Thank you. Can you kindly point to me the reference of the asymptotics for H(,)? I couldn't find anything for negative non-integer $\lambda$. Would really appreciate your help.

Thank you Carlo. Why does the $a_1$ term diverge as $e^{\frac{x^2}{2}}$? From en.wikipedia.org/wiki/… it seems that it should diverge as $e^{x^2}$? Also why does the $a_2$ term diverge as $e^{x^2}$? I couldn't find any pointers for this. I really appreciate your help!!

Also I get a much more complicated expression for $x_1'$.

How did you get Out[99] in the pdf image? I get "Limit[X[g], g -> [Infinity], Assumptions -> True]" when I run the command.

Actually please ignore my last comment. $F>0$ is what I wanted. I got confused.

Thanks for your reply! I'll spend some time to read it. Is it possible, though, to have $F>1$ (I just edited it) instead of $F>0$ in the old post ?

I see. Thank you!!!

Where does DR come from? And I guess you meant DR<0?