What is Beta Function?
The beta function, or the Euler integral of the first kind, is a function defined in the following way
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
The beta function can be expressed by the gamma function
in the following way
where the gamma function is defined for all complex numbers except the non-positive integers, and for any positive integer
For complex number $z$ with a positive real part the gamma function is defined by the formula
So, if $x$ and $y$ are positive integers, the beta function is
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
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
From the recurrence relation and using the symmetry of the beta function, it holds that
Let us calculate $B(3,2)$. If we use the relation of beta function with gamma function, we get
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.