along%20real%20line
(0.004 seconds)
1—10 of 684 matching pages
1: 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 -axis containing all real critical points of and is deformed outside this range so as to reach infinity along the asymptotic valleys of . … ►§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 , with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …2: 36.5 Stokes Sets
…
►where denotes a real critical point (36.4.1) or (36.4.2), and denotes a critical point with complex or , connected with by a steepest-descent path (that is, a path where ) in complex or space.
►In the following subsections, only Stokes sets involving at least one real saddle are included unless stated otherwise.
…
►The second sheet corresponds to and it intersects the bifurcation set (§36.4) smoothly along the line generated by , .
…
►the intersection lines with the bifurcation set are generated by , .
…
►The distribution of real and complex critical points in Figures 36.5.5 and 36.5.6 follows from consistency with Figure 36.5.1 and the fact that there are four real saddles in the inner regions.
…
3: Foreword
…
►The online version, the NIST Digital
Library of Mathematical Functions (DLMF), presents the same technical information along with extensions and innovative interactive features consistent with the new medium.
…
►November 20, 2009
…
4: 10.3 Graphics
…
►
§10.3(i) Real Order and Variable
… ►§10.3(ii) Real Order, Complex Variable
… ► … ► … ►§10.3(iii) Imaginary Order, Real Variable
…5: 20 Theta Functions
Chapter 20 Theta Functions
…6: 25.12 Polylogarithms
…
►The principal branch has a cut along the interval and agrees with (25.12.1) when ; see also §4.2(i).
…
►For real or complex and the polylogarithm
is defined by
…
►The series also converges when , provided that .
…
►valid when and , or and .
…
►valid when , or , .
…
7: 25.5 Integral Representations
§25.5 Integral Representations
… ►In (25.5.15)–(25.5.19), , is the digamma function, and is Euler’s constant (§5.2). (25.5.16) is also valid for , . … ►
25.5.19
.
…
►where the integration contour is a loop around the negative real axis; it starts at , encircles the origin once in the positive direction without enclosing any of the points , , …, and returns to .
…
8: Bibliography
…
►
Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
…
►
Evaluation of Coulomb wave functions along the transition line.
Physical Rev. (2) 96, pp. 77–79.
…
►
On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
…
►
Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics.
Comput. Phys. Comm. 150 (1), pp. 1–20.
…
►
Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
…
9: 4.3 Graphics
…
►