Beta Function – Definition, Formula, Properties and Solved Examples (2024)

In Mathematics, the two most popular functions are Beta and Gamma Function. Beta is a two-variable function, while Gamma is a single variable function. And the relation between the Beta Function and the Gamma Function will help solve many Physics and Mathematics problems. The Beta Function is a one-of-a-kind function, often known as the first type of Euler's integrals. β is the notation used to represent it. The Beta Function is represented by (p, q), where p and q are both real values.

It clarifies the relationship between the inputs and outputs. The Beta Function tightly associates each input value with one output value. Many Mathematical processes rely heavily on the Beta Function.

Functions are a very important part of Mathematics. A function acts as the link between a set of input and output values, such that if you pass a certain input value through a given function, it will always yield one specific output. Therefore, a function is a special correlation between two data sets. Now, we can have some special types of functions. These functions can act as solutions for integral and differential equations. One such set of functions is Euler's Integral Functions. This group consists of two types, namely Gamma and Beta Function. In this article, we are going to discuss the Beta Function, its definition, properties, the Beta Function formula, and some problems based on this topic.

Mathematical Functions can be represented in different ways, such as - in the form of an algorithm or formula that shows how to calculate an output for a given value or in the form of a graph or an image.

There is a term known as special functions. These are the specific Mathematical functions having special notations as well as established names because of their importance in various branches of Physics and Mathematics.

Few special functions appear as integrals of differential equation solutions at times. Some commonly studied special functions are step function, absolute value function, floor function, triangle wave function, error function, Bessel's function, Riemannian zeta function, Euler integral function, and more.

Definition of Beta Function

We would first like to define the Beta Function before we proceed with the properties and problems. A Beta Function is a special kind of function which we classify as the first kind of Euler's integrals. The function has real number domains. We express this function as B(x,y) where x and y are real and greater than 0. The Beta Function is also symmetric, which means B(x, y) = B(y ,x). The notation used for the Beta Function is "β". The Beta Function in calculus forms an association between the input and output sets in integral equations and many more Mathematical operations.

The Beta Function is a one-of-a-kind function, often known as the first type of Euler's integrals. "β" is the notation used to represent it. The Beta Function is represented by (p, q), where p and q are both real values.

It clarifies the relationship between the inputs and outputs. The Beta Function tightly associates each input value with one output value. Many Mathematical processes rely heavily on the Beta Function.

Different forms of special functions have become vital tools for scientists and engineers in many fields of applied Mathematics. The classical Beta Function β(α, β) is undoubtedly one of the most fundamental special functions due to its essential significance in a variety of fields such as Mathematics, Physics, statistics, and engineering. Various writers have produced numerous fascinating and valuable extensions of various special functions like the Gamma and Beta Functions, the Gauss hypergeometric function, and so on in the last few decades.

Many complex integrals in Calculus can be simplified to formulations involving the Beta Function. Because of its close relationship to the Gamma Function, which is an extension of the factorial function, the Beta Function is essential in calculus. Because of the following feature, the Gamma Function is related to the Factorial Function.

Γ (n+1)=nΓn

The only snag is that n must be a positive (+) integer.

Using the Beta Function to Integrate

When evaluating integrals in terms of the Gamma Function, the Beta Function comes in handy. We demonstrate the evaluation of various distinct forms of integrals that would otherwise be inaccessible to us in this article.

Beta Function Formula

The Beta Function formula is as follows:

Here, p and q are greater than 0 and real numbers.

The Beta Function plays a very important role in calculus as it has a very close relationship with the Gamma Function. The Gamma Function itself is a general expression of the factorial function in Mathematics. The application of the beta-Gamma Function lies in the simplification of many complex integral functions into simple integrals containing the Beta Function.

Relationship Between Beta and Gamma Functions

The beta-Gamma Function relationship is as follows:

B(p,q)=(Tp.Tq)/T(p+q)

Here, the Gamma Function formula is:

The Beta Function can also find expression as the factorial formula given below:

B(p,q)=(p−1)!(q−1)!(p−1)!(q−1)!/(p+q−1)!

Here, p! = p. (p-1). (p-2)… 3. 2. 1

These relationships formed by the beta-Gamma Function are extremely crucial in solving integrals and Beta Function problems.

Beta Function Properties

The following are some useful Beta Function properties that one should keep in mind:

  • The Beta Function is symmetric which means the order of its parameters does not change the outcome of the operation. In other words, B(p,q)=B(q,p).

  • B(p, q+1) = B(p, q). q/(p+q)q/(p+q).

  • B(p+1, q) = B(p, q). p/(p+q)p/(p+q).

  • B (p, q). B (p+q, 1-q) = π/ p sin (πq).

Incomplete Beta Function

The incomplete Beta Function is basically the formula expressed in a generalized form. We show it by the following relation:

B (z: a, b) =

The notation for the same is B(a,b). When we put z = 1, we obtain our normal Beta Function. Therefore, B(1:a,b) = B(a, b).

The incomplete Beta Function finds application in Physics, calculus, Mathematical analysis and many other domains.

Beta Function Applications

The Beta Function finds implementation in many areas of science and Mathematics. For instance, in string theory, which is a part of complex Physics, the function computes and represents the scattering amplitudes of the Regge trajectories. The beta-Gamma Function duo also has numerous applications in calculus.

