2003 I 6

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Write \phi\left(z,t\right)=-\left(t+x/t\right). There are saddles at t=\pm x^{1/2}. We want to integrate through the saddle at t = x1 / 2, as long as it is properly directed. The direction is \sqrt{-\phi_{0}^{\prime\prime}}=\sqrt{2x/t_{0}^{3}}=\sqrt{2x^{-1/2}}=\sqrt{2}x^{-1/4}, so it is along the real axis (for positive real x). Then the integral is:

\int e^{\phi\left(x,t\right)}dt\approx\frac{\sqrt{2\pi}}{\sqrt{-\phi_{0}^{\prime\prime}}}e^{\phi_{0}\left(z\right)}=\frac{\sqrt{2\pi}}{\sqrt{2}x^{-1/4}}e^{-2x^{1/2}}=\sqrt{\pi}x^{1/4}e^{-2\sqrt{x}}

This page was recovered in October 2009 from the Plasmagicians page on Generals_2003_I_6 dated 00:56, 7 May 2007.

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