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.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Nuclear Physics

Atoms and Radioactivity

Engineering Physics

Electricity

Electric Charge Field and Potential

Oscillations

Study anywhere. Anytime. Across all devices.

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

Nuclear Physics

Atoms and Radioactivity

Engineering Physics

Electricity

Electric Charge Field and Potential

Oscillations

Study anywhere. Anytime. Across all devices.

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