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Chapter 11: Q2P (page 543)

Prove equation (6.5).

Short Answer

Expert verified

It is proved that Bp,q=鈭�0鈭�yp-1(1+y)p+qdy.

Step by step solution

01

Given information.

The given equation (6.5), is Bp,q=鈭�0鈭�yp-1(1+y)p+qdy.

02

Definition of a Gamma function.

Beta function is defined as B(p,q)=鈭�01xp-1(1-x)q-1dx.

03

Perform necessary substitutions.

Make the following substitution and differentiate the terms.

y=x1-xdx=dy(1+y)2

04

Input these expressions in the integral.

Substitute these values in the integral.

Bp,q=鈭�0鈭�y1+yp-11-y1+yq-1dy(1+y)2=鈭�0鈭�yp-1(1+y)p-1(1+y)q-1(1+y)2dy=鈭�0鈭�yp-1(1+y)p+qdy

Hence proved.

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Most popular questions from this chapter

In A uniform solid sphere of density12is floating in water. (Compare Chapter 8, Problem (5.37).) It is pushed down just under water and released. Write the differential equation of motion (neglecting friction) and solve it to obtain the period in terms of K(5-12). Show that this period is approximately 1.16 times the period for small oscillations.

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