A lot of important functions in applied sciences are defined using improper integrals, and perhaps the most famous of these is the **Gamma Function**. Throughout my course I have heard mention of this function, and always wondered “*but what exactly is the Gamma Function?*” I finally decided to find out.

Mathematicians wanted to find a formula that generalised the factorial expression for the natural numbers. This lead to the discovery of following very well-known formula:

Say we replace the *n* but the *x* to create the function:

What is the domain of this function? The only possible ‘*bad points*‘ are 0 and ∞:

**t ≈ 0**: e^-t ≈ 1 and hence the expression inside the integral is approximately t^x, meaning we have convergence at around 0, only if x>-1.**t –> ∞:**No matter the value of x, the integral is convergent.

From this we can see that the domain of f(c) is (-1,∞).

To obtain (0,∞) as the domain, we need to shift the function one unit to the right. This defines the **Gamma Function**.

The above discussion explains why the Gamma Function has the x-1 power.

### Some Properties:

**Γ(n) = (n-1)!**where*n*is a natural number**Γ(x+1) = xΓ(x)**for x > 0. This follows easily from the definition of Γ(x) by performing integration by parts.- Weierstrass’s definition is also valid for all complex numbers (except non-positive integers):where is the Euler-Mascheroni Constant.
- Taking the logarithm of the above expression (we can do this as Γ(x) > 0 for x > 0), we get that:

- Differentiating this, we observe that:In fact, Γ(x) can be differentiated infinitely often.
- Γ(x) is connected with sin(x): For x > 0

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