Beta Function

The Beta function is a unique function and is also called the first kind of Euler’s integrals. The beta function is defined in the domains of real numbers. The notation to represent it is “β”. The beta function is denoted by β(p, q), Where the parameters p and q should be real numbers. It explains the association between the set of inputs and the outputs. Each input value the beta function is strongly associated with one output value. The beta function plays a major role in many mathematical operations.

Beta function is defined by-

\beta(p, q) = \int_{0}^{1}x^{p-1}(1-x)^{q-1}dx

where p>0 and q>0

Standard Forms and Results:

• Symmetry : \beta(p, q) = \beta(q, p)
• Beta Function and Substitution : By substituting x=1−y, we get: \beta(p, q) = \int_{0}^{1} x^{p-1}(1-x)^{q-1} \, dx = \int_{0}^{1} y^{q-1}(1-y)^{p-1} \, dy = \beta(q, p)
• Trigonometric Representation : \beta(p, q) = 2\int_{0}^{\pi/2}\sin^{2p-1}x.\cos^{2q-1}xdx
• Beta function expressed as improper integral : \beta(p, q) = \int_{0}^{\infty} \frac{y^{p-1}}{(1+y)^{p+q}}dy = \int_{0}^{\infty} \frac{y^{q-1}}{(1+y)^{p+q}}dy
• Relation between beta and gamma functions : \beta(p, q) = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)} . Here , \Gamma(n) = \int_{0}^{\infty} x^{n-1}e^{-x} \, dx is the Gamma Function.
  

Some different Properties of Beta Function :

1. Recurrence Relations : The Beta function satisfies the following recurrence relations:

\beta(p, q + 1) = \beta(p, q) \cdot \frac{q}{p + q} \\\beta(p + 1, q) = \beta(p, q) \cdot \frac{p}{p + q} \\\beta(p + 1, q) + \beta(p, q + 1) = \beta(p, q)

2. Functional Equation: This property shows how Beta functions with different arguments relate to each other :

\beta(p, q) \cdot \beta(p + q, r) = \beta(q, r) \cdot \beta(p, q + r)

3. Convexity Property: The Beta function is log-convex, meaning ln⁡ β(p,q) is a convex function in both variables p and q.

4. Asymptotic Behavior: For large values of the parameters, the Beta function exhibits specific asymptotic behavior:

\beta(p, q) \sim \frac{\Gamma(q)}{p^q} \quad \text{as} \quad p \to \infty

These properties make the Beta function extremely versatile in advanced mathematical analysis, probability theory, and special function theory.

Relationship to Gamma Function :

The beta function is closely related to the gamma function. For p,q>0 ,

\beta(p, q) = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}

And for integer values,

Γ(n)=(n−1)!

So, for positive integers m,n :

\beta(m, n) = \frac{(m - 1)!(n - 1)!}{(m + n - 1)!}

Applications of Beta Function :

The beta function plays an important role in:

• Probability and Statistics: Defines probability density functions such as the beta and Dirichlet distributions.
• Bayesian Inference: Used as conjugate priors for Bernoulli/binomial distributions.
• Physics and Engineering: Appears in formulas related to scattering and angular distributions.
• Pattern Recognition, Finance, Risk Management: Models diverse phenomena including portfolio allocation, asset returns, pattern frequencies in recognition tasks.

Solved Examples :

Example 1 : Evaluate 

\beta(5,3).

Solution :

Using the relationship between the Beta and Gamma functions:

\beta(5, 3) = \frac{\Gamma(5)\Gamma(3)}{\Gamma(5+3)} = \frac{\Gamma(5)\Gamma(3)}{\Gamma(8)}

We know:

• Γ(n)=(n−1)! for positive integers,

So,

• Γ(5) = 4! = 24
• Γ(3) = 2! = 2
• Γ(8) = 7! = 5040

Substitute these:

\beta(5, 3) = \frac{24 \times 2}{5040} = \frac{48}{5040} = \frac{1}{105}

Final Answer:

\beta(5, 3) = \frac{1}{105}

Example-2 : Evaluate

\int_{0}^{\pi/2} \sin^{10} \theta \, d\theta.

Solution :

This can be written using result for even positive integer :

\int_{0}^{\pi/2} \sin^{10} \theta \, d\theta = \frac{1 \times 3 \times 5 \times 7 \times 9}{2 \times 4 \times 6 \times 8 \times 10} \times \frac{\pi}{2}

= \frac{945}{3840} \cdot \frac{\pi}{2} = \frac{63\pi}{256}

Practice Questions :

Q.1 If β(p,q)=0.05 , find β(q,p) .

Q.2 \text{Evaluate: } \int_{0}^{\pi/2} \sin^{8} \theta \, d\theta.

Q.3 Find: β(2,3) using the definition and Gamma function.