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31—40 of 64 matching pages

32: 19.29 Reduction of General Elliptic Integrals

… ►Moreover, the requirement that one limit of integration be a branch point of the integrand is eliminated without doubling the number of standard integrals in the result. …

33: 36.12 Uniform Approximation of Integrals

… ►Branches are chosen so that

\Delta

is real and positive if the critical points are real, or real and negative if they are complex. …

§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 $\operatorname{Wp}\left(x\right)=W_{0}\left(x\right)$ and $\operatorname{Wm}\left(x\right)=W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$ .

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 $\frac{{\mathrm{d}}^{n}W}{{\mathrm{d}z}^{n}}$ , 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.

…

35: 2.5 Mellin Transform Methods

… ►

2.5.16

I(x)=\sum_{s=0}^{n-1}(c_{s}\ln x+d_{s})x^{-2s-1}+O\left(x^{-2n-1+\epsilon}% \right),

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\ln\NVar{z}$ : principal branch of logarithm function, $\epsilon$ : parameter, $c$ : point, $I(x)$ : convolution integral and $d$ : point
Permalink:: http://dlmf.nist.gov/2.5.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §2.5(i), §2.5(i), §2.5 and Ch.2

…

36: 10.41 Asymptotic Expansions for Large Order

… ►

10.41.7

\eta=(1+z^{2})^{\frac{1}{2}}+\ln\frac{z}{1+(1+z^{2})^{\frac{1}{2}}},

ⓘ

Defines:: $\eta$ (locally)
Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable
A&S Ref:: 9.7.11
Permalink:: http://dlmf.nist.gov/10.41.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §10.41(ii), §10.41 and Ch.10

… ►where the branches assume their principal values. … ►The expansions (10.41.3)–(10.41.6) also hold uniformly in the sector

|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta

(<\tfrac{1}{2}\pi)

, with the branches of the fractional powers in (10.41.3)–(10.41.8) extended by continuity from the positive real

z

-axis. ►Figures 10.41.1 and 10.41.2 show corresponding points of the mapping of the

z

-plane and the

\eta

-plane. …Thus

B

is the point

z=c

, where

c

is given by (10.20.18). …

37: 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,\infty)

. … ►

See accompanying text — Figure 4.3.3: $\ln\left(x+iy\right)$ (principal value). There is a branch cut along the negative real axis. Magnify 3D Help

…

38: 5.20 Physical Applications

… ►Suppose the potential energy of a gas of

n

point charges with positions

x_{1},x_{2},\dots,x_{n}

and free to move on the infinite line

-\infty<x<\infty

, is given by ►

5.20.1

W=\frac{1}{2}\sum_{\ell=1}^{n}x_{\ell}^{2}-\sum_{1\leq\ell<j\leq n}\ln|x_{\ell% }-x_{j}|.

ⓘ

Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, $j$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $W$ : potential energy
Permalink:: http://dlmf.nist.gov/5.20.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §5.20, §5.20 and Ch.5

… ►

5.20.4

W=-\sum_{1\leq\ell<j\leq n}\ln|e^{i\theta_{\ell}}-e^{i\theta_{j}}|,

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $j$ : nonnegative integer, $n$ : nonnegative integer and $W$ : potential energy
Permalink:: http://dlmf.nist.gov/5.20.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §5.20, §5.20 and Ch.5

…

39: 31.9 Orthogonality

… ►

31.9.2

\int_{\zeta}^{(1+,0+,1-,0-)}t^{\gamma-1}(1-t)^{\delta-1}(t-a)^{\epsilon-1}\*w_% {m}(t)w_{k}(t)\,\mathrm{d}t=\delta_{m,k}\theta_{m}.

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $\zeta$ : point and $\theta_{m}$ : normalization constant
Permalink:: http://dlmf.nist.gov/31.9.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §31.9(i), §31.9 and Ch.31

►Here

\zeta

is an arbitrary point in the interval

(0,1)

. …The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning. … ►

31.9.3

\theta_{m}=(1-{\mathrm{e}}^{2\pi i\gamma})(1-{\mathrm{e}}^{2\pi i\delta})\zeta% ^{\gamma}(1-\zeta)^{\delta}(\zeta-a)^{\epsilon}\*\frac{f_{0}(q,\zeta)}{f_{1}(q% ,\zeta)}\left.\frac{\partial}{\partial q}\mathscr{W}\left\{f_{0}(q,\zeta),f_{1% }(q,\zeta)\right\}\right|_{q=q_{m}},

ⓘ

Defines:: $\theta_{m}$ : normalization constant (locally)
Symbols:: $\mathscr{W}$ : Wronskian, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $\gamma$ : real or complex parameter, $\delta$ : real or complex parameter, $\epsilon$ : real or complex parameter, $m$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter, $\zeta$ : point and $f_{0}(q_{m},z)$ , $f_{1}(q_{m},z)$ : coefficients
Permalink:: http://dlmf.nist.gov/31.9.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §31.9(i), §31.9 and Ch.31

…

40: 9.16 Physical Applications

… ►Airy functions are applied in many branches of both classical and quantum physics. … ►The frequent appearances of the Airy functions in both classical and quantum physics is associated with wave equations with turning points, for which asymptotic (WKBJ) solutions are exponential on one side and oscillatory on the other. The Airy functions constitute uniform approximations whose region of validity includes the turning point and its neighborhood. … ►The KdV equation and solitons have applications in many branches of physics, including plasma physics lattice dynamics, and quantum mechanics. … ►This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point. …