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Chapter 11: Q2P (page 543)

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

Short Answer

Expert verified

The statementB(p,q)=B(q,p) is proved.

Step by step solution

01

Given information

Beta function is given. A hint to substitutex=1-y is given.

02

Definition of a Beta function

The beta function is defined as B(p,q)=∫01xp-1(1-x)q-1dx.

03

Begin with substitutions

Make the following substitution and differentiate the equation.

x=1-ydy=-dx

04

Evaluate the resulting integral.

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

∫10(1-y)p-1yq-1(-dy)=∫01(1-y)p-1yq-1dy=∫01(1-x)p-1xq-1dx=Bq,p

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

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Most popular questions from this chapter

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