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I am working with Whittaker functions for a project and have no experience with asymptotic analysis - how is the following expression, for $\kappa \rightarrow \infty$ through the real numbers \begin{equation}\mathop{W_{\kappa,\mu}}\nolimits\!\left(x\right)=\sqrt{x}\mathop{\Gamma}% \nolimits\!\left(\kappa+\tfrac{1}{2}\right)\left(\mathop{\sin}\nolimits\!% \left(\kappa\pi-\mu\pi\right)\mathop{J_{2\mu}}\nolimits\!\left(2\sqrt{x% \kappa}\right)-\mathop{\cos}\nolimits\!\left(\kappa\pi-\mu\pi\right)\mathop{% Y_{2\mu}}\nolimits\!\left(2\sqrt{x\kappa}\right)+\mathop{\mathrm{env}Y_{2\mu% }}\nolimits\!\left(2\sqrt{x\kappa}\right)\mathop{O}\nolimits\!\left(\kappa% ^{-\frac{1}{2}}\right)\right),\end{equation}

Derived? My goal is to extend this form for $\kappa$ imaginary and see something like an exponential decay.

Schwinger
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  • what is "env"?! – tired Aug 14 '15 at 16:42
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    Can you give a source for this expression? – Antonio Vargas Aug 14 '15 at 22:04
  • Sorry, this is from the NIST handbook of special functions: http://dlmf.nist.gov/13.21. Env is the so-called "envelope" function which is defined on the handbook as $\mathop{\mathrm{env}Y_{\nu}/}\nolimits(x)=\sqrt{2}\mathop{|Y_{\nu} (x)|}$ from zero to the smallest possible root $X_{\nu}$ and $env Y_{\nu} (x) = (J_{\nu}^2(x) + Y_{\nu}^2(x))^{1/2}$ for values greater than $X_{\nu}$ source: http://dlmf.nist.gov/2.8#E33 – Schwinger Aug 16 '15 at 22:42

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