Processing math: 100%
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

10.3: Gamma Function

( \newcommand{\kernel}{\mathrm{null}\,}\)

For ν not an integer the recursion relation for the Bessel function generates something very similar to factorials. These quantities are most easily expressed in something called a Gamma-function, defined as

Γ(ν)=0ettν1dt,ν>0.

Some special properties of Γ function now follow immediately:

Γ(1)=0etdt=e1|0=1e=1Γ(ν)=0ettν1dt=0detdttν1dt=ettν1|0+(ν1)0ettν2dt

The first term is zero, and we obtain Γ(ν)=(ν1)Γ(ν1)

From this we conclude that

Γ(2)=11=1,Γ(3)=211=2,Γ(4)=3211=2,Γ(n)=(n1)!.

Thus for integer argument the Γ function is nothing but a factorial, but it also defined for other arguments. This is the sense in which Γ generalises the factorial to non-integer arguments. One should realize that once one knows the Γ function between the values of its argument of, say, 1 and 2, one can evaluate any value of the Γ function through recursion. Given that Γ(1.65)=0.9001168163 we find

Γ(3.65)=2.65×1.65×0.9001168163=3.935760779.

Exercise 10.3.1

Evaluate Γ(3), Γ(11), Γ(2.65).

Answer

2!=2, 10!=3628800, 1.65×0.9001168163=1.485192746.

We also would like to determine the Γ function for ν<1. One can invert the recursion relation to read Γ(ν1)=Γ(ν)ν1,

Γ(0.7)=Γ(1.7)/0.7=0.909/0.7=1.30.

What is Γ(ν) for ν<0? Let us repeat the recursion derived above and find Γ(1.3)=Γ(0.3)1.3=Γ(0.7)1.3×0.3=Γ(1.7)0.7×0.3×1.3=3.33.

This works for any value of the argument that is not an integer. If the argument is integer we get into problems. Look at Γ(0). For small positive ϵ Γ(±ϵ)=Γ(1±ϵ)±ϵ=±1ϵ±.
Thus Γ(n) is not defined for n0. This can be easily seen in the graph of the Γ function, Fig. 10.3.1.

f12.png

Figure 10.3.1: A graph of the Γ function (solid line). The inverse 1/Γ is also included (dashed line). Note that this last function is not discontinuous.

Finally, in physical problems one often uses Γ(1/2), Γ(12)=0ett1/2dt=20etdt1/2=20ex2dx.

This can be evaluated by a very smart trick, we first evaluate Γ(1/2)2 using polar coordinates Γ(12)2=40ex2dx0ey2dy=40π/20eρ2ρdρdϕ=π.
(See the discussion of polar coordinates in Sec. 7.1.) We thus find Γ(1/2)=π,Γ(3/2)=12π,etc.


This page titled 10.3: Gamma Function is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?