A Bessel's functions of the second kind (Neumann' functions) deduction

[Rulonegger]

Homework Statement

I need to obtain the Bessel functions of the second kind, from the expressions of the Bessel functions of the first kind.

Homework Equations

Laplace equation in circular cylindrical coordinates reads
$$\nabla^2\phi(\rho,\varphi,z)=0$$ with $$\nabla^2=\frac{1}{\rho}\frac{\partial}{\partial \rho}\left(\rho \frac{\partial}{\partial \rho} \right)+\frac{1}{\rho^2}\frac{\partial^2}{\partial \varphi^2}+\frac{\partial^2}{\partial z^2}$$

The Attempt at a Solution

Supposing that $$\phi(\rho,\varphi,z)=R(\rho)\Phi(\varphi)Z(z)$$ i get $$\begin{eqnarray} Z_{m}(z)=A_{1_m}e^{m z}+A_{2_m}e^{-m z} && && \Phi_{k}(\varphi)=A_{3_k}\cos{k \varphi}+A_{4_k}\sin{k\varphi} \end{eqnarray}$$ with $m$ and $k$ being the eigenvalues from the ODE's associated with Z and $\Phi$, respectively.
Then the associated ODE for the radial component is $$\frac{d^2R}{d(k\rho)^2}+\frac{1}{k\rho}\frac{dR}{d(k\rho)}+\left(1-\frac{m^2}{(k\rho)^2}\right)R=0$$ and defining $x=k\rho$ i get $$\frac{d^2R}{d x^2}+\frac{1}{x}\frac{dR}{d x}+\left(1-\frac{m^2}{x^2}\right)R=0$$ which is Bessel's differential equation. Using Frobenius method to remove the singularity, I'm able to compute Bessel functions of the first kind $$J_{m}(x)=\sum_{s=0}^{\infty}{\frac{(-1)^s}{s!(m+s)!}\left(\frac{x}{2}\right)^{m+2s}}$$ for m integer. Changing n by -n and removing the zero terms in the series for $J_{-m}(x)$, we see that
$$J_{-m}(x)=(-1)^{m}J_{m}(x)$$ so both functions are linearly dependent, and we need to specify another function of x linearly independent of $J_{m}(x)$ because the differential equation is a second order one.
Then, a common way to find a second function of a differential equation of the form $$y''(x)+p(x)y'(x)+q(x)y(x)=0$$ with a known solution $y_{1}(x)$, we can suppose that $y_{2}(x)=y_{1}(x)g(x)$. After deriving $y_{2}$ and substituting in the differential equation, we get
$$\frac{g''}{g'}=-\frac{y_{1}'}{y_{1}}-p$$
but if i try to use this method to find the Neumann's functions $Y_{m}(x)$, i cannot find the expression
$$Y_{m}(x)=\frac{\cos{mx}J_{m}(x)-J_{-m}(x)}{\sin{mx}}$$ which commonly one finds that it is a "definition" for the Neumann's functions. Any ideas?

 

[Rulonegger]

Correction

I'm sorry, but the last expression was $$Y_{m}(x)=\frac{\cos{m\pi}J_{m}(x)-J_{-m}(x)}{\sin{m\pi}}$$ Any kind of help would be greatly appreciated!