derivatives with respect to order

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41: 10.20 Uniform Asymptotic Expansions for Large Order

§10.20 Uniform Asymptotic Expansions for Large Order

… ►all functions taking their principal values, with

\zeta=\infty,0,-\infty

, corresponding to

z=0,1,\infty

, respectively. … ►respectively. … ►respectively. … ►For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of

z

see §10.41(v).

42: 15.11 Riemann’s Differential Equation

… ►The importance of (15.10.1) is that any homogeneous linear differential equation of the second order with at most three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1). … ►

15.11.1

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\frac{1-a_{1}-a_{2}}{z-% \alpha}+\frac{1-b_{1}-b_{2}}{z-\beta}+\frac{1-c_{1}-c_{2}}{z-\gamma}\right)% \frac{\mathrm{d}w}{\mathrm{d}z}+{\left(\frac{(\alpha-\beta)(\alpha-\gamma)a_{1% }a_{2}}{z-\alpha}+\frac{(\beta-\alpha)(\beta-\gamma)b_{1}b_{2}}{z-\beta}+\frac% {(\gamma-\alpha)(\gamma-\beta)c_{1}c_{2}}{z-\gamma}\right)\frac{w}{(z-\alpha)(% z-\beta)(z-\gamma)}=0},

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\alpha$ : point, $\beta$ : point, $\gamma$ : point and $w$ : solution
Referenced by:: §15.11(i), §15.11(i), §15.17(v)
Permalink:: http://dlmf.nist.gov/15.11.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.11(i), §15.11 and Ch.15

… ►Here

\{a_{1},a_{2}\}

\{b_{1},b_{2}\}

\{c_{1},c_{2}\}

are the exponent pairs at the points

\alpha

\beta

\gamma

, respectively. … ►These constants can be chosen to map any two sets of three distinct points

\{\alpha,\beta,\gamma\}

and

\{\widetilde{\alpha},\widetilde{\beta},\widetilde{\gamma}\}

onto each other. …The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by …

43: 18.39 Applications in the Physical Sciences

… ►

\epsilon_{0}

is referred to as the ground state, all others,

n=1,2,\dots,

in order of increasing energy being excited states. … ►Namely the

k

th eigenfunction, listed in order of increasing eigenvalues, starting at

k=0

, has precisely

k

nodes, as real zeros of wave-functions away from boundaries are often referred to. …Thus the two missing quantum numbers corresponding to EOP’s of order

1

and

2

of the type III Hermite EOP’s are offset in the node counts by the fact that all excited state eigenfunctions also have two missing real zeros. … ►Orthogonality and normalization of eigenfunctions of this form is respect to the measure

r^{2}\,\mathrm{d}r\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\phi

. … ►Physical scientists use the

n

of Bohr as, to

0

th and

1

st order, it describes the structure and organization of the Periodic Table of the Chemical Elements of which the Hydrogen atom is only the first. …

44: 10.19 Asymptotic Expansions for Large Order

§10.19 Asymptotic Expansions for Large Order

►

§10.19(i) Asymptotic Forms

… ►

§10.19(ii) Debye’s Expansions

… ►

§10.19(iii) Transition Region

… ►See also §10.20(i).

45: 28.1 Special Notation

… ► ►►►►

$m, n$	integers.
…
$\nu$	order of the Mathieu function or modified Mathieu function. (When $\nu$ is an integer it is often replaced by $n$ .)
…
primes	unless indicated otherwise, derivatives with respect to the argument

… ►The radial functions

{\operatorname{Mc}^{(j)}_{n}}\left(z,h\right)

and

{\operatorname{Ms}^{(j)}_{n}}\left(z,h\right)

are denoted by

{\operatorname{Mc}^{(j)}_{n}}\left(z,q\right)

and

{\operatorname{Ms}^{(j)}_{n}}\left(z,q\right)

, respectively.

46: 9.9 Zeros

… ►They are denoted by

a_{k}

a^{\prime}_{k}

b_{k}

b^{\prime}_{k}

, respectively, arranged in ascending order of absolute value for

k=1,2,\ldots.

