Chapter 12: Q8P (page 593)

From equation (15.4) show that $\int_{0}^{\infty} J_{1} (x) dx = \int_{0}^{\infty} J_{3} (x) dx = . . . = \int_{0}^{\infty} J_{2 n + 1} (x) dx$ and
$\int_{0}^{\infty} J_{0} (x) dx = \int_{0}^{\infty} J_{2} (x) dx = . . . = \int_{0}^{\infty} J_{2 x} (x) dx$ . Then, by Problem 7, show that
$\int_{0}^{\infty} J_{n} (x) dx = 1$ for all integral. $n . (15.4) J_{p - 1} (x) - J_{P + 1} (x) = 2 J_{p}^{'} (x)$ .

Short Answer

Expert verified

This equation has been proved.

Step by step solution

Concept of Differential equations:

Differential equations are equations that connect one or more derivatives of a function. This implies that their answer is a function.

Determine equations and integrate them:

The equations are

$\int_{0}^{\infty} J_{1} (x) dx = \int_{0}^{\infty} J_{3} (x) dx = . . . = \int_{0}^{\infty} J_{2 n + 1} (x) dx$

And

$\int_{0}^{\infty} J_{0} (x) dx = \int_{0}^{\infty} J_{2} (x) dx = . . . = \int_{0}^{\infty} J_{2 n} (x) dx$

Calculate from each equation.

$2 J_{P} (x) = J_{P - 1} (x) - J_{P - 1} (x) 2 J_{P} (x) = J_{P - 1} (x) - J_{P - 1} (x) 2 J_{1} (x) = J_{0} (x) - J_{2} (x)$

...... (1)

Put P = 1 in equation (1) as follows:

$\int_{0}^{π} 2 J_{1} (x) dx = \int_{0}^{π} J_{0} (x) dx - \int_{0}^{π} J_{2} (x) dx$ ….. (2)

Integrating from limits 0 to 0 as follows:

role="math" localid="1659327415234" ${[2 J_{1} (x)]}_{0} = \int_{0}^{π} J_{0} (x) dx - \int_{0}^{m} J_{2} (x) dx 2 {[{limJ}_{1} (x) - J_{1} (0)]}_{0} = \int_{0 \to \infty}^{π} J_{0} (x) dx - \int_{0}^{m} J_{2} (x) dx$

Since. $J_{1} (x) \to 0$ as $J_{1} (x) \to 0$ .

Put 3 for p in the equation (1) as follows:

role="math" localid="1659328288884" $2 J_{3} (x) = J_{2} (x) - J_{4} (x) \int_{0}^{\infty} 2 J_{3} (x) dx = \int_{0}^{\infty} J_{2} (x) dx - \int_{0}^{\infty} J_{4} (x) dx$

….. (3)

Integrating with limits 0 to $\infty$ as follows:

From equation (2) and (3) obtain:

role="math" localid="1659328451089" $\int_{0}^{\infty} J_{0} (x) dx = \int_{0}^{\infty} J_{2} (x) dx = . . . = \int_{0}^{\infty} J_{4} (x) dx$

Similarly obtain:

$\int_{0}^{π} J_{4} (x) dx = \int_{0}^{π} J_{6} (x) dx \int_{0}^{π} J_{6} (x) dx = \int_{0}^{π} J_{8} (x) dx$

Hence,

$\int_{0}^{\infty} J_{0} (x) dx = \int_{0}^{\infty} J_{2} (x) dx = . . . = \int_{0}^{\infty} J_{2 n} (x) dx$ ...... (4)

Identify the equation:

Indentify the equation to show as follows:

$\int_{0}^{\infty} J_{1} (x) dx = \int_{0}^{\infty} J_{3} (x) dx = . . . \int_{0}^{\infty} J_{2 n + 1} (x) dx 2 J_{2} (x) = J_{1} (x) - J_{3} (x) \int_{0}^{\infty} 2 J_{2} (x) dx = \int_{0}^{\infty} J_{1} (x) dx = . . . \int_{0}^{\infty} J_{3} (x) dx 12 J_{4} (x) = J_{3} (x) - J_{5} (x)$

