How do you work out, the 'Delta Function Normalization', of a 'Regular Coulomb Wave Function'? A one dimensional momentum eigenfunction $ e^{ikx}$, can be normalized in terms of the 'Dirac Delta Function', because we can write that, for some value of $C$
\begin{equation*}
\int_{-\infty}^\infty e^{-ik^\prime x} e^{ikx} dx = C \delta( k-k^\prime)
\end{equation*}
$C$ can be evaluated, see the materials at
https://sites.google.com/site/rwhawksworthexamples/home/math/dirac-delta-function-various-pages/delta-function-normalization/momentum-eigenfunctions
within which it is proved, that $C$ has the value $2\pi$.
I think that a 'Regular Coulomb Wave Function', $F_L(\eta,~\rho)$, can also be normalised in terms of the delta function, as
\begin{equation*}
\int _0^\infty F_L(\eta^\prime,~k^\prime r) F_L(\eta,~kr) dr = N_L \delta(k-k^\prime)
\end{equation*}
NB: we could put
\begin{equation*}
kr=\rho,  ~~~~~~ k^\prime r = \rho^\prime
\end{equation*}
and also, my understanding is, that $F_L(\eta,~ \rho)$ is a real valued function, so it's complex conjugate function, is itself.
See $\mathbf{14.1.3}$ in 'Other Information', for the definition of $F_L(\eta, ~\rho)$, this equation is quoted from, Abramowitz and Stegun, see at
https://archive.org/details/AandS-mono600/page/n551/mode/2up
Could anyone provide a proof of what $N_L$ should be? I think a good guess is $N_L=1$ or $\frac{1}{k}$.
I have started a proof of what $N_L$ should be, but it looks as if it's going to be a very long one, I may never finish it.
Other Information
Some equations are quoted here, from Abramowitz and Stegun  see the provided link
$\mathbf{14.1.1}$
\begin{equation*}
\frac{  d^2 \omega   }{ d \rho^2     }
+ \left[ 1- \frac{ 2 \eta  }{\rho  } - \frac{ L(L+1)  }{ \rho^2  }  \right] \omega = 0
\end{equation*}
$(\rho >0, -\infty < \eta < \infty$, L a non-negative integer$)$
$\mathbf{14.1.2}$
\begin{equation*}
~~~~~~~~~~~~~~~~~~~~~~~\omega = C_1F_L(\eta, ~\rho)+C_2G_L(\eta,~ \rho)~~~~~~~~~(C_1, ~~C_2~~constants)
\end{equation*}
where $F_L(\eta, ~\rho)$ is the regular Coulomb wave function and $G_L(\eta,~ \rho)$ is the irregular (logarithmic) Coulomb wave function.
$\mathbf{14.1.3}$
\begin{equation*}
F_L(\eta,~ \rho)= C_L(\eta) \rho^{L+1} e^{  -i\rho  }~M( L+1-i\eta,~2L+2,~2i\rho   )
\end{equation*}
$\mathbf{14.1.7}$
\begin{equation*}
C_L(\eta)= ~\frac{ 2^L~e{ \frac{  -\pi\eta}{ 2  }  } ~ |~\Gamma(L+1+i\eta~) | } 
                            { \Gamma(2L+2)  }~~~~~~~~~~~~~~~~~~~
\end{equation*}
$\mathbf{14.1.14}$
\begin{equation*}
G_L(\eta,~ \rho)=\frac{ 2\eta  }  { C_0^2(\eta)  }~F_L(\eta,~ \rho)
\left[ ~ln~2\rho+ \frac{q_L(\eta)  }{ p_L(\eta)  } \right]
+\theta_L(\eta,~\rho)
\end{equation*}
To get on top of the definition of the irregular solution, quite a number of other results need to be taken into account.
Related Questions,
A problem with analysing the 'Delta Function Normalization', of an 'Irregular Coulomb Wave Function'.
Is this the way to 'Delta Function Normalise' a 'Continuum Wave Function'?
About the idea that, the "Normalization" of a "scattering wavefunction", being insensitive to the functions form near to the scattering centre.
https://physics.stackexchange.com/questions/635101/finding-two-remarks-in-diracs-the-principles-of-quantum-mechanics
 A: I have a link to what I thought was a proof, in a very restricted sense, of what $N_L$ is.
Please see at
https://sites.google.com/site/rwhawksworthexamples/home/math/dirac-delta-function-various-pages/delta-function-normalization/regular-coulomb-wave-function
Unfortunately my analysis of one of the limits involved, the one called $L_{2,3}$, is flawed.
Various results are incorrect for the special case, k= k’ ( essentially, “0 divided by 0” possibilities,  escaped my attention ).
I might, in the future, bring into the analysis of $L_{2,3}$, the idea of ‘the limit of a function’ and material on arg ( Gamma(z) ).
I thought I had proved that
\begin{equation*}
N_L = \frac{\pi}{2}
\end{equation*}
So that we would have had
\begin{equation*}
\int _0^\infty F_L(\eta,~kr)~F_L(\eta^\prime,~k^\prime r) ~ dr = \frac{\pi}{2} \delta(k-k^\prime)
\end{equation*}
NB:  $N_{L,F}$ is used in the  PDF found at the linked to web page, not $N_L$.
Originally, I thought there would be a lot more integrals, and hence limits, to be looked at in the proof.
NB: I think this value for $N_L$, can still be justified by a correct proof.
NB: I did not have to perform any integrals containing the '$ln$' function.