Now, the basic concepts are clear, we will look at Beta Function examples and Beta Function problems with solutions.

Beta Function – Definition, Formula, Properties and Solved Examples (2024)

FAQs

Beta Function – Definition, Formula, Properties and Solved Examples? ›

Beta

Beta
In probability and statistics, the Beta distribution is considered as a continuous probability distribution defined by two positive parameters. It is a type of probability distribution which is used to represent the outcomes or random behaviour of proportions or percentage.
functions are a special type of function, which is also known as Euler integral of the first kind. It is usually expressed as B(x, y) where x and y are real numbers greater than 0. It is also a symmetric function, such as B(x, y) = B(y, x).

What is the formula for the beta function in trigonometry? ›

Trigonometric Representation of the Beta Function

B ( x , y ) = ∫ 0 1 u y − 1 ( 1 − u ) x − 1 d u .

Why is the beta function symmetric? ›

The Beta Function is symmetric, which implies that the order of its arguments has no effect on the operation's output, B(p,q)=B(q,p). Ans. Beta and gamma are the two most popular functions in mathematics. Beta is a two-variable function, while Gamma is a single-variable function.

What is the formula for the beta function of the gamma function? ›

Claim: The gamma and beta functions are related as b(a, b) = Γ(a)Γ(b) Γ(a + b) . = -u. Also, since u = x + y and v = x/(x + y), we have that the limits of integration for u are 0 to с and the limits of integration for v are 0 to 1.

What are the properties of the beta function? ›

Beta Function Properties

The Beta Function is symmetric which means the order of its parameters does not change the outcome of the operation. In other words, B(p,q)=B(q,p). B(p, q+1) = B(p, q). q/(p+q)q/(p+q).

What is beta function definition and formula? ›

The beta function is defined in the domains of real numbers. The notation to represent the beta function is “β”. The beta function is meant by B(p, q), where the parameters p and q should be real numbers. The beta function in Mathematics explains the association between the set of inputs and the outputs.

What is the beta formula in maths? ›

There are some important integrals regarding beta functions that are given below: β(x, y)=∫∞0tx−1(1+t)x+ydt. β(x, y)=2∫π20(sin2x−1θ)(cos2y−1θ)dθ

What is the conclusion of the beta function? ›

Conclusion. The beta function aids in the creation of new extensions of the beta distribution, as well as new Gauss hypergeometric functions, confluent hypergeometric functions, and generating relations, as well as Riemann-Liouville derivatives.

How to calculate incomplete beta function? ›

I = betainc(X,Z,W) computes the incomplete beta function for corresponding elements of the arrays X , Z , and W . The elements of X must be in the closed interval [0,1]. The arrays Z and W must be nonnegative and real.

What is the special function of the beta function? ›

A function of two variables p and q which, for p,q>0, is defined by the equation B(p,q)=∫10xp−1(1−x)q−1dx. The values of the beta-function for various values of the parameters p and q are connected by the following relationships: B(p,q)=B(q,p), B(p,q)=q−1p+q−1B(p,q−1),q>1.

What is the role of the beta? ›

Beta (β) compares a stock or portfolio's volatility or systematic risk to the market. Beta provides an investor with an approximation of how much risk a stock will add to a portfolio. The S&P 500 has a beta of 1.0.

What is the formula for the variance of a beta function? ›

Properties of Beta Distributions

If X∼beta(α,β), then: the mean of X is E[X]=αα+β, the variance of X is Var(X)=αβ(α+β)2(α+β+1).

What are the properties of beta? ›

Beta particles have a mass which is half of one thousandth of the mass of a proton and carry either a single negative (electron) or positive (positron) charge. As they have a small mass and can be released with high energy, they can reach relativistic speeds (close to the speed of light).

How is beta calculated? ›

In this context, covariance refers to the measure of a stock's return relative to that of the market. And variance refers to the measure of how the market moves relative to its mean. Once those figures are identified, beta is calculated by dividing covariance by variance (Beta = Covariance/Variance).

What are the 5 values of beta? ›

One of North America's oldest fraternities. Beta Theta Pi is the oldest of the three fraternities that formed the Miami Triad, along with Phi Delta Theta and Sigma Chi. The five core values espoused by Beta Theta Pi are cultivation of intellect, responsible conduct, mutual assistance, integrity and trust.

What is the formula for calculating beta? ›

Beta = Covariance (Rs, RI) / Variance (RI)

For more data points, running the covariance and variance in a spreadsheet such as Excel is recommended. There are also online beta calculators available and stock beta is often listed on financial websites, along with other information about the stock.

What does β mean in trigonometry? ›

The angles opposite the sides of lengths a, b, and c are labeled α (alpha), β (beta), and γ (gamma), respectively. (Alpha, beta, and gamma are the first three letters in the Greek alphabet.) The small square with the angle γ indicates that this is the right angle in the right triangle.

What is the formula for beta transformation? ›

Basic Notion: If X=[0,1], then for all β>0 the associated β transformation is Tβx=βxmod(1), where x∈X. This can be easily embeded in the unit circle T by Tβe2πix=e2πiβx.

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