Question

From equation (15.4) show that ${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}{\mathbf{J}}_{\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{dx}\mathbf{=}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}{\mathbf{J}}_{\mathbf{3}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{dx}\mathbf{=}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{=}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}{\mathbf{J}}_{\mathbf{2}\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{dx}$ and

${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}{\mathbf{J}}_0\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{dx}\mathbf{=}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}{\mathbf{J}}_2\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{dx}\mathbf{=}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{=}{\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}{\mathbf{J}}_{\mathbf{2}\mathit{x}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{dx}$. Then, by Problem 7, show that

${\mathbf{\int }}_{\mathbf{0}}^{\mathbf{\infty }}{\mathbf{J}}_{\mathrm{n}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{dx}\mathbf{=}\mathbf{1}$for all integral.$\mathbf{n}\mathbf{.}\mathbf{}\mathbf{(}\mathbf{15}\mathbf{.}\mathbf{4}\mathbf{)}\mathbf{}{\mathbf{J}}_{\mathbf{p}\mathbf{-}\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{-}{\mathbf{J}}_{\mathbf{P}\mathbf{+}\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{2}{\mathbf{J}}_{\mathbf{p}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$ .

Short answer

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

${\int }_0^{\infty }{\mathrm{J}}_1\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\infty }{\mathrm{J}}_3\left(\mathrm{x}\right)\mathrm{dx}=...={\int }_0^{\infty }{\mathrm{J}}_{2\mathrm{n}+1}\left(\mathrm{x}\right)\mathrm{dx}$

And

${\int }_0^{\infty }{\mathrm{J}}_0\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\infty }{\mathrm{J}}_2\left(\mathrm{x}\right)\mathrm{dx}=...={\int }_0^{\infty }{\mathrm{J}}_{2\mathrm{n}}\left(\mathrm{x}\right)\mathrm{dx}$

Calculate from each equation.

$$\begin{gathered} 2{\mathrm{J}}_{\mathrm{P}}\left(\mathrm{x}\right)={\mathrm{J}}_{\mathrm{P}-1}\left(\mathrm{x}\right)-{\mathrm{J}}_{\mathrm{P}-1}\left(\mathrm{x}\right) \\ 2{\mathrm{J}}_{\mathrm{P}}\left(\mathrm{x}\right)={\mathrm{J}}_{\mathrm{P}-1}\left(\mathrm{x}\right)-{\mathrm{J}}_{\mathrm{P}-1}\left(\mathrm{x}\right) \\ 2{\mathrm{J}}_1\left(\mathrm{x}\right)={\mathrm{J}}_0\left(\mathrm{x}\right)-{\mathrm{J}}_2\left(\mathrm{x}\right) \end{gathered}$$

...... (1)

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

${\int }_0^{\mathrm{\pi }}2{\mathrm{J}}_1\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\mathrm{\pi }}{\mathrm{J}}_0\left(\mathrm{x}\right)\mathrm{dx}-{\int }_0^{\mathrm{\pi }}{\mathrm{J}}_2\left(\mathrm{x}\right)\mathrm{dx}$ ….. (2)

Integrating from limits 0 to 0 as follows:

role="math" localid="1659327415234" $$\begin{gathered} {\left[2{\mathrm{J}}_1\left(\mathrm{x}\right)\right]}_0={\int }_0^{\mathrm{\pi }}{\mathrm{J}}_0\left(\mathrm{x}\right)\mathrm{dx}-{\int }_0^{\mathrm{m}}{\mathrm{J}}_2\left(\mathrm{x}\right)\mathrm{dx} \\ 2{\left[{\mathrm{limJ}}_1\left(\mathrm{x}\right)-{\mathrm{J}}_1\left(0\right)\right]}_0={\int }_{0\to \infty }^{\mathrm{\pi }}{\mathrm{J}}_0\left(\mathrm{x}\right)\mathrm{dx}-{\int }_0^{\mathrm{m}}{\mathrm{J}}_2\left(\mathrm{x}\right)\mathrm{dx} \end{gathered}$$

Since. ${\mathrm{J}}_1\left(\mathrm{x}\right)\to 0$as ${\mathrm{J}}_1\left(\mathrm{x}\right)\to 0$.

