I am studying the following equation:
$$\dfrac{d^2}{dr^{*2}}R_k^l(r^*)=(V_{eff}-k^2)R_k^l(r^*)$$ where the potential is given by:
$$V_{eff}(r)=\left( 1-\dfrac{2M}{r} \right)\left( \dfrac{2M}{r^3} +\dfrac{l(l+1)}{r^2} \right)$$
and the coordinate $r^*$ is given by
$$r^*=r+2M \log(r/2M-1)$$
The constants $M$ and $l$ are both positive and $l$ can have only integer values. According to the text (Gravitation by Misner, Thorne and Wheeler page 869 if needed) the function $R_k^L$ has the following asymptotic expansion:
$$R_k^l=\exp(ikr^*)+\Gamma \exp(-ikr^*) \quad r^*\rightarrow-\infty$$ $$R_k^l=T \exp(ikr^*) \quad r^*\rightarrow +\infty$$
And the coefficients $\Gamma$ and $T$ are given by
$$\Gamma=-1+2i\alpha M k$$ $$T=\dfrac{\beta}{(2l-1)!!} (2Mik)^{l+1}$$ for $k\ll1/M$. I do not understand how to calculate the coefficients. Can anyone help?