integral representation of solutions

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21: 11.13 Methods of Computation

… ►For numerical purposes the most convenient of the representations given in §11.5, at least for real variables, include the integrals (11.5.2)–(11.5.5) for ${𝐊}_{\nu }\left(z\right)$ and ${𝐌}_{\nu }\left(z\right)$. …Other integrals that appear in §11.5(i) have highly oscillatory integrands unless $z$ is small. … ►To insure stability the integration path must be chosen so that as we proceed along it the wanted solution grows in magnitude at least as rapidly as the complementary solutions. … ►The solution ${𝐊}_{\nu }\left(x\right)$ needs to be integrated backwards for small $x$, and either forwards or backwards for large $x$ depending whether or not $\nu$ exceeds $\frac{1}{2}$. …

22: 4.13 Lambert $W$-Function

… ►The Lambert $W$-function $W\left(z\right)$ is the solution of the equation … ►Other solutions of (4.13.1) are other branches of $W\left(z\right)$. … ►Explicit representations for the $p_n(x)$ are given in Kalugin and Jeffrey (2011). …See Jeffrey and Murdoch (2017) for an explicit representation for the $c_n$ in terms of associated Stirling numbers. … ►For these and other integral representations of the Lambert $W$-function see Kheyfits (2004), Kalugin et al. (2012) and Mező (2020). …

23: 13.29 Methods of Computation

… ►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. … ►

§13.29(iii) Integral Representations

►The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. … ►with recessive solution … ►with recessive solution …

24: 22.15 Inverse Functions

… ►With real variables, the solutions of the equations …The general solutions of (22.15.1), (22.15.2), (22.15.3) are, respectively, … ►

§22.15(ii) Representations as Elliptic Integrals

… ►Other integrals, for example, … ►For representations of the inverse functions as symmetric elliptic integrals see §19.25(v). …

25: 36.5 Stokes Sets

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§36.5(ii) Cuspoids

… ►For , there are two solutions $u$, provided that $|Y|>{(\frac{2}{5})}^{1/2}$. … ►The first sheet corresponds to and is generated as a solution of Equations (36.5.6)–(36.5.9). … ►

§36.5(iii) Umbilics

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§36.5(iv) Visualizations

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26: 16.24 Physical Applications

… ►They are also potentially useful for the solution of more complicated restricted lattice walk problems, and the 3D Ising model; see Barber and Ninham (1970, pp. 147–148). ►

§16.24(ii) Loop Integrals in Feynman Diagrams

►Appell functions are used for the evaluation of one-loop integrals in Feynman diagrams. … ►For an extension to two-loop integrals see Moch et al. (2002). … ►The $\mathit{3}j$ symbols, or Clebsch–Gordan coefficients, play an important role in the decomposition of reducible representations of the rotation group into irreducible representations. …

27: 9.17 Methods of Computation

… ►As described in §3.7(ii), to ensure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows at least as fast as all other solutions of the differential equation. … ►

§9.17(iii) Integral Representations

►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing $\mathrm{Ai}\left(z\right)$ in the complex plane, once values of this function can be generated on the positive real axis. … ►In the first method the integration path for the contour integral (9.5.4) is deformed to coincide with paths of steepest descent (§2.4(iv)). …The second method is to apply generalized Gauss–Laguerre quadrature (§3.5(v)) to the integral (9.5.8). …

28: 10.74 Methods of Computation

… ►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. … ►

§10.74(iii) Integral Representations

►For evaluation of the Hankel functions $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ for complex values of $\nu$ and $z$ based on the integral representations (10.9.18) see Remenets (1973). … ►The integral representation used is based on (10.32.8). … ►

§10.74(vii) Integrals

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29: Errata

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§4.13 has been enlarged. The Lambert $W$-function is multi-valued and we use the notation $W_k\left(x\right)$, $k\in \mathbb{Z}$, for the branches. The original two solutions are identified via $\mathrm{Wp}\left(x\right)=W_0\left(x\right)$ and $\mathrm{Wm}\left(x\right)=W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$.

Other changes are the introduction of the Wright $\omega$-function and tree $T$-function in (4.13.1_2) and (4.13.1_3), simplification formulas (4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for $\frac{d^{n}W}{{dz}^n}$, additional Maclaurin series (4.13.5_1) and (4.13.5_2), an explicit expansion about the branch point at $z=-{\mathrm{e}}^{-1}$ in (4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10) and (4.13.11), and including several integrals and integral representations for Lambert $W$-functions in the end of the section.

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30: 33.14 Definitions and Basic Properties

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§33.14(i) Coulomb Wave Equation

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§33.14(ii) Regular Solution $f(ϵ,\mathrm{\ell };r)$

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§33.14(iii) Irregular Solution $h(ϵ,\mathrm{\ell };r)$

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§33.14(iv) Solutions $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$

… ►The function $s(ϵ,\mathrm{\ell };r)$ has the following properties: …