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Asymptotic analysis to the WKB approx

  1. Jan 27, 2013 #1
    Hi i have the WKB approx of:
    [itex]u_{+} = \sqrt{1-\frac{bm}{f}}e^{i\int f_{k} dt} + \sqrt{1+\frac{bm}{f}}e^{-i\int f_{k} dt}[/itex]

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

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


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

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


    contain M but surely that is [itex]M\rightarrow \pm ∞ [/itex] so why is it there?
     
  2. jcsd
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