Pragabhava
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 Nov28 reviewed Leave Open Assumed True until proven False. The Curious Case of the Vacuous Truth Nov25 revised Separating Partial Differential Eq added 143 characters in body Nov25 comment Separating Partial Differential Eq Nov25 revised Separating Partial Differential Eq texified it Nov25 comment Separating Partial Differential Eq The separation I've shown you is correct. What is $a$ and $b$? As I told you, the relationship between $C$ and the separation constant comes from the boundary conditions. Why don't you edit your question and show us how did you derived the relation $a^2 + b^2 = C^2$? Here is a quick guide on how to typeset equations. Nov25 awarded Informed Nov25 comment Separating Partial Differential Eq Well, if by a relationship you mean $C = C(\lambda)$, then no. The solution of the system is \begin{align}\Theta(\theta) &= A e^{\sqrt{\lambda} x} + B e^{-\sqrt{\lambda} x} \\ R(r) &= C J_\sqrt{\lambda}(C x) + D Y_\sqrt{\lambda}(C x)\end{align} where $J_\sqrt{\lambda}$ and $Y_\sqrt{\lambda}$ are Bessel and Neumann functions of the first kind, of order $\sqrt{\lambda}$. If no boundary conditions are given, then you have an infinite collection of solutions spanned by both parameters. Nov25 comment Separating Partial Differential Eq I'm guessing you're getting Helmoltz equation from separating time and space from a cylindrical wave equation. If you need to quantize $\lambda$ and $C$, then boundary conditions are needed. Nov25 answered Separating Partial Differential Eq Nov15 awarded Yearling Sep10 comment Solution of a differentiation in integral form @ComplexGuy Indeed. Sep10 comment Solution of a differentiation in integral form @ComplexGuy It gets absorbed in the definition of $\hat{c}_1(k)$. I abused the notation, sorry! Sep10 comment Solution of a differentiation in integral form @ComplexGuy When $t=0$, $\sin(\omega t) = 0$, and the equality follows. Sep10 revised Comparison between Bessel's coefficients deleted 21 characters in body Sep10 answered Comparison between Bessel's coefficients Jul22 comment Perturbation Theory on Finite Domains You should read about the WKB approximation. A great treatise on the asymptotics of $y'' + p(x) y = 0$ can be found in chapter 6 of Olver's Asymptotics and Special Functions. Jul14 comment Nonlinear equation (oscillon) comparison @ComplexGuy en.wikipedia.org/wiki/Virtual_work Jul13 revised General Solution of a Differential Equation using Green's Function added 255 characters in body Jul13 revised General Solution of a Differential Equation using Green's Function added 3 characters in body Jul13 revised General Solution of a Differential Equation using Green's Function added 3 characters in body