Chapter 11: Q1P (page 543)

Prove that $B (p, q) = B (q, p)$ . Hint:Put $x = 1 - y$ in Equation (6.1).

Short Answer

Expert verified

The statement $B (p, q) = B (q, p)$ is proved.

Step by step solution

Given information

Beta function is given. A hint to substitute $x = 1 - y$ is given.

Definition of a Beta function

The beta function is defined as $B (p, q) = \int_{0}^{1} x^{p - 1} {(1 - x)}^{q - 1} d x$ .

Begin with substitutions

Make the following substitution and differentiate the equation.

$\begin{matrix} x = 1 - y \\ d y = - d x \end{matrix}$

Evaluate the resulting integral.

Begin with the definition of a Beta function.

$B (p, q) = \int_{0}^{1} x^{p - 1} {(1 - x)}^{q - 1} d x$

Evaluate the resulting integral.

Simplify the resulting integral. Invert the boundary limits of the integral by negating the expression.

$\begin{matrix} \int_{1}^{0} {(1 - y)}^{p - 1} y^{q - 1} (- d y) = \int_{0}^{1} {(1 - y)}^{p - 1} y^{q - 1} d y \\ = \int_{0}^{1} {(1 - x)}^{p - 1} x^{q - 1} d x \\ = B (q, p) \end{matrix}$

Since this results in the same integral as in the beginning, the statement $B (p, q) = B (q, p)$ is proved.

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Prove that $B (p, q) = B (q, p)$ . Hint:Put $x = 1 - y$ in Equation (6.1).

Short Answer

Step by step solution

Given information

Definition of a Beta function

Begin with substitutions

Evaluate the resulting integral.

Evaluate the resulting integral.

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Prove that B(p,q)=B(q,p). Hint:Putx=1-yin Equation (6.1).

Short Answer

Step by step solution

Given information

Definition of a Beta function

Begin with substitutions

Evaluate the resulting integral.

Evaluate the resulting integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Scientific Method Physics

Radiation

Fluids

Geometrical and Physical Optics

Solid State Physics

Mechanics and Materials

Study anywhere. Anytime. Across all devices.

Prove that $B (p, q) = B (q, p)$ . Hint:Put $x = 1 - y$ in Equation (6.1).