Simplify further as follows:

${[2 J_{4} (x)]}_{0}^{\infty} = \int_{0}^{\infty} J_{3} (x) dx - \int_{0}^{\infty} J_{5} (x) dx 2 [l i m J_{4} (x) - J_{4} (0)] = \int_{0}^{\infty} J_{3} (x) - \int_{0}^{\infty} J_{5} (x) dx \int_{0}^{\infty} J_{3} (x) dx = \int_{0}^{\infty} J_{5} (x) dx = 0$

Put 2 for $β$ in equation (1).

$2 J_{2} (x) = J_{1} (x) - J_{3} (x) \int_{0}^{\infty} 2 J_{2} (x) dx = \int_{0}^{\infty} J_{1} (x) dx - \int_{0}^{\infty} J_{3} (x) dx$

...... (5)

Integrating with limits 0 to $\infty$ as follows:

Put for in equation (5) as follows:

$2 J_{4} (x) = J_{3} (x) - J_{5} (x)$

Integrating with limits 0 to $\infty$ as follows:

${[2 J_{4} (x)]}_{0}^{x} = \int_{0}^{\infty} J_{3} (x) dx - \int_{0}^{\infty} J_{5} (x) dx 2 [{limJ}_{4} (x) - J_{4} (0)] = \int_{0}^{\infty} J_{3} (x) - \int_{0}^{\infty} J_{5} (x) dx \int_{0}^{\infty} J_{3} (x) dx = \int_{0}^{\infty} J_{5} (x) dx = 0$

Since, $J_{4} (x) \to 0$ .

$x \to \infty J_{4} (0) \to \int_{0}^{\infty} J_{3} (x) dx = \int_{0}^{\infty} J_{5} (x) dx$ ...... (6)

From equation (5) and (6), obtain:

$\int_{0}^{\infty} J_{1} (x) dx = \int_{0}^{\infty} J_{3} (x) dx = . . . \int_{0}^{\infty} J_{5} (x) dx$

Similarly,

$\int_{0}^{\infty} J_{1} (x) dx = \int_{0}^{\infty} J_{7} (x) dx \int_{0}^{\infty} J_{7} (x) dx = \int_{0}^{\infty} J_{9} (x) dx$

Obtain the equation:

Hence,

$\int_{0}^{\infty} J_{1} (x) dx = \int_{0}^{\infty} J_{3} (x) dx = . . . \int_{0}^{\infty} J_{2 n + 1} (x) dx \int_{0}^{\infty} J_{n} (x) dx = 1$ ...... (7)

To show that, $\int_{0}^{\infty} J_{1} (x) dx = 1$ , from equation (7).

Since,

$\int_{0}^{\infty} J_{2 n + 1} (x) dx = 1 \int_{0}^{\infty} J_{0} (x) dx = 1$

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.

Short Answer

Step by step solution

Concept of Differential equations:

Determine equations and integrate them:

Identify the equation:

Obtain the equation:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fields in Physics

Space Physics

Electrostatics

Dynamics

Linear Momentum

Engineering Physics

Study anywhere. Anytime. Across all devices.

From equation (15.4) show that ∫0∞J1(x)dx=∫0∞J3(x)dx=...=∫0∞J2n+1(x)dx and ∫0∞J0(x)dx=∫0∞J2(x)dx=...=∫0∞J2x(x)dx. Then, by Problem 7, show that∫0∞Jn(x)dx=1for all integral.n.(15.4)Jp-1(x)-JP+1(x)=2Jp'(x) .

Short Answer

Step by step solution

Concept of Differential equations:

Determine equations and integrate them:

Identify the equation:

Obtain the equation:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fields in Physics

Space Physics

Electrostatics

Dynamics

Linear Momentum

Engineering Physics

Study anywhere. Anytime. Across all devices.