… ►They lie in the sectors

\tfrac{1}{3}\pi<\operatorname{ph}z<\tfrac{1}{2}\pi

and

-\tfrac{1}{2}\pi<\operatorname{ph}z<-\tfrac{1}{3}\pi

, and are denoted by

\beta_{k}

\beta^{\prime}_{k}

, respectively, in the former sector, and by

\overline{\beta_{k}}

\overline{\beta^{\prime}_{k}}

, in the conjugate sector, again arranged in ascending order of absolute value (modulus) for

k=1,2,\ldots.

See §9.3(ii) for visualizations. … ►

§9.9(ii) Relation to Modulus and Phase

… ►

§9.9(iii) Derivatives With Respect to $k$

… ►For error bounds for the asymptotic expansions of

a_{k}

b_{k}

a^{\prime}_{k}

, and

b^{\prime}_{k}

see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). …

47: 2.7 Differential Equations

… ►For corresponding definitions, together with examples, for linear differential equations of arbitrary order see §§16.8(i)–16.8(ii). … ►as

z\to\infty

in the sectors … ►For extensions to higher-order differential equations see Stenger (1966a, b), Olver (1997a, 1999), and Olde Daalhuis and Olver (1998). … ►The solutions

w_{1}(z)

and

w_{2}(z)

are respectively recessive and dominant as

\Re z\to-\infty

, and vice versa as

\Re z\to+\infty

. …In consequence, if a differential equation has more than one singularity in the extended plane, then usually more than two standard solutions need to be chosen in order to have numerically satisfactory representations everywhere. …

48: Errata

… ►The specific updates to Chapter 1 include the addition of an entirely new subsection §1.18 entitled “Linear Second Order Differential Operators and Eigenfunction Expansions” which is a survey of the formal spectral analysis of second order differential operators. … ►

Subsection 19.25(vi)

The Weierstrass lattice roots $e_{j},$ were linked inadvertently as the base of the natural logarithm. In order to resolve this inconsistency, the lattice roots $e_{j}$ , and lattice invariants $g_{2}$ , $g_{3}$ , now link to their respective definitions (see §§23.2(i), 23.3(i)).

Reported by Felix Ospald.

… ►

Equations (10.15.1), (10.38.1)

These equations have been generalized to include the additional cases of $\ifrac{\partial J_{-\nu}\left(z\right)}{\partial\nu}$ , $\ifrac{\partial I_{-\nu}\left(z\right)}{\partial\nu}$ , respectively.

… ►

Subsections 14.5(ii), 14.5(vi)

The titles have been changed to $\mu=0$ , $\nu=0,1$ , and Addendum to §14.5(ii): $\mu=0$ , $\nu=2$ , respectively, in order to be more descriptive of their contents.

… ►

Subsections 1.15(vi), 1.15(vii), 2.6(iii)

A number of changes were made with regard to fractional integrals and derivatives. In §1.15(vi) a reference to Miller and Ross (1993) was added, the fractional integral operator of order $\alpha$ was more precisely identified as the Riemann-Liouville fractional integral operator of order $\alpha$ , and a paragraph was added below (1.15.50) to generalize (1.15.47). In §1.15(vii) the sentence defining the fractional derivative was clarified. In §2.6(iii) the identification of the Riemann-Liouville fractional integral operator was made consistent with §1.15(vi).

…

49: 19.4 Derivatives and Differential Equations

§19.4 Derivatives and Differential Equations

►

§19.4(i) Derivatives

… ►

19.4.3

\frac{{\mathrm{d}}^{2}E\left(k\right)}{{\mathrm{d}k}^{2}}=-\frac{1}{k}\frac{% \mathrm{d}K\left(k\right)}{\mathrm{d}k}=\frac{{k^{\prime}}^{2}K\left(k\right)-% E\left(k\right)}{k^{2}{k^{\prime}}^{2}},

ⓘ

Symbols:: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/19.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.4(i), §19.4 and Ch.19

… ►Let

D_{k}=\ifrac{\partial}{\partial k}

. …An analogous differential equation of third order for

\Pi\left(\phi,\alpha^{2},k\right)

is given in Byrd and Friedman (1971, 118.03).

50: 3.8 Nonlinear Equations

… ►for all

n

sufficiently large, where

A

and

p

are independent of

n

, then the sequence is said to have convergence of the

p

th order. … … ►For other efficient derivative-free methods, see Le (1985). … ►This is useful when

f(z)

satisfies a second-order linear differential equation because of the ease of computing

f^{\prime\prime}(z_{n})

. … ►For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the Liouville–Green (WKB) approximation, see Segura (2013). …