I'm relatively new to Fourier transforms so apologize in advance if this problem seems trivial.
In order to solve a second order PDE I have defined the following sine Fourier transform
$$E(r,\lambda)=\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\eta(r,z)\cos(\lambda z)\,\textrm{d}z.$$
By doing so I arrive at the following solution for $E$ (which I know is correct)
$$E(r,\lambda)=\frac{1}{\sqrt{2\pi}}\frac{1}{\lambda}\left\{-\frac{rI_{0}(\lambda r)}{I_{1}(\lambda)}+\frac{I_{1}(\lambda)I_{1}(\lambda r)}{2I_{0}(\lambda)I_{1}(\lambda)-\lambda[I_{0}^{2}(\lambda)-I_{1}^{2}(\lambda)]}\right\},$$
where $I_{\alpha}$ is the modified Bessel function of the first kind.
My goal is to invert the Fourier transform and obtain the solution for $\eta$. So far I have that
\begin{align*} \eta(r,z)&=\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}E(r,\lambda)\cos(\lambda z)\,\textrm{d}\lambda \\ &=\frac{1}{\pi}\int_{0}^{\infty}\left\{-\frac{rI_{0}(\lambda r)}{I_{1}(\lambda)}+\frac{I_{1}(\lambda)I_{1}(\lambda r)}{2I_{0}(\lambda)I_{1}(\lambda)-\lambda[I_{0}^{2}(\lambda)-I_{1}^{2}(\lambda)]}\right\}\frac{\cos(\lambda z)}{\lambda}\,\textrm{d}\lambda. \end{align*}
This looks horrible so, to start, I've decided to focus on the first part of this integral
$$F_{1}=-\frac{r}{\pi}\int_{0}^{\infty}\frac{I_{0}(\lambda r)}{I_{1}(\lambda)}\frac{\cos(\lambda z)}{\lambda}\,\textrm{d}\lambda.$$
Using Erdelyi's 'Higher Transcendental Functions' I have the following
\begin{align*} F_{1}&=-\frac{r}{\pi}\int_{0}^{\infty}\frac{I_{0}(\lambda r)}{I_{1}(\lambda)}\frac{\cos(\lambda z)}{\lambda}\,\textrm{d}\lambda \\ &=-\frac{r}{\pi}\int_{0}^{\infty}\frac{J_{0}(\textrm{i}\lambda r)}{J_{1}(\textrm{i}\lambda)}\frac{\cos(\lambda z)}{\lambda}\,\textrm{d}\lambda \\ &=\frac{r}{\pi}\int_{0}^{\infty}\frac{J_{2}(\textrm{i}\lambda r)}{J_{1}(\textrm{i}\lambda)}\frac{\cos(\lambda z)}{\lambda}\,\textrm{d}\lambda+\frac{2\textrm{i}}{\pi}\int_{0}^{\infty}\frac{J_{1}(\textrm{i}\lambda r)}{J_{1}(\textrm{i}\lambda)}\frac{\cos(\lambda z)}{\lambda^{2}}\,\textrm{d}\lambda \\ &=\frac{2\textrm{i}r}{\pi}\int_{0}^{\infty}\sum_{n=1}^{\infty}\frac{J_{2}(\beta_{n}r)}{(\beta_{n}^{2}+\lambda^{2})J_{2}(\beta_{n})}\cos(\lambda z)\,\textrm{d}\lambda \\ &\quad+\frac{4\textrm{i}}{\pi}\int_{0}^{\infty}\sum_{n=1}^{\infty}\left[\frac{J_{1}(\beta_{n}r)}{\beta_{n}J_{2}(\beta_{n})}-\frac{\lambda^{2}J_{1}(\beta_{n}r)}{\beta_{n}(\beta_{n}^{2}+\lambda^{2})J_{2}(\beta_{n})}\right]\frac{\cos(\lambda z)}{\lambda^{2}}\,\textrm{d}\lambda \\ &=\frac{2\textrm{i}r}{\pi}\sum_{n=1}^{\infty}\frac{J_{2}(\beta_{n}r)}{J_{2}(\beta_{n})}\int_{0}^{\infty}\frac{\cos(\lambda z)}{\beta_{n}^{2}+\lambda^{2}}\,\textrm{d}\lambda \\ &\quad+\frac{4\textrm{i}}{\pi}\sum_{n=1}^{\infty}\frac{J_{1}(\beta_{n}r)}{\beta_{n}J_{2}(\beta_{n})}\int_{0}^{\infty}\frac{\cos(\lambda z)}{\lambda^{2}}\,\textrm{d}\lambda-\frac{4\textrm{i}}{\pi}\sum_{n=1}^{\infty}\frac{J_{1}(\beta_{n}r)}{\beta_{n}J_{2}(\beta_{n})}\int_{0}^{\infty}\frac{\cos(\lambda z)}{\beta_{n}^{2}+\lambda^{2}}\,\textrm{d}\lambda \\ &=-\frac{2\textrm{i}r}{\pi}\sum_{n=1}^{\infty}\frac{J_{0}(\beta_{n}r)}{J_{2}(\beta_{n})}\int_{0}^{\infty}\frac{\cos(\lambda z)}{\beta_{n}^{2}+\lambda^{2}}\,\textrm{d}\lambda +\frac{2\textrm{i}r}{\pi}\int_{0}^{\infty}\frac{\cos(\lambda z)}{\lambda^{2}}\,\textrm{d}\lambda \\ &=\textrm{i}r\sum_{n=1}^{\infty}\frac{J_{0}(\beta_{n}r)e^{-\beta_{n}z}}{\beta_{n}J_{0}(\beta_{n})}+\frac{2\textrm{i}r}{\pi}\int_{0}^{\infty}\frac{\cos(\lambda z)}{\lambda^{2}}\,\textrm{d}\lambda \end{align*}
The quoted solution I'm working towards is
$$\eta=-r\sum_{n=1}^{\infty}\frac{J_{0}(\beta_{n}r)e^{-\beta_{n}z}}{\beta_{n}J_{0}(\beta_{n})}+\frac{1}{2}\mathbb{R}\left[\textrm{i}\sum_{m=1}^{\infty}\frac{I_{1}(k_{m})I_{1}(rk_{m})e^{\textrm{i}k_{m}z}}{k_{m}I_{1}^{2}(k_{m})}\right],$$
where $k_{m}$ are the zeros of $2I_{0}(k)I_{1}(k)-k[I_{0}^{2}(k)-I_{1}^{2}(k)]$. I'm assuming that the first part of this solution comes from the first part of the integral that I have defined. I can see that I'm relatively close to the first part of the quoted solution but must be making some mistakes along the way.
One big problem is the
$$\int_{0}^{\infty}\frac{\cos(\lambda z)}{\lambda^{2}}\,\textrm{d}\lambda,$$
term that I've got - this integral doesn't converge!
Can anyone offer any help and advice?
Much appreciated. Thanks