How do I use the gamma function?

Question

Let's say I want to evaluate

$$\left(\frac{3}{4}\right) !$$

I know the related Gamma function is:

$$\Gamma(s)=\int_0^\infty t^{\frac34-1}e^{-t}\ \text{d}t$$

but I have no clue what $d$ and $t$ mean.

What would I put for $d$ and $t$? Also, let me know if I am using the wrong equation for this problem.

Answer 1

There is no $d$ in your integral/problems, and they use of Gamma function is incorrect.

The Gamma function $\Gamma(s)$ is the simplest function that gives you the factorial of a number when the argument is a Natural number, whilst gives otherwise the "generalisation" of the factorial, id est:

$$\Gamma(s) = \begin{cases} (s-1)! ~~~~~~~ \text{for} ~~~~~ s\in\mathbb{N} \\\\ \int_0^{+\infty} t^{s-1}e^{-t}\ \text{d}t ~~~~~ \text{for} ~~~~~ s\notin \mathbb{N}\end{cases}$$

Under the constraint $\Re(s) > 0$.

What you have, $3/4$, is clearly not a natural number, so you have to use the definition and calculate the integral.

Now be careful with the definition: we can, by analytic continuation, extend some great properties of the Gamma function to real numbers. In this way we use one of its biggest properties that reads

$$\Gamma(s) = (s-1)!$$

Adapted to our case: you want the "factorial" of $3/4$. Good, then you have to write down this:

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

hence

$$\left(\frac{3}{4}\right)\large ! = \Gamma\left(\frac{3}{4} + 1\right) = \Gamma\left(\frac{7}{4}\right)$$

Now just calculate the integral:

$$\Gamma\left(\frac{7}{4}\right) = \int_0^{+\infty} t^{7/4 - 1}e^{-t}\ \text{d}t$$

Tables for the Gamma function are everywhere. You will find out eventually

$$\left(\frac{3}{4}\right)\large ! = \Gamma\left(\frac{7}{4}\right) \approx 0.919603(...)$$

Study this: https://en.wikipedia.org/wiki/Gamma_function

Answer 2

It would seem you are in over your head with this question. In order to even understand what it is you are working with, you will need some calculus under your belt. This can be learned from various places online, such as on Khan Academy.

As for an alternative form of the Gamma function which does not require as much calculus to understand, we also have:

$$\Gamma(x-1)=x!\simeq\frac{\left(\frac x2+n\right)^x\times n!}{(x+1)\times (x+2)\times \dots\times (x+n)}$$

where the approximation becomes increasingly accurate for larger values of $n$.

See here for a graph of this for $n=5$, and here for a graph that shows the progression from $n=0$ to $n=25$.

For your example:

$$\left(\frac34\right)!\simeq\frac{\left(\frac38+5\right)^{3/8}\times 5!}{\left(\frac34+1\right)\times \left(\frac34+2\right)\times \left(\frac34+3\right)\times \left(\frac34+4\right)\times \left(\frac34+5\right)}\simeq0.859412434303$$