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31: 36.5 Stokes Sets

… ►Stokes sets are surfaces (codimension one) in $𝐱$ space, across which ${\mathrm{\Psi }}_K(𝐱;k)$ or ${\mathrm{\Psi }}^{(\mathrm{U})}(𝐱;k)$ acquires an exponentially-small asymptotic contribution (in $k$), associated with a complex critical point of ${\mathrm{\Phi }}_K$ or . …where $j$ denotes a real critical point (36.4.1) or (36.4.2), and $\mu$ denotes a critical point with complex $t$ or $s,t$, connected with $j$ by a steepest-descent path (that is, a path where $\mathrm{\Re }\mathrm{\Phi }=\mathrm{constant}$) in complex $t$ or $(s,t)$ space. … ►Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. …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. …

32: 1.10 Functions of a Complex Variable

… ►A function whose only singularities, other than the point at infinity, are poles is called a meromorphic function. If the poles are infinite in number, then the point at infinity is called an essential singularity: it is the limit point of the poles. … ►Then $a$ is a branch point of $F(z)$. For example, $z=0$ is a branch point of $\sqrt{z}$. …

33: 33.2 Definitions and Basic Properties

… ►

§33.2(i) Coulomb Wave Equation

… ►There are two turning points, that is, points at which $d^{2}w/{d\rho }^2=0$ (§2.8(i)). … ►

33.2.2 $${\rho }_{\mathrm{tp}}(\eta ,\mathrm{\ell })=\eta +{({\eta }^2+\mathrm{\ell }(\mathrm{\ell }+1))}^{1/2}.$$

ⓘ

Defines:

${\rho }_{\mathrm{tp}}(\eta ,\mathrm{\ell })$: outer turning point for Coulomb radial functions

Symbols:

$\mathrm{\ell }$: nonnegative integer and $\eta$: real parameter

Referenced by:

§33.12(i), §33.14(i)

Permalink:

http://dlmf.nist.gov/33.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.2(i), §33.2 and Ch.33

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34: 2.8 Differential Equations with a Parameter

… ►Zeros of $f(z)$ are also called turning points. … ►

§2.8(ii) Case I: No Transition Points

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§2.8(iii) Case II: Simple Turning Point

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§2.8(v) Multiple and Fractional Turning Points

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§2.8(vi) Coalescing Transition Points

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35: 33.14 Definitions and Basic Properties

… ►

§33.14(i) Coulomb Wave Equation

… ►When $ϵ>0$ the outer turning point is given by ►

33.14.3 $$r_{\mathrm{tp}}(ϵ,\mathrm{\ell })=\left(\sqrt{1+ϵ\mathrm{\ell }(\mathrm{\ell }+1)}-1\right)/ϵ;$$

ⓘ

Defines:

$r_{\mathrm{tp}}(ϵ,\mathrm{\ell })$: outer turning point for Coulomb functions

Symbols:

$\mathrm{\ell }$: nonnegative integer and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.14(i), §33.14 and Ch.33

…

36: 4.3 Graphics

… ►Corresponding points share the same letters, with bars signifying complex conjugates. …In the labeling of corresponding points $r$ is a real parameter that can lie anywhere in the interval $(0,\mathrm{\infty })$. …

37: 18.40 Methods of Computation

… ►The quadrature points and weights can be put to a more direct and efficient use. … ►This allows Stieltjes–Perron inversion for the $w(x_{i,N})$, given the quadrature weights and points. … ►

►►

Figure 18.40.2: Derivative Rule inversions for $w^{\mathrm{RCP}}(x)$ carried out via Lagrange and PWCF interpolations. Shown are the absolute errors of approximation (18.40.8) at the points $x_{i,N}$, $i=1,2,\mathrm{\dots },N$ for $N=40$. For the derivative rule Lagrange interpolation (red points) gives $\sim 15$ digits in the central region, while PWCF interpolation (blue points) gives $\sim 25$. Magnify

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38: 3.3 Interpolation

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Three-Point Formula

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Four-Point Formula

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Five-Point Formula

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Six-Point Formula

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Seven-Point Formula

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39: 22.9 Cyclic Identities

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Three Points

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Two Points

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Three Points

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Four Points

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Two Points

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40: 7.21 Physical Applications

… ►Carslaw and Jaeger (1959) gives many applications and points out the importance of the repeated integrals of the complementary error function ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$. …