Beta Function Formula
The formula for beta function is given below:
\( \beta\left(x,\ y\right)=\int_0^1t^{x-1}\left(1-x\right)^{y-1}dt \), where x, y>0
We can also use the below-mentioned formula to calculate the beta function.
\( \beta\left(x,\ y\right)=\frac{\left(x-1\right)!\left(y-1\right)!}{\left(x+y-1\right)!} \), where x!=x.(x-1).(x-2)… 3.2.1
Properties of Beta Function
The properties of the beta function are given below:
Let us consider a beta function \( \beta\left(x,\ y\right) \) then we get
- It is symmetric function, therefore \( \beta\left(x,\ y\right)=\beta\left(y,\ x\right) \)
- \( \beta\left(x,\ y\right)=\beta\left(x,\ y+1\right)+\beta\left(x+1,\ y\right) \)
- \( \beta\left(x,\ y+1\right)=\beta\left(x,\ y\right).\left[\frac{y}{\left(x+y\right)}\right] \)
- \( \beta\left(x+1,\ y\right)=\beta\left(x,\ y\right).\left[\frac{x}{\left(x+y\right)}\right] \)
- \( \beta\left(x,\ y\right).\beta\left(x+y,1-y\right)=\frac{\pi}{x\sin\left(\pi y\right)} \)
There are some important integrals regarding beta functions that are given below:
- \(\beta\left(x,\ y\right)=\int_0^{\infty}\frac{t^{x-1}}{\left(1+t\right)^{x+y}}dt\)
- \(\beta\left(x,\ y\right)=2\int_0^{\frac{\pi}{2}}\left(\sin^{2x-1}\theta\right)\left(\cos^{2y-1}\theta\right)d\theta \)
Incomplete Beta Function
In beta function, the generalized form is termed as the incomplete beta function. This can be expressed as:
B(z:a,b) = \(\int _0^zt^{a-1}\left(1-t\right)^{b-1}dt\)
This can also be denoted as \(B_z(a,b)\). Here when z becomes equal to 1, this incomplete beta function becomes the beta function, i.e. B(1:a,b) = B(a,b).
This incomplete beta function can be used in many areas like physics, integral calculus, functional analysis, etc.
Relation between Beta and Gamma Function
In generalization of factorial properties of function, beta function plays a major role with the association of gamma function. They help in solving calculus equations better, as they generalize many complex integral functions to normal form.
The relation between beta and gamma functions can be written as mentioned below:
\( \begin{array}{l}\beta(x,\ y)=\frac{\Gamma_x.\Gamma_y}{\Gamma(x+y)}\end{array} \), where \( \Gamma \) is the symbolic representation of gamma function which is given by
\( \Gamma\left(x\right)=\int_0^{\infty}t^{x-1}e^{-t}dt \)
Applications of Beta Function
The beta function has its application in both physics and mathematics.
- It is used in quantum hydrodynamics and string theory to compute and represent the scattering amplitude for Regge trajectories. The Beta function appeared in elementary particle physics as a model for the scattering amplitude in the so-called “dual resonance model”.
- The beta distribution is used to model things with outcomes in a limited range of 0 to 1 as the function is restricted by an interval with a minimum (0) and maximum (1) value. Thus, beta distribution is well suited to project/planning control systems like PERT and CPM.
- The united beta-gamma function can simplify difficult integral calculus functions into simple functions.
