Chapter 11: Q2P (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 $B (p, q) = {鈭�}_{0}^{1} x^{p - 1} {(1 - x)}^{q - 1} d x$ .

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.

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

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

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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91影视

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.

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91影视

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.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Thermodynamics

Rotational Dynamics

Linear Momentum

Fluids

Circular Motion and Gravitation

Measurements

Study anywhere. Anytime. Across all devices.

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