MHB Bessel Function: a^2-b^2 Integral Relationship

AI Thread Summary
The discussion centers on proving the integral relationship involving Bessel functions, specifically showing that the equation holds for the given parameters. The proof utilizes integration by parts and identities related to Bessel functions to derive the integral expression. Key steps include manipulating the integral of the product of Bessel functions and applying boundary conditions. The final expression confirms the relationship between the integral and the Bessel functions evaluated at specific points. This establishes a significant connection in the study of Bessel functions and their integrals.
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show that
$$(a^2-b^2)\int_{0}^{P} J_{v}(ax)J_{v}(bx)x\,dx=P\left\{bJ_{v}(aP)J^{'}_{v}(bP)-aJ^{'}_{v}(ap)J_{v}(bP)\right\}$$
when $$J^{'}_{v}(aP)=\d{J_{v}(ax)}{(ax)},(x=P)$$

I don, have idea
 
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Another said:
show that
$$(a^2-b^2)\int_{0}^{P} J_{v}(ax)J_{v}(bx)x\,dx=P\left\{bJ_{v}(aP)J^{'}_{v}(bP)-aJ^{'}_{v}(ap)J_{v}(bP)\right\}$$
when $$J^{'}_{v}(aP)=\d{J_{v}(ax)}{(ax)},(x=P)$$

(My thinking)
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identities

$$\d{}{x}\left\{ x^{-v}J_{v}(ax)\right\}=-ax^{-v}J_{v+1}(ax)$$
$$\d{}{x}\left\{ x^{v+1}J_{v+1}(ax)\right\}=ax^{v+1}J_{v}(ax) $$

$$\d{}{x}\left\{ x^{v}J_{v}(x)\right\}=x^{v}J_{v-1}(x) $$
$$\d{}{x}\left\{ x^{v+1}J_{v+1}(x)\right\}=x^{v+1}J_{v}(x) $$
$$x^{v+1}J_{v+1}(x)=\int \left\{ x^{v+1}J_{v}(x) \right\} dx $$
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soluion

$$\int J_{v}(ax)J_{v}(bx)x dx = \int \left[ x^{v+1}J_{v}(ax) \right] \left[ x^{-v}J_{v}(bx) \right]dx $$
$$uv-\int v du = \left[ x^{-v}J_{v}(bx) \right] \left[ \frac{x^{v+1}}{a}J_{v+1}(ax) \right]+\frac{b}{a}\int \left[ x^{-v}J_{v+1}(bx) \right] \left[ x^{v+1}J_{v+1}(ax) \right] dx $$

see (by parts again)
$$\int \left[ x^{-v}J_{v+1}(bx) \right] \left[ x^{v+1}J_{v+1}(ax) \right] dx=\int \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v+1}(ax) \right] dx $$
$$\int \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v+1}(ax) \right] dx=uv-\int vdu $$
$$\int \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v+1}(ax) \right] dx=-\frac{1}{a} \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] + \frac{b}{a}\int \left[ x^{v+1}J_{v}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] dx $$

So...
$$\int J_{v}(ax)J_{v}(bx)x dx = \left[ x^{-v}J_{v}(bx) \right] \left[ \frac{x^{v+1}}{a}J_{v+1}(ax) \right]+\frac{b}{a}\left[ -\frac{1}{a} \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] + \frac{b}{a}\int \left[ x^{v+1}J_{v}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] \right] dx $$
$$\int J_{v}(ax)J_{v}(bx)x dx = \frac{a}{a^2}\left[ x^{-v}J_{v}(bx) \right] \left[ x^{v+1}J_{v+1}(ax) \right] -
\frac{b}{a} \frac{1}{a} \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] + \frac{b}{a} \frac{b}{a}\int \left[ x^{v+1}J_{v}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] dx $$
$$a^{2}\int J_{v}(ax)J_{v}(bx)x dx = a\left[ x^{-v}J_{v}(bx) \right] \left[ x^{v+1}J_{v+1}(ax) \right] -
b \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] + b^2 \int J_{v}(bx) J_{v}(ax) xdx $$
$$(a^{2}-b^{2})\int J_{v}(ax)J_{v}(bx)x dx = a\left[ x^{-v}J_{v}(bx) \right] \left[ x^{v+1}J_{v+1}(ax) \right] -
b \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] $$
$$(a^{2}-b^{2})\int J_{v}(ax)J_{v}(bx)x dx = ax J_{v}(bx) J_{v+1}(ax) -
bx J_{v+1}(bx) J_{v}(ax) $$

And finally...
$$(a^{2}-b^{2}) \int_{0}^{P} J_{v}(ax)J_{v}(bx)x \,dx = P(a J_{v}(bP) J_{v+1}(aP) - b J_{v+1}(bP) J_{v}(aP)) $$

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