- #1
PLAGUE
- 15
- 0
- TL;DR Summary
- Proof of Beta Gamma function relation
So, my teacher showed me this proof and unfortunately it is vacation now. I don't understand what just happened in the marked line. Can someone please explain?
Relation Between Beta and Gamma functions
- I
- Thread starter PLAGUE
- Start date
So, my teacher showed me this proof and unfortunately it is vacation now. I don't understand what just happened in the marked line. Can someone please explain?
- #2
pasmith
Science Advisor
Homework Helper
- 3,072
- 1,639
The line asserts that [tex]
\int_0^\infty e^{-x(1+ y)}x^{m+n-1}\,dx = \frac{\Gamma(m + n)}{(1 + y)^{m + n}}.[/tex] How would you apply the given formulae to arrive at that?
\int_0^\infty e^{-x(1+ y)}x^{m+n-1}\,dx = \frac{\Gamma(m + n)}{(1 + y)^{m + n}}.[/tex] How would you apply the given formulae to arrive at that?
1. What is the relation between the Beta and Gamma functions?
The Beta function, denoted as B(x, y), is closely related to the Gamma function. The relation between the two functions is given by: B(x, y) = Γ(x)Γ(y) / Γ(x + y), where Γ(x) represents the Gamma function.
2. How are the Beta and Gamma functions defined?
The Beta function is defined as B(x, y) = ∫[0, 1] t^(x-1) * (1-t)^(y-1) dt, where x and y are positive real numbers. The Gamma function, denoted as Γ(x), is defined as Γ(x) = ∫[0, ∞] t^(x-1) * e^(-t) dt, where x is a positive real number.
3. What are the properties of the Beta and Gamma functions?
Some important properties of the Beta and Gamma functions include symmetry, reflection, duplication, and multiplication. These functions have various identities and relationships that are useful in mathematical analysis and applications.
4. How are the Beta and Gamma functions used in mathematics?
The Beta and Gamma functions are widely used in various branches of mathematics, such as probability theory, statistics, and integral calculus. They are essential for solving problems involving integrals, series, and special functions.
5. Can the relation between the Beta and Gamma functions be generalized?
Yes, the relation between the Beta and Gamma functions can be generalized to higher dimensions using the multivariate Beta and Gamma functions. These generalized functions are used in multivariate analysis and have applications in diverse fields such as physics, engineering, and economics.
Similar threads
-
Calculus
-
Calculus
-
High Energy, Nuclear, Particle Physics
-
Programming and Computer Science
-
High Energy, Nuclear, Particle Physics