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31: 6.18 Methods of Computation

§6.18 Methods of Computation

… ►For large $x$ or $|z|$ these series suffer from slow convergence or cancellation (or both). … ►Quadrature of the integral representations is another effective method. … ►Lastly, the continued fraction (6.9.1) can be used if $|z|$ is bounded away from the origin. … ►

§6.18(iii) Zeros

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32: 36.15 Methods of Computation

… ►Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. Close to the bifurcation set but far from $𝐱=\mathrm{𝟎}$, the uniform asymptotic approximations of §36.12 can be used. … ►(For the umbilics, representations as one-dimensional integrals (§36.2) are used.) … ►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. … ►For numerical solution of partial differential equations satisfied by the canonical integrals see Connor et al. (1983).

33: 10.9 Integral Representations

§10.9 Integral Representations

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§10.9(ii) Contour Integrals

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§10.9(iii) Products

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34: 5.9 Integral Representations

§5.9 Integral Representations

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Binet’s Formula

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§5.9(ii) Psi Function, Euler’s Constant, and Derivatives

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35: 10.32 Integral Representations

§10.32 Integral Representations

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§10.32(i) Integrals along the Real Line

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§10.32(ii) Contour Integrals

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§10.32(iii) Products

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§10.32(iv) Compendia

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36: 1.10 Functions of a Complex Variable

… ►The recurrence relation for $C_n^{(\lambda )}\left(x\right)$ in §18.9(i) follows from $(1-2xz+z^2)\frac{\partial }{\partial z}F(x,\lambda ;z)=2\lambda (x-z)F(x,\lambda ;z)$, and the contour integral representation for $C_n^{(\lambda )}\left(x\right)$ in §18.10(iii) is just (1.10.27).

37: 5.12 Beta Function

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Euler’s Beta Integral

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Pochhammer’s Integral

►When $a,b\in ℂ$ …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 $P$. … ►

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Figure 5.12.3: $t$-plane. Contour for Pochhammer’s integral. Magnify

38: 5.19 Mathematical Applications

… ►Hence from (5.7.6), (5.4.13), and (5.4.19) … ►

§5.19(ii) Mellin–Barnes Integrals

►Many special functions $f(z)$ can be represented as a Mellin–Barnes integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of $z$, the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends. …By translating the contour parallel to itself and summing the residues of the integrand, asymptotic expansions of $f(z)$ for large $|z|$, or small $|z|$, can be obtained complete with an integral representation of the error term. …

39: 3.3 Interpolation

… ►and according to Berrut and Trefethen (2004) it is the most efficient representation of $P_n(z)$. … ►

3.3.6 $$R_n(z)=\frac{{\omega }_{n+1}(z)}{2\pi \mathrm{i}}{\int }_C\frac{f(\zeta )}{(\zeta -z){\omega }_{n+1}(\zeta )}d\zeta ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $f(z)$: function, ${\omega }_{n+1}(z)$: nodal polynomial, $R_n(x)$: remainder and $C$: simple closed contour in $D$

A&S Ref:

25.2.5

Permalink:

http://dlmf.nist.gov/3.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §3.3(i), §3.3 and Ch.3

… ►If $f$ is analytic in a simply-connected domain $D$, then for $z\in D$, ►

3.3.37 $$\left[z_0,z_1,\mathrm{\dots },z_n\right]f=\frac{1}{2\pi \mathrm{i}}{\int }_C\frac{f(\zeta )}{{\omega }_{n+1}(\zeta )}d\zeta ,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\left[z_0,z_1,\mathrm{\dots },z_n\right]f$: divided difference, $\mathrm{i}$: imaginary unit, $\int$: integral, $f(z)$: function, $C$: simple closed contour in $D$ , $z_k$: nodes of function and ${\omega }_{n+1}(z)$: nodal polynomial

Permalink:

http://dlmf.nist.gov/3.3.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §3.3(iii), §3.3 and Ch.3

…

40: 9.17 Methods of Computation

… ►In the case of $\mathrm{Ai}\left(z\right)$, for example, this means that in the sectors we may integrate along outward rays from the origin with initial values obtained from §9.2(ii). But when the integration has to be towards the origin, with starting values of $\mathrm{Ai}\left(z\right)$ and ${\mathrm{Ai}}^{\prime }\left(z\right)$ computed from their asymptotic expansions. … ►

§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. … ►The second method is to apply generalized Gauss–Laguerre quadrature (§3.5(v)) to the integral (9.5.8). …