for contour integrals
(0.003 seconds)
31—40 of 53 matching pages
31: 12.14 The Function
…
►These follow from the contour integrals of §12.5(ii), which are valid for general complex values of the argument and parameter .
…
32: 1.9 Calculus of a Complex Variable
33: 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.
…
34: 36.3 Visualizations of Canonical Integrals
…
►In Figure 36.3.13(a) points of confluence of phase contours are zeros of ; similarly for other contour plots in this subsection.
…
35: 5.9 Integral Representations
…
►
…
36: 31.10 Integral Equations and Representations
…
►
31.10.1
…
►For suitable choices of the branches of the -symbols in (31.10.9) and the contour
, we can obtain both integral equations satisfied by Heun functions, as well as the integral representations of a distinct solution of Heun’s equation in terms of a Heun function (polynomial, path-multiplicative solution).
…
►
31.10.12
…
37: 3.4 Differentiation
…
►
3.4.17
…
38: 15.6 Integral Representations
…
►In (15.6.2) the point lies outside the integration contour, and assume their principal values where the contour cuts the interval , and at .
►In (15.6.3) the point lies outside the integration contour, the contour cuts the real axis between and , at which point and .
►In (15.6.4) the point lies outside the integration contour, and at the point where the contour cuts the negative real axis and .
►In (15.6.5) the integration contour starts and terminates at a point on the real axis between and .
…However, this reverses the direction of the integration contour, and in consequence (15.6.5) would need to be multiplied by .
…
39: 3.3 Interpolation
40: 36.15 Methods of Computation
…
►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 , with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints.
…