I have a question in Tom M Apostol ( Mathematical Analysis) Example on Page 278.
I beg for every one's pardon, by mistake a different image was added. I am really sorry for wastage of other people's time.
It's image is ->
I understood the example completely, but I want to ask: Is the functional equation derived valid only when $a$ is small positive and $b$ tends to $\infty$?
I think I have seen the function equation for any $a, b$ satisfying $0<a< b$.
The functional equation, $\Gamma(y+1)=y\Gamma(y)$, involves integrals from $0$ to $\infty$ on both sides of the equation; there are no $a$ and $b$. (Indeed, the Gamma function has other definitions as well, not involving integrals, and this functional equation of course applies to equivalent definitions as well.)
If you wanted to consider an incomplete Gamma function with endpoints at $a>0$ and $b<\infty$, that is a perfectly valid function, and its value at $y+1$ is related to its value at $y$, by exactly the equation in the OP. Notice that they are not exactly equal though, so I would hesitate to call this a "functional equation" for this incomplete Gamma function.
Basically, the introduction of $a$ and $b$ is a tool for the proof of the functional equation for the (complete) Gamma function, used because the definition of $\Gamma(y)$ is a doubly improper integral. The numbers $a$ and $b$ are not part of the definition of $\Gamma(y)$.