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  "body": "\n## Summary\n\nThe method of steepest descent, also called the saddle point method, finds the asymptotic behaviour of contour integrals $\\int_C g(z)\\, e^{\\lambda f(z)}\\,dz$ as the parameter $\\lambda$ grows.\nOne deforms the contour so that it passes through [[saddle points|0JW6MDHAZD]] of the holomorphic function $f$ along paths on which $\\operatorname{Im} f$ is constant.\nOn these paths the integrand does not oscillate, and [[Laplace's method|9G7K8DBT9Y]] gives the leading terms.\nIt yields the asymptotics of many special functions, such as the Airy and Bessel functions, and of coefficients of generating functions.\n\n## Explanations\n\n### L1 human\n\nPicture $\\operatorname{Re} f$ as a landscape over the complex plane, with mountains and valleys.\nA holomorphic function has no peaks or pits, only passes, the saddle points.\nFor large $\\lambda$ the integrand is tiny everywhere except where $\\operatorname{Re} f$ is largest along the path.\nThe trick is to walk from one valley to the next over the lowest pass, along the steepest route.\nThen the integrand is concentrated at the pass and does not oscillate.\nFor example, rotating the line of integration of $\\int_{\\R} e^{i\\lambda x^2}\\,dx$ by $45$ degrees turns it into a real Gaussian integral and gives $\\sqrt{\\pi/\\lambda}\\, e^{i\\pi/4}$.\n",
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