Another said:show that
\(\displaystyle (a^2-b^2)\int_{0}^{P} J_{v}(ax)J_{v}(bx)x\,dx=P\left\{bJ_{v}(aP)J^{'}_{v}(bP)-aJ^{'}_{v}(ap)J_{v}(bP)\right\}\)
when \(\displaystyle J^{'}_{v}(aP)=\d{J_{v}(ax)}{(ax)},(x=P)\)
(My thinking)
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identities
\(\displaystyle \d{}{x}\left\{ x^{-v}J_{v}(ax)\right\}=-ax^{-v}J_{v+1}(ax)\)
\(\displaystyle \d{}{x}\left\{ x^{v+1}J_{v+1}(ax)\right\}=ax^{v+1}J_{v}(ax) \)
\(\displaystyle \d{}{x}\left\{ x^{v}J_{v}(x)\right\}=x^{v}J_{v-1}(x) \)
\(\displaystyle \d{}{x}\left\{ x^{v+1}J_{v+1}(x)\right\}=x^{v+1}J_{v}(x) \)
\(\displaystyle x^{v+1}J_{v+1}(x)=\int \left\{ x^{v+1}J_{v}(x) \right\} dx \)
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soluion
\(\displaystyle \int J_{v}(ax)J_{v}(bx)x dx = \int \left[ x^{v+1}J_{v}(ax) \right] \left[ x^{-v}J_{v}(bx) \right]dx \)
\(\displaystyle uv-\int v du = \left[ x^{-v}J_{v}(bx) \right] \left[ \frac{x^{v+1}}{a}J_{v+1}(ax) \right]+\frac{b}{a}\int \left[ x^{-v}J_{v+1}(bx) \right] \left[ x^{v+1}J_{v+1}(ax) \right] dx \)
see (by parts again)
\(\displaystyle \int \left[ x^{-v}J_{v+1}(bx) \right] \left[ x^{v+1}J_{v+1}(ax) \right] dx=\int \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v+1}(ax) \right] dx \)
\(\displaystyle \int \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v+1}(ax) \right] dx=uv-\int vdu \)
\(\displaystyle \int \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v+1}(ax) \right] dx=-\frac{1}{a} \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] + \frac{b}{a}\int \left[ x^{v+1}J_{v}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] dx \)
So...
\(\displaystyle \int J_{v}(ax)J_{v}(bx)x dx = \left[ x^{-v}J_{v}(bx) \right] \left[ \frac{x^{v+1}}{a}J_{v+1}(ax) \right]+\frac{b}{a}\left[ -\frac{1}{a} \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] + \frac{b}{a}\int \left[ x^{v+1}J_{v}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] \right] dx \)
\(\displaystyle \int J_{v}(ax)J_{v}(bx)x dx = \frac{a}{a^2}\left[ x^{-v}J_{v}(bx) \right] \left[ x^{v+1}J_{v+1}(ax) \right] -
\frac{b}{a} \frac{1}{a} \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] + \frac{b}{a} \frac{b}{a}\int \left[ x^{v+1}J_{v}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] dx \)
\(\displaystyle a^{2}\int J_{v}(ax)J_{v}(bx)x dx = a\left[ x^{-v}J_{v}(bx) \right] \left[ x^{v+1}J_{v+1}(ax) \right] -
b \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] + b^2 \int J_{v}(bx) J_{v}(ax) xdx \)
\(\displaystyle (a^{2}-b^{2})\int J_{v}(ax)J_{v}(bx)x dx = a\left[ x^{-v}J_{v}(bx) \right] \left[ x^{v+1}J_{v+1}(ax) \right] -
b \left[ x^{v+1}J_{v+1}(bx) \right] \left[ x^{-v}J_{v}(ax) \right] \)
\(\displaystyle (a^{2}-b^{2})\int J_{v}(ax)J_{v}(bx)x dx = ax J_{v}(bx) J_{v+1}(ax) -
bx J_{v+1}(bx) J_{v}(ax) \)
And finally...
\(\displaystyle (a^{2}-b^{2}) \int_{0}^{P} J_{v}(ax)J_{v}(bx)x \,dx = P(a J_{v}(bP) J_{v+1}(aP) - b J_{v+1}(bP) J_{v}(aP)) \)
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A Bessel function is a type of special function that is used to solve differential equations that arise in physics and engineering. It is named after the mathematician Friedrich Bessel and is denoted by the letter J.
The a^2-b^2 integral is a special type of integral that is closely related to Bessel functions. It is used to calculate the values of Bessel functions for specific values of its parameters a and b. This integral relationship is important in many applications, such as solving differential equations and in signal processing.
The a^2-b^2 integral relationship is significant because it provides a way to calculate Bessel functions, which are important in many areas of science and engineering. It also allows for the approximation of Bessel functions for values of its parameters that are outside the range of known values.
Yes, the a^2-b^2 integral relationship can be used for complex values of a and b. In fact, it is often used in complex analysis to solve problems involving Bessel functions and other special functions.
Yes, there are many practical applications of the a^2-b^2 integral relationship. Some examples include solving heat transfer problems, analyzing electromagnetic fields, and calculating the resonance frequencies of mechanical systems.