Context
I am working with three-dimensional Green's functions. A Green's function must satisfies $$ 1 = \lim_{\boldsymbol{\varepsilon}\to0 } \int_{\mathbf{x}^{\prime}-\boldsymbol{\varepsilon}}^{\mathbf{x}^{\prime}+\boldsymbol{\varepsilon}} L[G(\mathbf{x};\mathbf{x}^\prime)], $$
In the problem that I am working, which is a three-dimensional Laplacian operator with cylindrical boundary conditions, this requirement becomes \begin{align*} 1 &= \lim_{\varepsilon\to 0} \int_{\rho = \rho^\prime -\varepsilon}^{ \rho^\prime +\varepsilon} \int_{\varphi = 0}^{ 2\,\pi} \int_{z = 0}^{L} \nabla^2 G \, dV. \end{align*} After some effort, this requirement reduces to the following integral: \begin{align*} 1&= \lim_{\varepsilon\to 0} \int\limits_{-\infty}^{\infty} \int\limits_{0}^{2\,\pi} \left[ \frac{\partial G }{\partial \rho} \right]_{\rho=\rho^\prime + \varepsilon} - \left[ \frac{\partial G }{\partial \rho} \right]_{\rho=\rho^\prime - \varepsilon} \, \rho\, d\varphi\,dz \,. \end{align*} I now substitute in my Green's function, which I almost positive is correct up to multiplicative constants (i.e., $k$ in the equation below). When I substitute in, the requirement becomes \begin{align*} 1&= \int\limits_{0}^{2\,\pi} \sum\limits_{m=0}^\infty \dfrac{ \left[ a_m\cos{\left(m\,\varphi\right)} + b_m\sin{\left(m\,\varphi\right)} \right] \left[ a_m\cos{\left(m\,\varphi^\prime\right)} + b_m\sin{\left(m\,\varphi^\prime\right)} \right] } { 2\,\pi \left|a_m\right|^2 \,\delta_{m,0} + \pi \left(\left|a_m\right|^2 + \left|b_m\right|^2\right) \left(1-\delta_{m,0}\right) } \, d\varphi \\ &\times \int\limits_{-\infty}^{\infty} \int\limits_{0}^{\infty} k\, \exp{\left(-l\bigl[\left|z\right|+ \left|z^\prime\right|\bigr]\right)} \, dl \,dz \\ &\times \lim_{\rho \to \rho^\prime} \dfrac{ \begin{bmatrix} + \left[ J_m{\left(l\,\rho^\prime\right)} \, Y_m{\left(l\,a\right)} - J_m{\left(l\,a\right)} \,Y_m{\left(l\,\rho^\prime\right)} \right] \left[ \frac{dJ_m{\left(l\,\rho \right)}}{\rho} Y_m{\left(l\,b\right)} - J_m{\left(l\,b\right)} \frac{dY_m{\left(l\,\rho \right)}}{\rho} \right] \\ - \left[ \frac{dJ_m{\left(l\,\rho \right)}}{dr} \, Y_m{\left(l\,a\right)} - J_m{\left(l\,a\right)} \frac{dY_m{\left(l\,\rho\right)}}{dr} \right] \left[ J_m{\left(l\,\rho^\prime \right)} \, Y_m{\left(l\,b\right)} - J_m{\left(l\,b\right)} \,Y_m{\left(l\,\rho^\prime \right)} \right] \end{bmatrix} } { \frac{2 }{\pi\,\rho }\, \left[ J_m{\left(l\,a\right)} \, Y_m{\left(l\,b\right)} - J_m{\left(l\,b\right)} \, Y_m{\left(l\,a\right)} \right] } \,. \end{align*} To show that the this equation is consistent, I have to show a few things. This includes that (derivation not included for pithiness) $$ \lim_{\rho \to \rho^\prime} \dfrac{ \begin{bmatrix} + \left[ J_m{\left(l\,\rho^\prime\right)} \, Y_m{\left(l\,a\right)} - J_m{\left(l\,a\right)} \,Y_m{\left(l\,\rho^\prime\right)} \right] \left[ \frac{dJ_m{\left(l\,\rho \right)}}{\rho} Y_m{\left(l\,b\right)} - J_m{\left(l\,b\right)} \frac{dY_m{\left(l\,\rho \right)}}{\rho} \right] \\ - \left[ \frac{dJ_m{\left(l\,\rho \right)}}{dr} \, Y_m{\left(l\,a\right)} - J_m{\left(l\,a\right)} \frac{dY_m{\left(l\,\rho\right)}}{dr} \right] \left[ J_m{\left(l\,\rho^\prime \right)} \, Y_m{\left(l\,b\right)} - J_m{\left(l\,b\right)} \,Y_m{\left(l\,\rho^\prime \right)} \right] \end{bmatrix} } { \frac{2 }{\pi\,\rho }\, \left[ J_m{\left(l\,a\right)} \, Y_m{\left(l\,b\right)} - J_m{\left(l\,b\right)} \, Y_m{\left(l\,a\right)} \right] } =1\,.$$ This also includes that $$ \int\limits_{0}^{2\,\pi} \sum\limits_{m=0}^\infty \dfrac{ \left[ a_m\cos{\left(m\,\varphi\right)} + b_m\sin{\left(m\,\varphi\right)} \right] \left[ a_m\cos{\left(m\,\varphi^\prime\right)} + b_m\sin{\left(m\,\varphi^\prime\right)} \right] } { 2\,\pi \left|a_m\right|^2 \,\delta_{m,0} + \pi \left(\left|a_m\right|^2 + \left|b_m\right|^2\right) \left(1-\delta_{m,0}\right) } \, d\varphi = \int\limits_{0}^{2\,\pi} \delta{(\varphi-\varphi^\prime)}\, d\varphi, $$ which i explain by the resolution of the identity. Ultimately, the requirement on the Green's function is reduced to showing that \begin{align*} 1&= \int\limits_{-\infty}^{\infty} \left[ \int\limits_{0}^{\infty} k\, \exp{\left(-l\bigl[\left|z\right|+ \left|z^\prime\right|\bigr]\right)} \, dl \right]dz \,. \tag{A} \end{align*} So, from here I ask myself the questions that follow.
Questions
Q.1. Can I use the resolution of the identity on exponential functions (cf, the the bracketed factor in Eq. A), such as
$$
\int\limits_{l=0}^{\infty}
k\,
\exp{\left(-l\bigl[\left|z\right|+ \left|z^\prime\right|\bigr]\right)}
\,
dl
\, ?
$$
Q.2. Why or why not?