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.
- 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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