Asymptotic analysis to the WKB approx

[pleasehelpmeno]

Hi i have the WKB approx of:
$u_{+} = \sqrt{1-\frac{bm}{f}}e^{i\int f_{k} dt} + \sqrt{1+\frac{bm}{f}}e^{-i\int f_{k} dt}$

to the differential equation:
$\frac{d^{2}u_{+}}{dt^2} + [f_{k}^{2} + i(\frac{d(bm)}{dt})]u_{+} =0$

This equation can be written as:
$\frac{d^{2}u_{+}}{dN^2} + [p^2 - i + M^2]u_{+} =0$


by using $p=\frac{k}{\sqrt{\frac{\partial x}{\partial t}Cb_{*}(t_{*1})}}$ and $M=\sqrt{\frac{\partial x}{\partial t}Cb_{*}(t_{*1})}(t-t_{*})^{2}$

How to i find the asymptotic solutions for when $M\rightarrow \pm ∞$ do i need to sub in using: $f^{2}_{k}=k^2 + b^2(t_{*1})(\frac{\partial x}{\partial t}|_{*1})^2 (t-t_{*})^2$ and the above and then take the limit for M, i am a bit confused though because the broken down solutions :
$\sqrt{1+\frac{bm}{f}}\rightarrow \sqrt{2}$
$\sqrt{1-\frac{bm}{f}}\rightarrow \frac{p}{-\sqrt{2}M}$
$e^{+i\int f_{k}dt}\rightarrow (\frac{p}{-2M})^{\frac{ip}{2}}e^{-\frac{iM^2}{2}}e^{\frac{-ip^2}{4}}\pm$


contain M but surely that is $M\rightarrow \pm ∞$ so why is it there?

 

[jvicens]

Thank you for your post. It seems like you are working on a complex and interesting problem. To find the asymptotic solutions for when M\rightarrow \pm ∞, you will indeed need to substitute the expressions for p and M that you have derived into the equation. This will give you a new equation that can be solved for u_{+}.

The broken down solutions that you have mentioned do contain M, but they also contain p. As you take the limit for M, p will also approach a certain value. Therefore, the expressions for u_{+} will still contain both M and p. This is because these variables are related and cannot be separated completely.

I hope this helps clarify your confusion. Keep up the good work on your research. Good luck!