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

role="math" localid="1659328288884" $$\begin{gathered} 2{\mathrm{J}}_3\left(\mathrm{x}\right)={\mathrm{J}}_2\left(\mathrm{x}\right)-{\mathrm{J}}_4\left(\mathrm{x}\right) \\ {\int }_0^{\infty }2{\mathrm{J}}_3\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\infty }{\mathrm{J}}_2\left(\mathrm{x}\right)\mathrm{dx}-{\int }_0^{\infty }{\mathrm{J}}_4\left(\mathrm{x}\right)\mathrm{dx} \end{gathered}$$

….. (3)

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

From equation (2) and (3) obtain:

role="math" localid="1659328451089" ${\int }_0^{\infty }{\mathrm{J}}_0\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\infty }{\mathrm{J}}_2\left(\mathrm{x}\right)\mathrm{dx}=...={\int }_0^{\infty }{\mathrm{J}}_4\left(\mathrm{x}\right)\mathrm{dx}$

Similarly obtain:

$$\begin{gathered} {\int }_0^{\mathrm{\pi }}{\mathrm{J}}_4\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\mathrm{\pi }}{\mathrm{J}}_6\left(\mathrm{x}\right)\mathrm{dx} \\ {\int }_0^{\mathrm{\pi }}{\mathrm{J}}_6\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\mathrm{\pi }}{\mathrm{J}}_8\left(\mathrm{x}\right)\mathrm{dx} \end{gathered}$$

Hence,

${\int }_0^{\infty }{\mathrm{J}}_0\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\infty }{\mathrm{J}}_2\left(\mathrm{x}\right)\mathrm{dx}=...={\int }_0^{\infty }{\mathrm{J}}_{2\mathrm{n}}\left(\mathrm{x}\right)\mathrm{dx}$ ...... (4)

Identify the equation:

Indentify the equation to show as follows:

$$\begin{gathered} {\int }_0^{\infty }{\mathrm{J}}_1\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\infty }{\mathrm{J}}_3\left(\mathrm{x}\right)\mathrm{dx}=...{\int }_0^{\infty }{\mathrm{J}}_{2\mathrm{n}+1}\left(\mathrm{x}\right)\mathrm{dx} \\ 2{\mathrm{J}}_2\left(\mathrm{x}\right)={\mathrm{J}}_1\left(\mathrm{x}\right)-{\mathrm{J}}_3\left(\mathrm{x}\right) \\ {\int }_0^{\infty }2{\mathrm{J}}_2\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\infty }{\mathrm{J}}_1\left(\mathrm{x}\right)\mathrm{dx}=...{\int }_0^{\infty }{\mathrm{J}}_3\left(\mathrm{x}\right)\mathrm{dx} \\ 12{\mathrm{J}}_4\left(\mathrm{x}\right)={\mathrm{J}}_3\left(\mathrm{x}\right)-{\mathrm{J}}_5\left(\mathrm{x}\right) \end{gathered}$$

Simplify further as follows:

$$\begin{gathered} {\left[2J_4\left(x\right)\right]}_0^{\infty }={\int }_0^{\infty }{\mathrm{J}}_3\left(\mathrm{x}\right)\mathrm{dx}-{\int }_0^{\infty }{\mathrm{J}}_5\left(\mathrm{x}\right)\mathrm{dx} \\ 2\left[lim{J}_4\left(x\right)-J_4\left(0\right)\right]={\int }_0^{\infty }{\mathrm{J}}_3\left(\mathrm{x}\right)-{\int }_0^{\infty }{\mathrm{J}}_5\left(\mathrm{x}\right)\mathrm{dx} \\ {\int }_0^{\infty }{\mathrm{J}}_3\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\infty }{\mathrm{J}}_5\left(\mathrm{x}\right)\mathrm{dx}=0 \end{gathered}$$

