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,

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

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Short Answer

Step by step solution

Concept of Differential equations:

Determine equations and integrate them:

Identify the equation:

Obtain the equation:

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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

Electricity and Magnetism

Solid State Physics

Engineering Physics

Conservation of Energy and Momentum

Translational Dynamics

Geometrical and Physical Optics

Study anywhere. Anytime. Across all devices.