This question is from an assignment sheet which I am solving and I need help in solving following problem.
Show that Gamma Function can be expressed $$\Gamma(z) =\int_{1}^{\infty} e^{-t} t^{z-1} dt + \sum_{n=0}^{\infty} \frac{(-1)^n } { n! (z+n)}$$
Using Definition of Gamma Function , $\Gamma(z) = \int_{1}^{\infty} e^{-t} t^{z-1} dt + \int_{0}^{1} e^{-t} t^{z-1} dt $. but I am not able to co-orelate $\int_{0}^{1} e^{-t} t^{z-1} dt $ with $\sum_{n=0}^{\infty} \frac{(-1)^n } { n! (z+n)}$ .
So, Can you please shed some light on this.
I shall be really thankful.
From the integral definition of the gamma function we have $$ \Gamma(z) =\int_0^\infty e^{-t} t^{z-1}\, \mathrm dt =\int_1^\infty e^{-t} t^{z-1}\, \mathrm dt+\int_0^1 e^{-t} t^{z-1}\, \mathrm dt. $$ All that is left is to find the series expansion for the second integral. Expanding the exponential term gives $$ \int_0^1 e^{-t} t^{z-1}\, \mathrm dt=\int_0^1 \lim_{N\to\infty}\underbrace{\sum_{n=0}^N\frac{(-t)^n}{n!} t^{z-1}}_{f_n(z)}\, \mathrm dt. $$ Now note that on $t\in[0,1]$: $$ |f_n(z)|<\sum_{n=0}^N\frac{t^n}{n!} |t^{z-1}|<e^t |t^{z-1}|\leq e t^{\Re z-1}, $$ which is integrable in $t\in[0,1]$ for all $\Re z>0$. Hence, by argument of dominated convergence $$ \int_0^1 e^{-t} t^{z-1}\, \mathrm dt=\sum_{n=0}^\infty\frac{(-1)^n}{n!}\int_0^1 t^{z+n-1}\, \mathrm dt=\sum_{n=0}^\infty\frac{(-1)^n}{n! (z+n)}, $$ which is the desired result.