Generals 2003 I 6

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Write $\int e^{\phi\left(x,t\right)}dt$ , so $\phi\left(z,t\right)=-\left(t+x/t\right)$ . There are saddles at $t=\pm x^{1/2}$ . We want to integrate through the saddle at $t = x 1 / 2$ , 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:

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

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Category: Generals Problems