geqslant \int_0^\infty \liminf_{x\to 0^+}  t^{x^2-1} e^{-2t} \,dt=\int_0^\infty   t^{-1} e^{-2t} \,dt =+\infty. $$ Hence, $F$ does not have a removable singularity at zero, so it cannot be represented as a power series centered at $x=0$.


*For the final question, in its domain, $F(x) = \frac 1 {2^{x^2}} \Gamma(x^2)$ as you established. Since the gamma function is analytic except at $x=0,-1,-2,\dots$, it follows that $F$ is analytic in its domain.
