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21: 10.40 Asymptotic Expansions for Large Argument

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10.40.11 $$|R_{\mathrm{\ell }}(\nu ,z)|\le 2|a_{\mathrm{\ell }}(\nu )|{𝒱}_{z,\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)\exp \left(|{\nu }^2-\frac{1}{4}|{𝒱}_{z,\mathrm{\infty }}\left(t^{-1}\right)\right),$$

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Defines:

$R_{\mathrm{\ell }}(\nu ,z)$: remainder (locally)

Symbols:

$\exp z$: exponential function, ${𝒱}_{a,b}\left(f\right)$: total variation, $z$: complex variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

Permalink:

http://dlmf.nist.gov/10.40.E11

Encodings:

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See also:

Annotations for §10.40(iii), §10.40 and Ch.10

►where $𝒱$ denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that $|\mathrm{\Re }t|$ changes monotonically. Bounds for ${𝒱}_{z,\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$ are given by ►

10.40.12

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, ${𝒱}_{a,b}\left(f\right)$: total variation and $z$: complex variable

Referenced by:

§10.40(i), §10.40(iii), Erratum (V1.0.11) for Equation (10.40.12)

Permalink:

http://dlmf.nist.gov/10.40.E12

Encodings:

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Correction (effective with 1.0.11):

Originally the third constraint was written incorrectly as $\pi \le |\mathrm{ph}z|\le \frac{3}{2}\pi$.

Suggested 2014-11-05 by Gergő Nemes

See also:

Annotations for §10.40(iii), §10.40 and Ch.10

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22: 35.2 Laplace Transform

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35.2.3

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Defines:

$\ast$: convolution operator (locally)

Symbols:

$\int$: integral, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $d$: differential of matrix, : less-than for matrices, $f(𝐗)$: complex-valued function of matrix argument and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.6(ii)

Permalink:

http://dlmf.nist.gov/35.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.2, §35.2 and Ch.35

23: Bibliography R

… ►

Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators. Academic Press, New York.

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External Links:

ISBN 0-12-585004-2(vol.4), MathReview (P. R. Chernoff)

Referenced by:

§1.18(x).

Encodings:

BibTeX

… ►

On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.

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External Links:

ISSN 0044-2267, Document, MathReview Entry, ZentralBlatt

Referenced by:

§29.20(ii).

Encodings:

BibTeX

… ►

On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42, pp. 684–760.

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External Links:

ISSN 0022-247X, Document, Link, MathReview (P. Saphar), ZentralBlatt

Referenced by:

§18.2(xii).

Encodings:

BibTeX

… ►

On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

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External Links:

ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

…

24: 17.2 Calculus

… ►

17.2.41

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Defines:

${𝒟}_q$: $q$-differential operator (locally)

Symbols:

$q$: complex base and $z$: complex variable

Referenced by:

§17.2(iv), §18.27(i)

Permalink:

http://dlmf.nist.gov/17.2.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(iv), §17.2 and Ch.17

… ►

17.2.42

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $j$: nonnegative integer, $n$: nonnegative integer, $z$: complex variable and ${𝒟}_q$: $q$-differential operator

Referenced by:

§17.2(iv)

Permalink:

http://dlmf.nist.gov/17.2.E42

Encodings:

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See also:

Annotations for §17.2(iv), §17.2 and Ch.17

… ►

17.2.43 $${𝒟}_q(f(z)g(z))=g(z)f^{[1]}(z)+f(zq)g^{[1]}(z).$$

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Symbols:

$q$: complex base, $z$: complex variable and ${𝒟}_q$: $q$-differential operator

Referenced by:

§17.2(iv)

Permalink:

http://dlmf.nist.gov/17.2.E43

Encodings:

pMML, png, TeX

See also:

Annotations for §17.2(iv), §17.2(iv), §17.2 and Ch.17

… ►

17.2.44 $${𝒟}_q^n(f(z)g(z))=\sum _{j=0}^n{\left[\genfrac{}{}{0.0pt}{}{n}{j}\right]}_q{f}^{[n-j]}(z{q}^j)g^{[j]}(z).$$

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $q$: complex base, $j$: nonnegative integer, $n$: nonnegative integer, $z$: complex variable and ${𝒟}_q$: $q$-differential operator

Referenced by:

§17.2(iv)

Permalink:

http://dlmf.nist.gov/17.2.E44

Encodings:

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See also:

Annotations for §17.2(iv), §17.2(iv), §17.2 and Ch.17

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25: 16.8 Differential Equations

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16.8.3 $$\left(\mathit{\vartheta }(\mathit{\vartheta }+b_1-1)\mathrm{\cdots }(\mathit{\vartheta }+b_q-1)-z(\mathit{\vartheta }+a_1)\mathrm{\cdots }(\mathit{\vartheta }+a_p)\right)w=0.$$

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Symbols:

$p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $\mathit{\vartheta }$: differential operator and $w$: solution

Referenced by:

§16.8(ii), §16.8(ii), §16.8(ii)

Permalink:

http://dlmf.nist.gov/16.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

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16.8.4 $$z^q{𝐷}^{q+1}w+\sum _{j=1}^q{z}^{j-1}({\alpha }_{j}z+{\beta }_j){𝐷}^{j}w+{\alpha }_{0}w=0,$$ $p\le q$,

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Symbols:

