1). Properties of Beta Function :
(1). B(m,n) = B(n,m)
Substituting x =1 – y in the definition of Beta Function &space;=&space;%5Cint_%7B1%7D%5E%7B0%7D&space;(1-y)%5E%7Bm-1%7D&space;y%5E%7Bn-1%7D(-dy) "https://latex.codecogs.com/svg.image? B (m,n) = int_{1}^{0} (1-y)^{m-1} y^{n-1}(-dy)")
&space;=&space;%5Cint_%7B0%7D%5E%7B1%7Dx%5E%7Bm-1%7D&space;(1-x)%5E%7Bn-1%7Ddx&space;&space;%5D "Euler's First Integrals For Beta Function")
%5E%7Bm-1%7D&space;y%5E%7Bn-1%7Ddy "https://latex.codecogs.com/svg.image? = int_{0}^{1} (1-y)^{m-1} y^{n-1}dy")
B (m,n) = B (n,m)
Substituting x = sin^2 θ in the definition of Beta Function
2). Euler’s First Integrals For Beta Function :
This is for positive values of m and n, following definite integral is called Beta Function and It is denoted by notation B(m,n)
3). Euler’s Second Integrals For Gamma Function :
The following infinite integral n € N is called Gamma Functions and It is denoted by notation Γ(n).