Recommend me a text or webpage introducting gamma function throughly

Question

Till now, i have learned abstract Integration, all basic properties of the (n-dimensional) Lebesgue(-Stieltjes) measure and the lebesgue integral is an extension of Riemann integral.

Here's an illustration i'm curious about:

Let $\Gamma$ be a complex-valued gamma function.

Often, it's written in texts $\Gamma(z)=\int_{0}^\infty t^{z-1}e^{-t} dt$. ($Re(z)>0$)

If i understand this integral as the Lebesgue-integral, it makes sense to me. However if this integral is the Riemann-integral, i wonder which limit should be taken first. That is, is it taking limit first to $0$ and then to $\infty$? Or first to $\infty$ then to $0$? Or an order of taking limit doesn't matter? What are relations between the Lebesgue integral and Riemann improper integral?

Please suggest me a text which could answer these questions. However, i don't want to spend very much time on this. I hope i can learn these topics within 2 or 3 days. Thank you :)

Answer 1

Perhaps "The Gamma Function" by Emil Artin, very short, and in pdf format, ( just google it.)

Otherwise just about any book on the theory of functions such as Ahlfors' Complex Analysis. I'm guessing this list is fairly long.

Answer 2

I learned all I know (well maybe not all, but quite a bit) about the gamma function from Frank Jones' Lebesgue Integration on Euclidean Spaces. Chapter 11 I want to say. I feel that the text introduces the concepts in a relatively straight forward manner accessible by an upper division undergrad, beginning grad student.

Jones also addresses topics on Lebesgue versus Riemann integration.

Answer 3

I don't know a good reference for the gamma function (maybe Abramowitz and Stegun?) but I can answer your integration question. In general there are improper Riemann integrals that are not Lebesgue integrals, and vice versa. (Wikipedia has an example of an improper Riemann integral that is not Lesbegue.

In this particular case, where the integrand is positive, they coincide. You can prove that via the monotone convergence theorem.