$p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $𝐷$: differential operator, $w$: solution, ${\alpha }_j$: constant and ${\beta }_j$: constant

Referenced by:

§16.8(ii)

Permalink:

http://dlmf.nist.gov/16.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

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16.8.5 $$z^q(1-z){𝐷}^{q+1}w+\sum _{j=1}^q{z}^{j-1}({\alpha }_{j}z+{\beta }_j){𝐷}^{j}w+{\alpha }_{0}w=0,$$ $p=q+1$,

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Symbols:

$p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $𝐷$: differential operator, $w$: solution, ${\alpha }_j$: constant and ${\beta }_j$: constant

Referenced by:

§16.8(ii)

Permalink:

http://dlmf.nist.gov/16.8.E5

Encodings:

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See also:

Annotations for §16.8(ii), §16.8 and Ch.16

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26: 18.39 Applications in the Physical Sciences

… ►The fundamental quantum Schrödinger operator, also called the Hamiltonian, $\mathscr{H}$, is a second order differential operator of the form … ►Analogous to (18.39.7) the 3D Schrödinger operator is …where ${\mathrm{L}}^2$ is the (squared) angular momentum operator (14.30.12). … ►Here tridiagonal representations of simple Schrödinger operators play a similar role. The radial operator (18.39.28) …

27: 19.4 Derivatives and Differential Equations

… ►Then ►

19.4.8 $$(kk_{}^{\prime }{}_{}{}^2{D}_k^2+(1-3k^2)D_k-k)F(\varphi ,k)=\frac{-k\sin \varphi \cos \varphi }{{(1-k^2{\sin }^2\varphi )}^{3/2}},$$

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Symbols:

$\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $D_k$: differential operator

Referenced by:

§19.4(ii)

Permalink:

http://dlmf.nist.gov/19.4.E8

Encodings:

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See also:

Annotations for §19.4(ii), §19.4 and Ch.19

►

19.4.9 $$(kk_{}^{\prime }{}_{}{}^2{D}_k^2+k_{}^{\prime }{}_{}{}^2{D}_k+k)E(\varphi ,k)=\frac{k\sin \varphi \cos \varphi }{\sqrt{1-k^2{\sin }^2\varphi }}.$$

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Symbols:

$\cos z$: cosine function, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $D_k$: differential operator

Permalink:

http://dlmf.nist.gov/19.4.E9

Encodings:

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See also:

Annotations for §19.4(ii), §19.4 and Ch.19

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28: 24.4 Basic Properties

… ►

§24.4(viii) Symbolic Operations

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29: 1.4 Calculus of One Variable

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1.4.33 $${𝒱}_{a,b}\left(f\right)=sup\sum _{j=1}^n\left|f(x_j)-f(x_{j-1})\right|,$$

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Defines:

$𝒱\left(f\right)$: total variation and ${𝒱}_{a,b}\left(f\right)$: total variation

Symbols:

$sup$: least upper bound (supremum), $j$: integer, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.4.E33

Encodings:

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See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

… ►If , then $f(x)$ is of bounded variation on $(a,b)$. In this case, $g(x)={𝒱}_{a,x}\left(f\right)$ and $h(x)={𝒱}_{a,x}\left(f\right)-f(x)$ are nondecreasing bounded functions and $f(x)=g(x)-h(x)$. … ►

1.4.34 $${𝒱}_{a,b}\left(f\right)={\int }_a^b\left|f^{\prime }(x)\right|dx,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, ${𝒱}_{a,b}\left(f\right)$: total variation and $\left|x\right|$: absolute value of $x$

Referenced by:

§1.4(v), §1.4(v), Erratum (V1.0.28) for Equation (1.4.34)

Permalink:

http://dlmf.nist.gov/1.4.E34

Encodings:

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See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

… ►Lastly, whether or not the real numbers $a$ and $b$ satisfy , and whether or not they are finite, we define ${𝒱}_{a,b}\left(f\right)$ by (1.4.34) whenever this integral exists. …

30: Bibliography K

… ►

Lowering and Raising Operators for Some Special Orthogonal Polynomials. In Jack, Hall-Littlewood and Macdonald Polynomials, Contemp. Math., Vol. 417, pp. 227–238.

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External Links:

FullText, MathReview Entry, ZentralBlatt

Referenced by:

§18.9(iii).

Encodings:

BibTeX

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Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§15.14, §15.14.

Encodings:

BibTeX

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Jacobi polynomials, Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator. Adv. Math. 333, pp. 796–821.

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External Links:

ISSN 0001-8708, Document, Link, MathReview (Martin Axel Hallnäs)

Referenced by:

§18.14(i).

Encodings:

BibTeX

… ►

Algebras of Virasoro type, the energy-momentum tensor, and operator expansions on Riemann surfaces. Funktsional. Anal. i Prilozhen. 23 (1), pp. 24–40 (Russian).

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Notes:

In Russian. English translation: Funct. Anal. Appl. 23(1989), no. 1, pp. 19–33

External Links:

ISSN 0374-1990, Document, MathReview (Fedor Malikov), ZentralBlatt

Referenced by:

§20.12(ii).

Encodings:

BibTeX

… ►

Construction of differential operators having Bochner-Krall orthogonal polynomials as eigenfunctions. J. Math. Anal. Appl. 324 (1), pp. 285–303.

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External Links:

ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt

Referenced by:

§18.36(i).

Encodings:

BibTeX