relation between beta and gamma function
relation between beta and gamma function
beta function is an area function that means it has two variable π (m,n). on the other hand gamma function is one dimensional function that means it has one variable. so the relation between beta and gamma function says that the beta function of two variable is always equal to the multiplication of two variable gamma function divided by the addition of two gamma function. that is given by,
π (m,n) = (πͺm πͺn) ∕ πͺm + πͺn
What is Beta function?
Beta function is a two variable function. the beta function is denoted by π (m,n). It's value does not depends upon x and y. it's value depends upon π (m,n). Beta function is one of the function that can be written in integral form and also converted into trigonometrical Integral.
In other words, beta function is an area function. having two variable. that means its value changes when π (m,n) changes.
Beta function is defined integral as,
π (m,n) = ∫ x (m-1) (1-x) n-1 dx
Properties of beta function
- π (m,n) = π (n,m) read as symmetry of beta function
- π (m,n) = ∫ x^ (n-1) ∕ (1+x)^ (m+n)
- π (m,n) = 2 ∫ (sinΞΈ)^ (2m-1) (cosΞΈ)^ (2n-1) dΞΈ
what is gamma function?
It is defined as the definite integral of e to the power minus x multiplied by x to the power (n-1) dx.
it is one dimensional function πͺn that means it has only one variable. there are six properties of gamma function. but here we use only five.
properties of gamma function
- πͺ1 = 1
- πͺn+1 = n πͺn
- πͺn = Z^n ∫ e^ -2x x^(n-1) dx
- πͺn = ∫ log (1/y)^(n-1) dy
- πͺn+1 = ∫ e^ -y
state and prove of relation between beta and gamma function
This is the derivation of relation between beta and gamma function. the relation between beta and gamma function states that the beta function of two variable π (m,n) is equal to the gamma function of 'm' and 'n' divided by addition of two variable.
Hence, the formula of relation between beta and gamma function is π (m,n) = (πͺm πͺn) ∕ πͺm + πͺn.