Intuition of Beta and Gamma functions

Question

What is the intuition behind beta and gamma functions?I know that the gamma fucntion is related to the factorial fucntion but that is not it's definition as it exists for other numbers also which aren't integers. I saw the graph of the gamma function and this got me thinking about this even further

Are these functions simply without any intuition and just integrals which pop up everywhere in physics and math and hence are dissemnated in math textbooks and have been found out using usual integration techniques or is there something behind that curtain?

Answer 1

Suppose you tried to invent an integral representation $n!=\int_0^\infty f_n(x)dx$, so$$0=n!-n\cdot(n-1)!=\int_0^\infty(f_n(x)-nf_{n-1}(x))dx.$$Comparing that integrand to familiar differential equations, one can't help but try the Ansatz $f_n(x)=g_n(x)e^{-x}$ viz.$$0=\int_0^\infty(g_n(x)-ng_{n-1}(x))e^{-x}dx=[ng_{n-1}(x)e^{-x}]_0^\infty$$provided $g_n=ng_{n-1}^\prime$, which is exactly what happens with $g_n=Cx^n$. The case $n=0$ tells you $C=1$.

With$$\Gamma(s):=(s-1)!=\int_0^\infty x^{s-1}e^{-x}dx=2\int_0^\infty x^{2s-1}e^{-x^2}dx$$in hand, one can't help but try leveraging Pythagoras in a 2D Cartesian-to-polar conversion:$$\begin{align}\Gamma(a)\Gamma(b)&=4\int_{[0,\,\infty)^2}x^{2a-1}y^{2b-1}e^{-r^2}dxdy\\&=4\int_0^{\pi/2}d\theta\cos^{2a-1}\theta\sin^{2b-1}\theta\int_0^\infty r^{2(a+b-1)}e^{-r^2}dr\\&=\operatorname{B}(a,\,b)\Gamma(a+b)\end{align}$$with$$\operatorname{B}(a,\,b):=2\int_0^{\pi/2}d\theta\cos^{2a-1}\theta\sin^{2b-1}\theta.$$This new function is clearly something special. Leaning on Pythagoras a bit more, this time in the form $\cos^2\theta+\sin^2\theta=1$, one can't help but try the order-preserving $t=\sin^2\theta$, so$$\operatorname{B}(a,\,b)=\int_0^1t^{a-1}(1-t)^{b-1}dt.$$We can't hope to give a complete account of why these two functions show up so much, but:

• You can think of Gamma distributions as a generalization of exponential distributions, and (with quadratic transformations) of Gaussian ones too. So the usual motivations for these can in certain contexts motivate Gamma distributions. For example, thermodynamics mandates an exponential distribution of particulate energy, which in the ideal gas model means a Gaussian distribution of each of its velocity's Cartesian components, while the speed (the velocity vector's modulus) ends up Gamma-distributed, the exponent of $v$ depending on the dimension of space.
• Many integrands are derived from differential equations, so polynomials multiplied by decaying exponentials are at least approximately valid PDFs for many non-negative continuous variables.
• Where there is combinatorics (which there often is with power series, not to mention many other things), there are factorials; where there are factorials, there are Gamma functions, not to mention binomial coefficients, which can be written in terms of Beta functions.
• Equations such as$$\Gamma(s)\zeta(s)=\sum_{n\ge1}\int_0^\infty x^{s-1}e^{-nx}dx=\int_0^\infty\frac{x^{s-1}dx}{e^x-1}$$only make Gamma functions more common in complex analysis.
• The Beta function is useful because the Gamma function is; we can say much the same of Beta distributions. For example, consider $X/(X+Y)$ for independent Gammas $X,\,Y$.