I have been reading a lot of high mathematics lately and i am getting a bit confused...
so here goes !

1) What is the 't' in the definition of the gamma function stand for ?
2) What precisely is the relationship between the zeta and the gamma functions ?
3) With respect to the Gamma function, f(x) = x.f(x-1)
how is this proved ?
4)Because of 3) ,the gamma function is used to compute factorials of non intgral and negative numbers. HOW ? (to some extent i know this,but am not totally clear)
5) I have this major doubt...in the definition of the zeta function, how are 1^s , 2^s ,etc defined when s is complex ? Does it mean that 1 is multiplied by itself an imaginary number of times?

Please answer these questions...
and please remember that I do NOT KNOW much calculus.
Also,please provide one to one answers,instead of suggesting more material - which is what,to some extent,confused me.
THANKS
cheers
Niranjan

Niranjan,
About the Gamma function:
here is a little history:
Consider the following integral:

Where $n$ is a certain integer.

and $x$ is the variable (your $t$)

When we try to integrate it we get the most interesting result.

We integrate it using the following integral formula:

$\int u\text{'}v\mathrm{dx}=uv-\int uv\text{'}\mathrm{dx}$

Where $u$ and $v$ are two functions, $u\text{'}$ is the derivative of $u$, and $v\text{'}$ is the derivative of $v$.

Let's choose:

$u\text{'}=e^{-x}$

so, $u=-e^{-x}$

$v=x^n$

so, $v\text{'}=n{x}^{n-1}$

So we get:

${\int }_0^{\infty }{x}^n{e}^{-x}\mathrm{dx}=[-x^n{e}^{-x}{]}_0^{\infty }+n{\int }_0^{\infty }{x}^{n-1}{e}^{-x}\mathrm{dx}$

as $x$ tends to $\infty$ $-x^n{e}^{-x}$ tends to 0.

Let's define $I_n={\int }_0^{\infty }{x}^n{e}^{-x}\mathrm{dx}$

We get $I_n=n{I}_{n-1}$

$I_1=1I_0$

$I_2=2I_1=2\times 1I_0$

$I_3=3I_2=3\times 2\times 1I_0$

Or we can generalize: $I_n=n!I_0$

Now we only have to answer what is $I_0$

$I_0={\int }_0^{\infty }{x}^0{e}^{-x}\mathrm{dx}=1$

So our integral is really equal to $n!$

$I_n=n!$

Now, the gamma function is defined as the following:

$\Gamma (n)=I_{n-1}$

$\Gamma (n+1)=n!$

$\Gamma (n+1)=n\Gamma (n)$

An interesting result is:

$\Gamma (1/2)=\sqrt{\pi }$
Hope it helped.
If you don't understand it, don't hesitate to ask.

Yatir

I think Yatir has answered everything quite appropriately

However,for the last question you have posed about zeta function...

check this site
as you have mentioned you need no extra material,you may just read the first para which quite gives you the answer...

love arun

To answer your questions (i hope i can ):


1) What is the 't' in the definition of the gamma function stand for ?

I assume you mean the definition as:

I would kinda like to solve the Riemann hypothesis....would do wonders for the start of one's maths career

And earn you a million dollars in the process!

I have yet to meet a mathematician who wouldn't want to solve it :-)