Gamma Function Specifics and properties

Question

I'm a bit confused on a few aspects of the gamma function. I've listed my (7) main questions below (they're all short questions):

1) I read it is a "generalized factorial for arbitrary arguments", but where does the $\sqrt{\frac {\pi} 2}$ term come from? What is the relationship $\pi$ has with the gamma function?

2) What is happening when you use a gamma function? I see things like $\int{t^{z-1}e^{-t}dt}$ and $\int\bigg[{ln\big(\frac 1 t \big)^{z-1}\bigg]}dt$ whenever I try to research the "function", but neither seem to be integrals with actual closed form solutions.

3) If the gamma function is just made to extend to complex numbers, then is $z-1$ only referring to the decreasing real part of $z$, or, if $z=a-bi$, could you get away with either using $\big((a-bi)(a+bi)-1\big)$ or the magnitude $\sqrt{a^2+b^2}-1$ instead of $z-1$?

4) Does the gamma function just become a regular factorial when used with regular integers? If so, do the integrals above only come into play with imaginary numbers, or are they also necessary with real, non-integer numbers as well?

5) Speaking of the integrals above, if $z$ is the number being evaluated in the gamma function, what is $t$? Is it an integer like $n$ or $i$ in sumations and series? Or is it dependant on whatever type of number $z$ is?

6) What is meant by lower and upper gamma function? Are they the value of a gamma function as $t$ goes from negative infinity to 0 and 0 to infinity respectively?

7) Lastly, if this function can be applied to imaginary numbers, does that necessarily mean it can also be applied to quaternions and octonions (and every other number system) as well?

I want the answer to all these questions, but If I can get the answer to just a couple (preferably from the first 5 as 6 doesn't really matter as much and 7 is just out of curiosity) that would help. I really want to understand this function better as it seems to keep popping up every time I try to research, well, anything in depth.

Answer 1

I cannot really talk about 7) (and maybe the link provided by J.G. is of help for that) and from what I can tell 4) does not make that much sense.

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1) I read it is a "generalized factorial for arbitrary arguments", but where does the $\sqrt\frac\pi2$ term come from? What is the relationship $\pi$ has with the gamma function?

First off, do not take $\pi$ popping up somewhere too seriously. Of course, you can draw a connection to circles most of the time but often these connections appear kind of contrived (there are cases where the relation is beautiful, see e.g. $3$Blue$1$Browns video on the Basel Problem).
For the Gamma Function, this relation is 'simple' insofer that we have the following (note that you are missing a precise defintion of the Gamma Function, but I will come to that back later) $$\Gamma\left(\frac12\right)=\int_0^\infty t^{-1/2}e^{-t}{\rm d}t=\sqrt\pi\overset{\sqrt t\mapsto t}\iff2\int_0^\infty e^{-t^2}{\rm d}t=\int_{-\infty}^\infty e^{-t^2}{\rm d}t=\sqrt\pi$$ The integral appearing on the right is the so-called Gaussian Integral, which gives rise to another well-known special function: the Error Function.

2) What is happening when you use a gamma function? I see things like $\int t^{z-1}e^{-t}dt$ and $\int[ln(1/t)^{z−1}]dt$ whenever I try to research the "function", but neither seem to be integrals with actual closed form solutions.

An important detail you are missing: the Gamma Function is the closed-form of these integrals, or rather is defined to be. There are integrals of elementary functions (that are polynomials, exponentials, etc.) whose anti-derivatives are non-elementary, the most (in)famous example being $f(x)=e^{-x^2}$ (which, again, leads to the Error Function). Another example is the function $f_z(t)=t^{z-1}e^{-t}$. There is no finite combination of elementary functions which gives you a function $F_z(t)$ such that $F_z'(t)=f(t)$. So, we define $$\Gamma(z):=\int_0^\infty t^{z-1}e^{-t}{\rm d}t\tag1$$ Now we have an expression which appears like a closed-form but really is just an integral in disguise. However, we derive properties like $\Gamma(n)=(n-1)!$ for $n\in\Bbb N$ or $\Gamma(1/2)=\sqrt\pi$ and use them whenever we can break down problem to an integral of the form $(1)$ without doing the calculations again. The Gamma Function (and other special functions) help us to shorten things up and presenting them more concisely.

4) Does the gamma function just become a regular factorial when used with regular integers? If so, do the integrals above only come into play with imaginary numbers, or are they also necessary with real, non-integer numbers as well?

Yes. For $n\in\Bbb N$ we have $\Gamma(n)=(n-1)!$ which can be proven from $(1)$ by induction. That the Gamma Function is shifted by $1$ has historical reasons. There is also a function, called the $\Pi$-function for which $\Pi(n)=n!$; but this is just $\Pi(z)=\Gamma(z+1)$ and the Gamma Function has, IMO, more desirable properties when related to other special functions.
Also $\Gamma(1/2)=\sqrt\pi$ but there are no known alternative forms for $\Gamma(1/3)$ or $\Gamma(1/4)$ (they are in fact transcendental aswell) and these are just as accepted as closed-form as $\pi$ and $e$. The Gamma Function is already a useful tool for real arguments, and not because it extends the factorial.

5) Speaking of the integrals above, if $z$ is the number being evaluated in the gamma function, what is $t$? Is it an integer like $n$ or $i$ in sumations and series? Or is it dependant on whatever type of number $z$ is?

As I mentioned earlier, you are not using a precise definition of the Gamma Function, from which it is clear that confusion arised. Take $(1)$ as the defintion. Clearly, $t^{z-1}e^{-t}$ is a function of $t$ and $z$. We integrate w.r.t. $t$, and in fact we have a definite integral. So the result will not depend on $t$ any longer, but still on $z$. $t$ is only used as integration variable.

6) What is meant by lower and upper gamma function? Are they the value of a gamma function as t goes from negative infinity to 0 and 0 to infinity respectively?

As you can see from the definition I gave: the integral is defined (or improper if you want) with lower bound $0$ and upper bound $\infty$. The so-called Incomplete Gamma Functions are defined by adjusting the lower and upper bound respectively, that is \begin{align*} \Gamma(z,a)&:=\int_a^\infty t^{z-1}e^{-t}{\rm d}t\\ \gamma(z,a)&:=\int_0^a t^{z-1}e^{-t}{\rm dt} \end{align*} There are used when you can get your integrand to the form $t^{z-1}e^{-t}$ (e.g. via suitable substitutions) but fail to obtain the interval $[0;\infty)$ as domain of integration. Again, they are defined for the purpose of expressing non-elementary integrals and for particular choices of $a$ this is just the Gamma Function.

Answer 2

You shouldn't ask so many questions at once, but I'll ignore that this once:

1. $\Gamma(\tfrac12)=\sqrt{\pi}$.
2. The Gamma function is special; it is the closed form. You won't in general do better than that.
3. No, the exponent really is the complex number $z-1$.
4. You can prove $\Gamma(n)=(n-1)!$ for all integers $n\ge1$.
5. It's the dummy integration variable (well, Wikipedia would call it bound).
6. Both are defined here; you just adjust the integration range.
7. Much of your confusion comes from not understanding the point of the usual definition of the Gamma function as an integral. There are other definitions of $\Gamma(z)$, and they're all equivalent for complex $z$. I'm not sure if that remains true e.g. for quaternions, because they don't commute, and that raises a number of complications. So which definition should be used for the generalization? I'll leave that to an expert on this.