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Beta function is one of the special functions and it is well-known as Euler’s integral. Beta Function can be written as B(x, y) in which x and y are denoted as real numbers and β is the symbol to denote Beta Function. The value of x and y should be more than zero. This is a proportional function like B(x, y) = B(y, x) and this is a special mathematical function. This special type of function of mathematics, the Beta function, was invented by the Swiss mathematician Leonhard Euler.
Read Also: Class 12 Chapter – Integrals
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Key Terms: Beta function, Euler function, gamma function, real number, domain, range, codomain, beta distribution
What is Function?
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The function is one of the most important parts of mathematics because, in every part of Maths, function comes like in Algebra, Geometry, Trigonometry, set theory etc. Beta function and gamma function are the most important part of Euler integral functions. Beta function co-relates the input and output function.
Beta and Gamma Functions Video Explaination
Definition of Beta Function
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Beta Function is an eccentric function and it is also identified as a particular kind of Euler”s integral. The symbol of Beta Function is “β”. It can also represent B(x, y) and where x and y are real numbers. In mathematics, Beta Function is the combination or set of input and output. It is one of the basic special functions which is used in physics, statistics, and engineering. Beta Function is the combination of domain, range and codomain.
Example:
f(x) = x2, where the input is domain and output is co-domain
Suppose the value of x as input is 2, x=2 then the
Output will be 4 because f(2) = 4
So, it can be written as pair of (2,4)
Check Important Notes for Eccentricity
Example of beta function
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Ques. Solve \(\int^{1}_{0} t^{4} (1-t)^{3}dt\)
Ans. Above equation can be written as:\(\int^{1}_{0} t^{5-1} (1-t)^{4-1}dt\)
Standard beta function isB (p, q) = \(\int^{1}_{0} t^{p-1} (1-t)^{q-1}dt\)
If you put t p= 5 and q = 4
And if you will put (p, q) = (4!. 3!) / 8!
You will get (4!. 6) /8! = 1/ 280
So the solution of this expression is 1/280
Beta Function is 1/280.
The Formula of Beta Function
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There are different ways by which you can write the formula of beta function:
As you know x and are the real numbers that are greater than 0 and they can be written as
B (p, q) = \(\int^{1}_{0} t^{p-1} (1-t)^{q-1}dt\)
Beta Function is very important in calculus and the gamma function is also very close to calculus. So, the combination of the beta-gamma function simplifies the difficult integral function and converts it into a simple integral.
Check Also: Trapezoid Formula
Relation of Beta Function with Gamma Function
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Beta Function can be represented like this -
B(p,q)=(Tp. Tq)/T(p+q)
and the gamma function can be represented like this -
B(p,q)=(p−1)!(q−1)!(p−1)!(q−1)!/(p+q−1)!
Where, p! = p. (p-1). (p-2)… 3. 2. 1
The formula of the beta-gamma function is very useful in solving calculus beta function problems.
Properties of Beta Function
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Listed below are some of the salient properties of Beta Function which can be applicable in many parts:
- Beta Function is proportional which means if the order of the variables will be changed it doesn’t mean the output of the operation will also change. The output will not change. You can also write like this: B(p,q)=B(q,p).
- B(p+1, q) = B(p, q). p/(p+q)p/(p+q).
- B(p, q) = B(p, q+1) + B(p+1, q)
- B(p, q+1) = B(p, q). [q/(p+q)]
Also Check: Sin2x Formula
Application Of Beta Function
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- Beta Function can be used in computing and representing the scattering amplitude for Regge trajectories in physics.
- It has great importance and is used in calculus with gamma function.
- The united beta-gamma function can simplify difficult integral functions into simple functions.
Things to remember
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- Beta Function is one of the special functions and it is well known as Euler”s integral
- Beta Function was originated by the Swiss mathematician Leonhard Euler.
- Beta function satisfies the truth that each input value has one output value.
- The role of the application of the beta-gamma function is very important in mathematical calculus.
- Beta Function is an association of domain, range and codomain.
Read Further: Trigonometric Identities
Sample Questions
Ques. What is the definition of Beta Function? (2 marks)
Ans. Beta Function is one of the important special functions and it is also known as Euler’s function. The symbol of Beta Function is “β”. This is the set and association of input and output which is used in solving mathematical problems.
Ques. Who studied Beta Function? (2 marks)
Ans. Beta Function was studied by the Swiss mathematician Euler and Legendre who was an astronomer, logician, engineer etc. For Regge trajectories, Beta Function is useful for determining and describing the scattering amplitude.
Ques. What is the difference between beta and gamma functions? (1 marks)
Ans. Beta and gamma function are the two types of Euler’s integral where Beta Function is a two-variable function and the gamma is a one-variable function.
Ques: How does beta function math work? (2 marks)
Ans. In Beta Function there are two real numbers and one of them is domain and it is denoted by “β”. Beta functions, also known as Euler integrals of the first kind, are a specific sort of function. It's generally written as B(x, y), where x and y are both positive real values.
B(x, y) = B(x, y) = B(x, y) = B(x, y) = B(x, y) = B(x, y) (y, x).
Ques: What are the uses of alpha, beta and gamma in maths? (2 marks)
Ans. Alpha, beta and gamma are very important notations in maths and this denotes the constant value and constant expressions which help a lot solve the simple as well as complicated problems..
Ques: Explain the uses of the beta distribution. (3 marks)
Ans. The use of beta distribution is very fixed which is 0 and 1. Example chances of success in any demonstration can be only two either pass or fail. The most typical application of this distribution is to simulate uncertainty in the likelihood of a random experiment's outcome. A three-point approach known as "beta distribution" is used in project management to assess the uncertainty in project time prediction.
Ques: Define symmetric function? Is Beta Function symmetric? (2 marks)
Ans. The definition of symmetric function is a function having many variables which remain unchanged for any type of permutation of the variables. Beta Function is symmetric. In other words, we can say that,
β(x,y) = β(y,x)
Ques: How can a function be defined? (5 marks)
Ans. The definition of function is the process or a relation between an element of one non-empty set with one element of another non-empty set. The other way of defining this is for every single input, there will be one unique output.
A function is a relationship between a group of inputs that each have one output. A function, in simple terms, is a relationship between inputs in which each input is associated with just one output. Each function has a domain and a codomain, often known as a range. The general notation for a function is f(x), where x represents the input.
Ques: Explain domain in beta function. (4 marks)
Ans. A domain is the set of values that behaves as input in the function while it can be defined. Beta Function is specified in real-number domains. Beta Function is denoted by the symbol "\(\beta\)". B(p, q) denotes Beta Function, with the parameters p and q being real values. In mathematics, Beta Function describes the relationship between the set of inputs and the set of outputs.
Ques: What is gamma function? (2 marks)
Ans. Gamma function is a single variable function and Beta is a two-variable function. The relation between beta and gamma functions will help to solve many problems in physics and mathematics.
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