Integral -- Beta function, Bessel function?

In summary, the Beta function is a special function in mathematics, denoted by B(x,y), that is used to calculate probabilities and has applications in fields such as economics, engineering, and physics. It is closely related to the Gamma function and the Bessel function, and has real-world applications in finance, economics, physics, engineering, and biology. It is also used in Bayesian statistics to calculate posterior distributions.
  • #1
LagrangeEuler
717
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Integral
[tex]\int^{\pi}_0\sin^3xdx=\int^{\pi}_0\sin x \sin^2xdx=\int^{\pi}_0\sin x (1-\cos^2 x)dx=\frac{4 \pi}{3}[/tex]
Is it possible to write integral ##\int^{\pi}_0\sin^3xdx## in form of Beta function, or even Bessel function?
 
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  • #2
Since [itex]B(\frac12,\frac12) = \pi[/itex] you have [tex]
\int_0^{\pi} \sin^3 x\,dx = \tfrac43B(\tfrac12,\tfrac12) = \tfrac43 \int_0^1 u^{-1/2}(1 - u)^{-1/2}\,du.[/tex]
 

1. What is the Integral-Beta function?

The Integral-Beta function, also known as the Beta function, is a special function that is used to evaluate definite integrals in mathematics. It is defined as B(x,y) = ∫01 tx-1(1-t)y-1 dt, where x and y are real numbers.

2. What is the relationship between the Integral-Beta function and the Gamma function?

The Integral-Beta function is closely related to the Gamma function, as it can be expressed in terms of the Gamma function. Specifically, B(x,y) = Γ(x)Γ(y)/Γ(x+y), where Γ(x) is the Gamma function. This relationship is known as the Beta-Gamma identity.

3. How is the Integral-Beta function used in statistics?

The Integral-Beta function is commonly used in statistics to calculate probabilities and cumulative distribution functions for continuous probability distributions, such as the beta distribution. It is also used in Bayesian analysis and in calculating confidence intervals.

4. What are Bessel functions and how are they related to the Integral-Beta function?

Bessel functions are a family of special functions that are used to solve differential equations in physics and engineering. They are closely related to the Integral-Beta function, as they can be expressed in terms of the Beta function. Specifically, the Bessel function of the first kind, Jα(x), can be written as Jα(x) = (x/2)α B(α+1/2, 1/2).

5. Can the Integral-Beta function be evaluated numerically?

Yes, the Integral-Beta function can be evaluated numerically using various methods, such as the Gauss-Legendre quadrature or the Simpson's rule. There are also several software packages, such as MATLAB and Mathematica, that have built-in functions for calculating the Integral-Beta function.

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