Beta Function Calculator

What is the Beta Function?

The Beta function, denoted as B(x,y), is a special function that is closely related to the Gamma function and factorials. It is defined for positive real numbers x and y as:

\[B(x,y) = \int_0^1 t^{x-1}(1-t)^{y-1} dt\]

The Beta function has numerous applications in probability theory, statistics, and various areas of mathematics.

Formula and Its Meaning

The Beta function can also be expressed in terms of the Gamma function:

\[B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\]

Where:

• \(x\) and \(y\) are positive real numbers
• \(\Gamma\) is the Gamma function

This relationship between the Beta and Gamma functions is particularly useful for computational purposes.

Calculation Steps

1. Calculate \(\Gamma(x)\) and \(\Gamma(y)\) separately.
2. Calculate \(\Gamma(x+y)\).
3. Divide the product of \(\Gamma(x)\) and \(\Gamma(y)\) by \(\Gamma(x+y)\).

Example Calculation

Let's calculate B(2,3):

1. \(\Gamma(2) = 1\)
2. \(\Gamma(3) = 2\)
3. \(\Gamma(2+3) = \Gamma(5) = 24\)
4. \[B(2,3) = \frac{\Gamma(2)\Gamma(3)}{\Gamma(2+3)} = \frac{1 \cdot 2}{24} = \frac{1}{12} \approx 0.0833\]

Visual Representation

This graph represents the Beta function B(x,y). The curve shows how the function value changes as x and y vary. The area under this curve between 0 and 1 gives the value of the Beta function.