# Beta Function Calculator

X
Y
Beta Function β(X,Y)  =  0.0833
CALCULATE

Beta Function calculator calculates the beta function for the given two positive real numbers $x$ and $y$ by applying the beta function formula.
It is necessary to follow the next steps:

1. Enter the values of two parameters in the boxes. These values must be positive real numbers;
2. Press the "GENERATE WORK" button to make the computation;
3. Beta function calculator will find the beta function, given values of the parameters $x$ and $y$.
Input : Two positive real numbers;
Output : A positive real number.

Beta Function Formulas:
Beta Function Formula: The beta function is a defined by the formula
$$B(x,y)=\int_0^1 t^{x-1}(1-t)^{y-1}dt$$
for $Re(x)>0$ and $Re(y)>0$.
Beta and Gamma Relation: For positive integers $x$ and $y$, {the beta function} is a defined in terms of the gama function as
$$B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x y)}$$
Trigonometric Beta Function Formula: The trigonometric representation of the beta function is
$$B(x,y)=2\int_0^{\frac{\pi}{2}} (\sin\theta)^{2x-1}(\cos\theta)^{2y-1}d\theta$$

## What is Beta Function?

The beta function, or the Euler integral of the first kind, is a function defined in the following way

$$B(x,y)=\int_0^1 t^{x-1}(1-t)^{y-1}dt$$
for $Re(x)>0$ and $Re(y)>0$. The beta function is the named by Legendre and Whittaker and Watson (1990) for the Euler integral of the first kind.
For any $x,y$ such that $Re(x)>0$ and $Re(y)>0$, the beta function is symmetric $$B(x,y)=B(y,x)$$ The beta function can be expressed by the gamma function in the following way $$B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$$ where the gamma function is defined for all complex numbers except the non-positive integers, and for any positive integer $$\Gamma(n)=(n-1)!$$ For complex number $z$ with a positive real part the gamma function is defined by the formula
$$\Gamma(z)=\int_0^{+\infty}x^{z-1}e^{-x}dz,\quad Re(z)>0$$
So, if $x$ and $y$ are positive integers, the beta function is
$$B(x,y)=\frac{(x-1)!(y-1)!}{(x+y-1)!}$$
In the picture below is shown the beta function for real positive values
For $Re(x)>0$ and $Re(y)>0$, the trigonometric representation of the beta function is
$$B(x,y)=2\int_0^{\frac{\pi}{2}} (\sin\theta)^{2x-1}(\cos\theta)^{2y-1}d\theta$$
In the picture below is shown the beta function for the absolute value in the complex plane:
The recurrence relation of the beta function is determined by the following formula
$$B(x+1,y)=B(x,y)\frac{x}{x+y}$$
From the recurrence relation and using the symmetry of the beta function, it holds that
$$B(x+1,y)+B(x,y+1)=B(x,y)$$
Let us calculate $B(3,2)$. If we use the relation of beta function with gamma function, we get
$$B(3,2)=\frac{2!1!}{4!}=\frac{1}{12}\approx0.833$$
The Beta Function Calculator work with steps shows the complete step-by-step calculation for finding the beta function of $3$ and $2$ using the beta function formulas. For any other values, just supply two positive real numbers and click on the "GENERATE WORK" button. The grade school students may use this Beta function calculator to generate the work, verify the results of evaluating integrals or do their homework problems efficiently.

### Practice Problems for Beta Function

The beta function is one of the most fundamental special functions, due to its important role in various fields of mathematics, physics, engineering, statistics, etc. The beta function is important in calculus and analysis due to its close connection to the gamma function, which is itself a generalization of the factorial function. Many complex integrals can be reduced to expressions involving the beta function. In statistics and probability, the beta distribution is defined by the density

$$\frac{1}{B(x,y)}t^{x-1}(1-t)^{y-1},$$
where $0 < x < 1$ and $B(x,y)$ is the beta function. The beta function $B(x,y)$ appears as a constant also known as the normalizing constant.
The beta probability distribution is useful in the statistical analysis of trustworthiness, life testing models and in many other applications. Since
$$B(n,n+1)=\frac{1}{n\binom{2n}{n}}$$
the beta function is also used to calculate some binomial coefficients and summations.

Practice Problem 1:
Find $$\frac{B(\frac{5}{2},\frac{7}{2})}{B(4,5)}$$ where $B(x,y)$ is the Beta function.

Practice Problem 2:
Evaluate integral $\int_0^1\sqrt[3]{x^2(1-x^4)}dx$.

The Beta function calculator, work with steps, formula and practice problems would be very useful for grade school students of K-12 education to understand the concept of the beta function. This concept can be of significance in many fields of mathematics, physics, engineering, statistics, etc, especially in evaluating integrals and beta distributions.