# steepest-descent paths

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##### 1: 5.21 Methods of Computation
Another approach is to apply numerical quadrature (§3.5) to the integral (5.9.2), using paths of steepest descent for the contour. …
For example, steepest descent paths can be used; see §2.4(iv). … The steepest descent path is given by $\Im(t-2\sqrt{t})=0$, or in polar coordinates $t=re^{i\theta}$ we have $r={\sec^{2}}\left(\frac{1}{2}\theta\right)$. … A special case is the rule for Hilbert transforms (§1.14(v)): …
where $j$ denotes a real critical point (36.4.1) or (36.4.2), and $\mu$ denotes a critical point with complex $t$ or $s,t$, connected with $j$ by a steepest-descent path (that is, a path where $\Re\Phi=\mathrm{constant}$) in complex $t$ or $(s,t)$ space. …
Paths on which $\Im(zp(t))$ is constant are also the ones on which $|\exp\left(-zp(t)\right)|$ decreases most rapidly. …However, for the purpose of simply deriving the asymptotic expansions the use of steepest descent paths is not essential. …