# contour

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## 1—10 of 58 matching pages

##### 1: 9.14 Incomplete Airy Functions
Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …
##### 2: 9.15 Mathematical Applications
Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …
##### 3: 36.15 Methods of Computation
###### §36.15(iii) Integration along Deformed Contour
Direct numerical evaluation can be carried out along a contour that runs along the segment of the real $t$-axis containing all real critical points of $\Phi$ and is deformed outside this range so as to reach infinity along the asymptotic valleys of $\exp\left(i\Phi\right)$. …
###### §36.15(iv) Integration along Finite Contour
This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of $\Phi$, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …
##### 4: 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. …
##### 5: 12.16 Mathematical Applications
PCFs are used as basic approximating functions in the theory of contour integrals with a coalescing saddle point and an algebraic singularity, and in the theory of differential equations with two coalescing turning points; see §§2.4(vi) and 2.8(vi). …
##### 6: 13.4 Integral Representations
###### §13.4(ii) Contour Integrals
The contour of integration starts and terminates at a point $\alpha$ on the real axis between $0$ and $1$. …At the point where the contour crosses the interval $(1,\infty)$, $t^{-b}$ and the ${{}_{2}{\mathbf{F}}_{1}}$ function assume their principal values; compare §§15.1 and 15.2(i). …The contour cuts the real axis between $-1$ and $0$. …
##### 7: 5.12 Beta Function
with the contour as shown in Figure 5.12.1. … when $\Re b>0$, $a$ is not an integer and the contour cuts the real axis between $-1$ and the origin. … where the contour starts from an arbitrary point $P$ in the interval $(0,1)$, circles $1$ and then $0$ in the positive sense, circles $1$ and then $0$ in the negative sense, and returns to $P$. It can always be deformed into the contour shown in Figure 5.12.3.
##### 8: 27.18 Methods of Computation: Primes
An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). …
##### 9: 15.6 Integral Representations
In (15.6.2) the point $\ifrac{1}{z}$ lies outside the integration contour, $t^{b-1}$ and $(t-1)^{c-b-1}$ assume their principal values where the contour cuts the interval $(1,\infty)$, and $(1-zt)^{a}=1$ at $t=0$. In (15.6.3) the point $\ifrac{1}{(z-1)}$ lies outside the integration contour, the contour cuts the real axis between $t=-1$ and $0$, at which point $\operatorname{ph}t=\pi$ and $\operatorname{ph}\left(1+t\right)=0$. In (15.6.4) the point $\ifrac{1}{z}$ lies outside the integration contour, and at the point where the contour cuts the negative real axis $\operatorname{ph}t=\pi$ and $\operatorname{ph}\left(1-t\right)=0$. In (15.6.5) the integration contour starts and terminates at a point $A$ on the real axis between $0$ and $1$. …However, this reverses the direction of the integration contour, and in consequence (15.6.5) would need to be multiplied by $-1$. …
##### 10: 12.5 Integral Representations
###### §12.5(ii) Contour Integrals
For the particular loop contour, see Figure 5.9.1. … where the contour separates the poles of $\Gamma\left(t\right)$ from those of $\Gamma\left(\tfrac{1}{2}+a-2t\right)$. … where the contour separates the poles of $\Gamma\left(t\right)$ from those of $\Gamma\left(\tfrac{1}{2}-a-2t\right)$. …