1. Notes are from D. R. Wilton, Dept. of ECE
Fall 2016
David R. Jackson
Notes 13
The Gamma Function
Notes are from D. R. Wilton, Dept. of ECE

2. The Gamma Function Definition # 1
The Gamma function appears in many expressions, including asymptotic series.
It generalizes the factorial function n! to non-integer values and even complex values.
Definition # 1
This definition gives the Gamma function a nice property, as shown on the next slide.

3. The Gamma Function
Factorial property:
Hence or

4. The Gamma Function (cont.)
Definition # 2
This is the Euler-integral form of the definition.
Note:
Leonard Euler
Note: Definition 1 is the analytic continuation of definition 2 into the entire complex plane (except at the negative integers).

5. The Gamma Function (cont.)

6. The Gamma Function (cont.)
Equivalence of definitions #1 and #2

7. The Gamma Function (cont.)
Definition # 3
The Weierstrass product form can be shown to be equivalent to definitions #1 and #1.

8. The Gamma Function (cont.)
Euler Reflection Formula
Note: If we can calculate (z) for Re(z) > 0,
we can use this formula to find (z) for Re(z) < 0.
The two points are reflections about the x = 1/2 line.

9. The Gamma Function (cont.)
A special result that occurs frequently is (1/2).
To calculate this, use the reflection formula:
Set z = 1/2:

10. The Gamma Function (cont.)
Pole Behavior
Use

11. The Gamma Function (cont.)
Residues at Poles
Use
Hence
In general:

12. The Gamma Function (cont.)
Note: There are simple poles at z = 0, -1, -2,…

13. The Gamma Function (cont.)
(x) and 1 / (x)
Note: (x) never goes to zero.
In fact, (z) never goes to zero (1 / (z) is analytic everywhere).

14. The Gamma Function (cont.)
Sterling’s formula (asymptotic series for large argument):
Taking the ln of both sides, we also have
Valid for

15. The Gamma Function (cont.)
Summary of Factorial Generalization
Integers
Real numbers
Complex numbers

16. The Gamma Function (cont.)
Summary of Factorial Generalization
Complex numbers
Complex numbers