1 00:00:00,729 –> 00:00:04,571 The mathematical Symbol of today is the so called Gamma function.

2 00:00:04,686 –> 00:00:06,309 Written with a capital Gamma.

3 00:00:07,429 –> 00:00:11,709 and the name of the variable we put in is usually given by z.

4 00:00:12,443 –> 00:00:18,077 Now the explicit definition of the Gamma function is not so simple, because it’s given by an integral.

5 00:00:19,014 –> 00:00:22,408 Indeed we integrate from 0 to infinity.

6 00:00:23,171 –> 00:00:27,033 and the variable for the integration is given by x.

7 00:00:27,800 –> 00:00:32,706 and the number z from the left hand side we find here in the exponent of x.

8 00:00:33,614 –> 00:00:36,663 Because it’s x to the power (z-1).

9 00:00:37,214 –> 00:00:41,794 Moreover this is then multiplied with e to the power (-x).

10 00:00:42,686 –> 00:00:48,471 So you see this is not a simple definition for a function and to make it even more complicated

11 00:00:48,472 –> 00:00:52,357 i can tell you this z could even be a complex number.

12 00:00:53,071 –> 00:00:57,202 However then we need that the real part of z is positive.

13 00:00:58,114 –> 00:01:02,640 Now this means it’s a allowed to put in for example natural numbers.

14 00:01:03,514 –> 00:01:07,713 and indeed there we find a nice property for Gamma of n.

15 00:01:08,529 –> 00:01:13,434 Because one can show this is exactly (n-1)!

16 00:01:13,871 –> 00:01:18,593 Or in other words the Gamma function is a generalisation for the factorial.

17 00:01:19,500 –> 00:01:24,884 and exactly this gets even more apparent when we prove a similar recursive formula.

18 00:01:25,729 –> 00:01:33,627 Which reads like Gamma of (z+1) is equal to z time Gamma of z.

19 00:01:34,786 –> 00:01:40,744 Of course the Gamma function has also other nice properties we can discuss in another video.

20 00:01:41,514 –> 00:01:45,300 However now you already know what the definition of the Gamma function is.

21 00:01:45,886 –> 00:01:49,260 So if this was helpful, then i see you next time.

Q1: What is the value of $\Gamma(1)$?

A1: $1$

A2: $0$

A3: $-1$

A4: $2$

Q2: Is $\Gamma(1+i)$ defined?

A1: Yes!

A2: No!

A3: One needs more information.