Let $f(z)=\sum_{n=0}^\infty a_nz^n$ be a power series of radius of convergence $R>0.$
Denote, now $F(z)=\sum_{n=0}^\infty \frac{a_n}{n!}z^n$. I proved that the radius of convergence for this power series is $+\infty.$
Now I would like to prove that $f(z)=\int_{0}^\infty F(tz)e^{-t}dt$ for $\vert z\vert First I tried to prove that the integral converges. I didn't succeed, my main problem is that I get always $a_n$ somewhere. Any idea ?
Let us show that the integral converges absolutely, if $|z| $$\int_0^\infty |F(tz)e^{-t}| dt \le \int_0^\infty \sum_{n=0}^\infty \frac{|a_n z^n|}{n!} t^n e^{-t} dt = \sum_{n=0}^\infty \frac{|a_n z^n|}{n!} \int_0^\infty t^n e^{-t} dt.$$ Note that the interchange of summation and integral is justified by Fubini-Tonelli (since the integrand is non-negative). Also please take care to understand that in case of divergence (which happens if $|z|>R$), this calculation still makes perfect sense, by positivity. That is, in case of divergence, all the terms simply equal $\infty$. Now we recognize the Gamma function,
$$\int_0^\infty t^n e^{-t} dt =: \Gamma(n+1) = n!,$$
which can be verified using repeated integration by parts. Thus the sum on the right in the previous display equals $$\sum_{n=0}^\infty |a_n z^n|$$ which converges to a finite value by assumption since $|z| Also you see, that to prove your claimed relation $f(z)=\int_0^\infty F(tz) e^{-t} dt$, you just need to follow this argument without the absolute values.
See that $$ n! =\int_{0}^\infty t^n e^{-t}dt $$ then
\begin{split} f(z)=\sum_{n=0}^\infty \frac{a_n}{n!}z^n n!&=& \sum_{n=0}^\infty \int_{0}^\infty \frac{a_n}{n!}(zt)^ne^{-t} dt\\ &= &\int_{0}^\infty\sum_{n=0}^\infty \frac{a_n}{n!}(zt)^ne^{-t} dt\\ &=&\int_{0}^\infty F(tz)e^{-t}dt \end{split} Since your Domain is $D(0,R)$ we can exchange the sum with integral sign.