Put 2 for $\mathrm{\beta }$in equation (1).

$$\begin{gathered} 2{\mathrm{J}}_2\left(\mathrm{x}\right)={\mathrm{J}}_1\left(\mathrm{x}\right)-{\mathrm{J}}_3\left(\mathrm{x}\right) \\ {\int }_0^{\infty }2{\mathrm{J}}_2\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\infty }{\mathrm{J}}_1\left(\mathrm{x}\right)\mathrm{dx}-{\int }_0^{\infty }{\mathrm{J}}_3\left(\mathrm{x}\right)\mathrm{dx} \end{gathered}$$

...... (5)

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

Put for in equation (5) as follows:

$2{\mathrm{J}}_4\left(\mathrm{x}\right)={\mathrm{J}}_3\left(\mathrm{x}\right)-{\mathrm{J}}_5\left(\mathrm{x}\right)$

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

$$\begin{gathered} {\left[2{\mathrm{J}}_4\left(\mathrm{x}\right)\right]}_0^{\mathrm{x}}={\int }_0^{\infty }{\mathrm{J}}_3\left(\mathrm{x}\right)\mathrm{dx}-{\int }_0^{\infty }{\mathrm{J}}_5\left(\mathrm{x}\right)\mathrm{dx} \\ 2\left[{\mathrm{limJ}}_4\left(\mathrm{x}\right)-{\mathrm{J}}_4\left(0\right)\right]={\int }_0^{\infty }{\mathrm{J}}_3\left(\mathrm{x}\right)-{\int }_0^{\infty }{\mathrm{J}}_5\left(\mathrm{x}\right)\mathrm{dx} \\ {\int }_0^{\infty }{\mathrm{J}}_3\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\infty }{\mathrm{J}}_5\left(\mathrm{x}\right)\mathrm{dx}=0 \end{gathered}$$

Since, ${\mathrm{J}}_4\left(\mathrm{x}\right)\to 0$.

$\mathrm{x}\to \infty {\mathrm{J}}_4\left(0\right)\to {\int }_0^{\infty }{\mathrm{J}}_3\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\infty }{\mathrm{J}}_5\left(\mathrm{x}\right)\mathrm{dx}$ ...... (6)

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

${\int }_0^{\infty }{\mathrm{J}}_1\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\infty }{\mathrm{J}}_3\left(\mathrm{x}\right)\mathrm{dx}=...{\int }_0^{\infty }{\mathrm{J}}_5\left(\mathrm{x}\right)\mathrm{dx}$

Similarly,

$$\begin{gathered} {\int }_0^{\infty }{\mathrm{J}}_1\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\infty }{\mathrm{J}}_7\left(\mathrm{x}\right)\mathrm{dx} \\ {\int }_0^{\infty }{\mathrm{J}}_7\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\infty }{\mathrm{J}}_9\left(\mathrm{x}\right)\mathrm{dx} \end{gathered}$$

Obtain the equation:

Hence,

$$\begin{gathered} {\int }_0^{\infty }{\mathrm{J}}_1\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\infty }{\mathrm{J}}_3\left(\mathrm{x}\right)\mathrm{dx}=...{\int }_0^{\infty }{\mathrm{J}}_{2\mathrm{n}+1}\left(\mathrm{x}\right)\mathrm{dx} \\ {\int }_0^{\infty }{\mathrm{J}}_{\mathrm{n}}\left(\mathrm{x}\right)\mathrm{dx}=1 \end{gathered}$$ ...... (7)

To show that, ${\int }_0^{\infty }{\mathrm{J}}_1\left(\mathrm{x}\right)\mathrm{dx}=1$, from equation (7).

Since,

$$\begin{gathered} {\int }_0^{\infty }{\mathrm{J}}_{2\mathrm{n}+1}\left(\mathrm{x}\right)\mathrm{dx}=1 \\ {\int }_0^{\infty }{\mathrm{J}}_0\left(\mathrm{x}\right)\mathrm{dx}=1 \end{gathered}$$