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Chapter 12: Q6P (page 590)

To show the following equation shown in the problem

2ddxJa(x)=Ja-1(x).

Short Answer

Expert verified

The answer is,2ddxJa(x)=Ja-1(x) .

Step by step solution

01

 Step 1: Concept of Bessel Function:

The Bessel functions of the first kindJn(x) are defined as the solutions to the Bessel differential equation.

x2d2ydx2+xdydx+(x2-n2)y=0

02

Solve the equation 2ddxJa(x)=Ja-1(x)-Ja+1(x)  about the Bessel Function:

The given equation for the following equation is shown as,

2ddxJa(x)=Ja-1(x)-Ja+1(x)

By the use of the following proposition about the zeros of Bessel functions solve the equation as follows:

ddxJ伪(x)=ddxxa(x-伪J伪(x))=伪x伪-1J伪x+x伪ddxx伪J伪x=-伪x伪-1Jax+x-伪x伪J伪-1x=-伪x伪-1J伪x+Ja-1x
03

Calculation of the equation ddx(xαx-αJα(x) :

By adding the two expressions as follows:

ddxJa(x)=ddxx伪x-伪J伪x=伪x伪-1J伪x+x伪ddxx伪J伪(x)=J伪-1(x)-J伪+1x/22ddxJ伪(x)=Ja-1x-Ja+1(x)

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Most popular questions from this chapter

Find the norm of each of the following functions on the given interval and state the normalized functionP2(x)on(-1,1)

The equation for the associated Legendre functions (and for Legendre functions when m=0) usually arises in the form (see, for example, Chapter 13, Section 7) 1/sin胃 d/d胃 (sin胃 dy/d胃)+[l (l+1)-m2/sin2胃] y=0.

Make the change of variable x=cos胃, and obtain (10.1):

(1-x2) y"-2xy'+[l (l+1) -m2/1-x2] y=0

Expand each of the following polynomials in a Legendre series. You should get the same results that you got by a different method in the corresponding problems in Section 5.

3x2+x-1

Expand each of the following polynomials in a Legendre series. You should get the same results that you got by a different method in the corresponding problems in Section 5.

x-x3
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