Eigenfunction integral
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HallsofIvy
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The image is not at all "self explanatory"! What is "S" (the interval the integraton is over)? The result clearly depends upon that. Also, \(\displaystyle \psi(x,y)\) and \(\displaystyle \phi(x,y,z)\) are defined as functions of x and y (and z) while in the integral they have \(\displaystyle \psi(x)\) and \(\displaystyle \phi(x)\), functions of x only.
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For any others who can't make out the image in the thumbnail, the text in the image is as follows:
\(\displaystyle \displaystyle{\mbox{Let }\, \phi_m(x,\,y)\, =\, \sin\left(\frac{m\pi x}{a}\right)\sin\left(\frac{n\pi y}{b}\right)\, \mbox{ and }\, \psi_r(x,\, y\, z)\, =\, \cos\left(\frac{p\pi x}{a}\right)\cos\left(\frac{q\pi y}{b}\right)\cos\left(\frac{r\pi z}{c}\right)}\)
\(\displaystyle \displaystyle{\mbox{To solve }\, B_{mr}\, =\, \left[\frac{C_o^2 \rho}{V\epsilon_r \epsilon_m M}\right]^{1/2}\int_S\, \psi_r (x)\phi_m (x)\, dx,\, \mbox{ let }\, K\, =\, \left(\frac{c_o^2 p}{V\epsilon_r \epsilon_m M}\right)^{1/2}}\)
\(\displaystyle \displaystyle{\mbox{My question is how to get the following:}}\)
\(\displaystyle \displaystyle{B_{mr}\, =\, K(-1)^r \left[\frac{ma}{p^2 \pi}\, \left\{\frac{(-1)^m (-1)^p\, -\, 1}{1\, -\, (m/p)^2}\right\}\right]\left[\frac{nb}{q^2 \pi}\left\{\frac{(-1)^n (-1)^q \, -\, 1}{1\, -\, (n/q)^2}\right\}\right]\, \mbox{ for }\, m\, \neq\, p,\, n \, \neq\, q}\)
\(\displaystyle \displaystyle{B_{mr}\, =\, 0\, \mbox{ for }\, (m\,+\,p)\, \mbox{ even or }\, (n\, +\, q)\, \mbox{ even.}}\)
\(\displaystyle \displaystyle{\mbox{Let }\, \phi_m(x,\,y)\, =\, \sin\left(\frac{m\pi x}{a}\right)\sin\left(\frac{n\pi y}{b}\right)\, \mbox{ and }\, \psi_r(x,\, y\, z)\, =\, \cos\left(\frac{p\pi x}{a}\right)\cos\left(\frac{q\pi y}{b}\right)\cos\left(\frac{r\pi z}{c}\right)}\)
\(\displaystyle \displaystyle{\mbox{To solve }\, B_{mr}\, =\, \left[\frac{C_o^2 \rho}{V\epsilon_r \epsilon_m M}\right]^{1/2}\int_S\, \psi_r (x)\phi_m (x)\, dx,\, \mbox{ let }\, K\, =\, \left(\frac{c_o^2 p}{V\epsilon_r \epsilon_m M}\right)^{1/2}}\)
\(\displaystyle \displaystyle{\mbox{My question is how to get the following:}}\)
\(\displaystyle \displaystyle{B_{mr}\, =\, K(-1)^r \left[\frac{ma}{p^2 \pi}\, \left\{\frac{(-1)^m (-1)^p\, -\, 1}{1\, -\, (m/p)^2}\right\}\right]\left[\frac{nb}{q^2 \pi}\left\{\frac{(-1)^n (-1)^q \, -\, 1}{1\, -\, (n/q)^2}\right\}\right]\, \mbox{ for }\, m\, \neq\, p,\, n \, \neq\, q}\)
\(\displaystyle \displaystyle{B_{mr}\, =\, 0\, \mbox{ for }\, (m\,+\,p)\, \mbox{ even or }\, (n\, +\, q)\, \mbox{ even.}}\)
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