# contour

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## 1—10 of 58 matching pages

##### 1: 9.14 Incomplete Airy Functions

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►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.
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##### 2: 9.15 Mathematical Applications

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►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.
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##### 3: 36.15 Methods of Computation

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###### §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 $\mathrm{\Phi}$ and is deformed outside this range so as to reach infinity along the asymptotic valleys of $\mathrm{exp}\left(\mathrm{i}\mathrm{\Phi}\right)$. … ►###### §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 $\mathrm{\Phi}$, 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

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►Another approach is to apply numerical quadrature (§3.5) to the integral (5.9.2), using paths of steepest descent for the contour.
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##### 5: 12.16 Mathematical Applications

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►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).
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##### 6: 13.4 Integral Representations

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###### §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 $(1,\mathrm{\infty})$, ${t}^{-b}$ 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

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►with the contour as shown in Figure 5.12.1.
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►when $\mathrm{\Re}b>0$, $a$ is not an integer and the contour cuts the real axis between $-1$ and the origin.
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►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.
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##### 8: 27.18 Methods of Computation: Primes

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►An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000).
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##### 9: 15.6 Integral Representations

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►In (15.6.2) the point $1/z$ lies outside the integration contour, ${t}^{b-1}$ and ${(t-1)}^{c-b-1}$ assume their principal values where the contour cuts the interval $(1,\mathrm{\infty})$, and ${(1-zt)}^{a}=1$ at $t=0$.
►In (15.6.3) the point $1/(z-1)$ lies outside the integration contour, the contour cuts the real axis between $t=-1$ and $0$, at which point $\mathrm{ph}t=\pi $ and $\mathrm{ph}\left(1+t\right)=0$.
►In (15.6.4) the point $1/z$ lies outside the integration contour, and at the point where the contour cuts the negative real axis $\mathrm{ph}t=\pi $ and $\mathrm{ph}\left(1-t\right)=0$.
►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 $-1$.
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##### 10: 12.5 Integral Representations

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