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21: 18.38 Mathematical Applications

… ►The monic Chebyshev polynomial

2^{1-n}T_{n}\left(x\right)

n\geq 1

, enjoys the ‘minimax’ property on the interval

[-1,1]

, that is,

|2^{1-n}T_{n}\left(x\right)|

has the least maximum value among all monic polynomials of degree

n

. In consequence, expansions of functions that are infinitely differentiable on

[-1,1]

in series of Chebyshev polynomials usually converge extremely rapidly. … ►

18.38.4

K_{2}=[K_{0},K_{1}]_{q},

ⓘ

Symbols:: $[\NVar{a},\NVar{b}]$ : closed interval and $q$ : real variable
Referenced by:: §18.38(iii), §18.38(i), Erratum (V1.2.0) Section 18.38
Permalink:: http://dlmf.nist.gov/18.38.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §18.38(iii), §18.38(iii), §18.38 and Ch.18

… ►

18.38.5

[X,Y]_{q}=q^{\frac{1}{2}}XY-q^{-\frac{1}{2}}YX.

ⓘ

Symbols:: $[\NVar{a},\NVar{b}]$ : closed interval, $X$ : subset of $\mathbb{R}$ and $q$ : real variable
Permalink:: http://dlmf.nist.gov/18.38.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §18.38(iii), §18.38(iii), §18.38 and Ch.18

… ►SUSY leads to algebraic simplifications in generating excited states, and partner potentials with closely related energy spectra, from knowledge of a single ground state wave function. …

22: 9.5 Integral Representations

… ►

9.5.3

\operatorname{Bi}\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}\exp\left(-{% \tfrac{1}{3}}t^{3}+xt\right)\,\mathrm{d}t+\frac{1}{\pi}\int_{0}^{\infty}\sin% \left(\tfrac{1}{3}t^{3}+xt\right)\,\mathrm{d}t.

ⓘ

Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b})$ : half-closed interval, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $(\NVar{a},\NVar{b}]$ : half-closed interval, $\sin\NVar{z}$ : sine function and $x$ : real variable
Proof sketch:: Use (9.5.5) with the substitions: for the paths $(-\infty,0]$ use $t=-\tau$ , for the paths $[0,\infty{\mathrm{e}}^{\pm\pi\mathrm{i}/3})$ use $t={\mathrm{e}}^{\pm\pi\mathrm{i}/3}\tau$ . For the resulting integrals use (4.14.1) and (4.14.2).
A&S Ref:: 10.4.33
Referenced by:: (9.12.19)
Permalink:: http://dlmf.nist.gov/9.5.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §9.5(i), §9.5 and Ch.9

…

23: 18.39 Applications in the Physical Sciences

… ►where the orthogonality measure is now

\,\mathrm{d}r

r\in[0,\infty).

… ►Orthogonality, with measure

\,\mathrm{d}r

for

r\in[0,\infty)

, for fixed

l

… ►normalized with measure

r^{2}\,\mathrm{d}r

r\in[0,\infty)

. … ►is tridiagonalized in the complete

L^{2}

non-orthogonal (with measure

\,\mathrm{d}r

r\in[0,\infty)

) basis of Laguerre functions: … ►which maps

\epsilon\in[0,\infty)

onto

x\in[-1,1]

. …

24: 23.20 Mathematical Applications

… ►There is a unique point

z_{0}\in[\omega_{1},\omega_{1}+\omega_{3}]\cup[\omega_{1}+\omega_{3},\omega_{3}]

such that

\wp\left(z_{0}\right)=0

. … ►The two pairs of edges

[0,\omega_{1}]\cup[\omega_{1},2\omega_{3}]

and

[2\omega_{3},2\omega_{3}-\omega_{1}]\cup[2\omega_{3}-\omega_{1},0]

R

are each mapped strictly monotonically by

\wp

onto the real line, with

0\to\infty

\omega_{1}\to e_{1}

2\omega_{3}\to-\infty

; similarly for the other pair of edges. … ►If

a,b\in\mathbb{R}

, then

C

intersects the plane

{\mathbb{R}}^{2}

in a curve that is connected if

\Delta\equiv 4a^{3}+27b^{2}>0

; if

\Delta<0

, then the intersection has two components, one of which is a closed loop. …

25: 28.30 Expansions in Series of Eigenfunctions

… ►Then every continuous

2\pi

-periodic function

f(x)

whose second derivative is square-integrable over the interval

[0,2\pi]

can be expanded in a uniformly and absolutely convergent series …

27: 36.15 Methods of Computation

… ►Close to the origin

\mathbf{x}=\boldsymbol{{0}}

of parameter space, the series in §36.8 can be used. … ►Close to the bifurcation set but far from

\mathbf{x}=\boldsymbol{{0}}

, the uniform asymptotic approximations of §36.12 can be used. …

28: 4.2 Definitions

… ►where the path does not intersect

(-\infty,0]

; see Figure 4.2.1. … ►We regard this as the closed definition of the principal value. ►In contrast to (4.2.5) the closed definition is symmetric. … ►However, in the absence of any indication to the contrary it is assumed that the definition is the closed one. …

29: 31.13 Asymptotic Approximations

… ►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). …

30: 5.20 Physical Applications

… ►

5.20.5

\psi_{n}(\beta)=\frac{1}{(2\pi)^{n}}\int_{[-\pi,\pi]^{n}}e^{-\beta W}\,\mathrm% {d}\theta_{1}\cdots\,\mathrm{d}\theta_{n}=\Gamma\left(1+\tfrac{1}{2}n\beta% \right)(\Gamma\left(1+\tfrac{1}{2}\beta\right))^{-n}.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$ : closed interval, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $n$ : nonnegative integer, $W$ : potential energy, $\psi_{n}(\beta)$ : partition function and $\beta=1/(kT)$
Permalink:: http://dlmf.nist.gov/5.20.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §5.20, §5.20 and Ch.5

…