reparametrization of integration paths

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11: 16.17 Definition

… ►where the integration path $L$ separates the poles of the factors $\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-s\right)$ from those of the factors $\mathrm{\Gamma }\left(1-a_{\mathrm{\ell }}+s\right)$. … ►

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Figure 16.17.1: s-plane. Path $L$ for the integral representation (16.17.1) of the Meijer $G$-function.

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12: 4.37 Inverse Hyperbolic Functions

… ►In (4.37.1) the integration path may not pass through either of the points $t=\pm \mathrm{i}$, and the function ${(1+t^2)}^{1/2}$ assumes its principal value when $t$ is real. In (4.37.2) the integration path may not pass through either of the points $\pm 1$, and the function ${(t^2-1)}^{1/2}$ assumes its principal value when $t\in (1,\mathrm{\infty })$. Elsewhere on the integration paths in (4.37.1) and (4.37.2) the branches are determined by continuity. In (4.37.3) the integration path may not intersect $\pm 1$. … ►The principal values (or principal branches) of the inverse $\sinh$, $\cosh$, and $\tanh$ are obtained by introducing cuts in the $z$-plane as indicated in Figure 4.37.1(i)-(iii), and requiring the integration paths in (4.37.1)–(4.37.3) not to cross these cuts. …

13: 10.74 Methods of Computation

… ►A comprehensive and powerful approach is to integrate the differential equations (10.2.1) and (10.25.1) by direct numerical methods. As described in §3.7(ii), to insure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows in magnitude at least as fast as all other solutions of the differential equation. ►In the interval , $J_{\nu }\left(x\right)$ needs to be integrated in the forward direction and $Y_{\nu }\left(x\right)$ in the backward direction, with initial values for the former obtained from the power-series expansion (10.2.2) and for the latter from asymptotic expansions (§§10.17(i) and 10.20(i)). In the interval either direction of integration can be used for both functions. ►Similarly, to maintain stability in the interval the integration direction has to be forwards in the case of $I_{\nu }\left(x\right)$ and backwards in the case of $K_{\nu }\left(x\right)$, with initial values obtained in an analogous manner to those for $J_{\nu }\left(x\right)$ and $Y_{\nu }\left(x\right)$. …

14: 13.29 Methods of Computation

… ►A comprehensive and powerful approach is to integrate the differential equations (13.2.1) and (13.14.1) by direct numerical methods. As described in §3.7(ii), to insure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows in magnitude at least as fast as all other solutions of the differential equation. ►For $M(a,b,z)$ and $M_{\kappa ,\mu }\left(z\right)$ this means that in the sector $|\mathrm{ph}z|\le \pi$ we may integrate along outward rays from the origin with initial values obtained from (13.2.2) and (13.14.2). … ►In the sector the integration has to be towards the origin, with starting values computed from asymptotic expansions (§§13.7 and 13.19). On the rays $\mathrm{ph}z=\pm \frac{1}{2}\pi$, integration can proceed in either direction. …

15: 31.6 Path-Multiplicative Solutions

§31.6 Path-Multiplicative Solutions

►A further extension of the notation (31.4.1) and (31.4.3) is given by …These solutions are called path-multiplicative. …

16: 11.5 Integral Representations

… ►In (11.5.8) and (11.5.9) the path of integration separates the poles of the integrand at $s=0,1,2,\mathrm{\dots }$ from those at $s=-1,-2,-3,\mathrm{\dots }$. …

17: 8.21 Generalized Sine and Cosine Integrals

… ►(obtained from (5.2.1) by rotation of the integration path) is also needed. … ►In these representations the integration paths do not cross the negative real axis, and in the case of (8.21.4) and (8.21.5) the paths also exclude the origin. …

18: 3.5 Quadrature

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§3.5(iii) Romberg Integration

►Further refinements are achieved by Romberg integration. … ►For these cases the integration path may need to be deformed; see §3.5(ix). … ►

§3.5(ix) Other Contour Integrals

… ►A special case is the rule for Hilbert transforms (§1.14(v)): …

19: 16.5 Integral Representations and Integrals

… ►where the contour of integration separates the poles of $\mathrm{\Gamma }\left(a_k+s\right)$, $k=1,\mathrm{\dots },p$, from those of $\mathrm{\Gamma }\left(-s\right)$. … ►In (16.5.2)–(16.5.4) all many-valued functions in the integrands assume their principal values, and all integration paths are straight lines. … ►Lastly, the restrictions on the parameters can be eased by replacing the integration paths with loop contours; see Luke (1969a, §3.6). …

20: 19.13 Integrals of Elliptic Integrals

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§19.13(i) Integration with Respect to the Modulus

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§19.13(ii) Integration with Respect to the Amplitude

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