高中毕业证等级h《做证微fuk7778》100

SearchAdvancedHelp

(0.002 seconds)

21—30 of 305 matching pages

21: Bibliography K

… ►

Separation of Variables for Riemannian Spaces of Constant Curvature. Longman Scientific & Technical, Harlow.

ⓘ

External Links:

ISBN 0-582-98807-1, MathReview (H. Kilp), ZentralBlatt

Referenced by:

§31.17(i).

Encodings:

BibTeX

… ►

Zur Theorie der Modulformen $n$-ten Grades. I. Math. Z. 59, pp. 399–416 (German).

ⓘ

External Links:

ISSN 0025-5874, Document, MathReview (H. S. Zuckerman), ZentralBlatt

Referenced by:

§35.6(i).

Encodings:

BibTeX

… ►

The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.

ⓘ

External Links:

Document, MathReview (M. Mikolas), ZentralBlatt

Referenced by:

(18.14.8), §18.14(i), §18.17(ii), §18.18(ii).

Encodings:

BibTeX

… ►

Meixner-Pollaczek polynomials and the Heisenberg algebra. J. Math. Phys. 30 (4), pp. 767–769.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§18.21(ii).

Encodings:

BibTeX

… ►

Askey-Wilson polynomial. Scholarpedia 7 (7), pp. 7761.

ⓘ

Notes:

Electronic only

External Links:

Document, FullText

Referenced by:

§18.28(i), §18.28(i).

Encodings:

BibTeX

…

22: 28.2 Definitions and Basic Properties

… ►

28.2.2 $$\zeta (1-\zeta )w^{\prime \prime }+\frac{1}{2}(1-2\zeta )w^{\prime }+\frac{1}{4}(a-2q(1-2\zeta ))w=0.$$

ⓘ

Symbols:

$q=h^2$: parameter, $a$: parameter, $w(z)$: Mathieu’s equation solution and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/28.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(i), §28.2 and Ch.28

… ►

28.2.3 $$(1-{\zeta }^2)w^{\prime \prime }-\zeta w^{\prime }+\left(a+2q-4q{\zeta }^2\right)w=0.$$

ⓘ

Symbols:

$q=h^2$: parameter, $a$: parameter, $w(z)$: Mathieu’s equation solution and $\zeta$: change of variable

A&S Ref:

20.1.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(i), §28.2 and Ch.28

… ►

28.2.9 $$w_{\text{I}}(\pi ;a,q)=w_{\text{II}}^{\prime }(\pi ;a,q),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $q=h^2$: parameter, $a$: parameter, $w(z)$: Mathieu’s equation solution and $w_{\text{I}}(z;a,q)$: fundamental solution

Referenced by:

§28.2(iv)

Permalink:

http://dlmf.nist.gov/28.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(ii), §28.2 and Ch.28

… ►

28.2.26 $$a_{2n}\left(-q\right)=a_{2n}\left(q\right),$$

ⓘ

Symbols:

$a_n\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $n$: integer

A&S Ref:

20.8.3

Permalink:

http://dlmf.nist.gov/28.2.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(v), §28.2(v), §28.2 and Ch.28

… ►

28.2.28 $$b_{2n+2}\left(-q\right)=b_{2n+2}\left(q\right).$$

ⓘ

Symbols:

$b_n\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $n$: integer

A&S Ref:

20.8.3

Permalink:

http://dlmf.nist.gov/28.2.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(v), §28.2(v), §28.2 and Ch.28

…

23: 10.2 Definitions

… ►These solutions of (10.2.1) are denoted by $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$, and their defining properties are given by … ►The principal branches of $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ are two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. … ►For fixed $z$ $(\ne 0)$ each branch of $H_{\nu }^{(1)}\left(z\right)$ and $H_{\nu }^{(2)}\left(z\right)$ is entire in $\nu$. … ►Except where indicated otherwise, it is assumed throughout the DLMF that the symbols $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, and $H_{\nu }^{(2)}\left(z\right)$ denote the principal values of these functions. … ►The notation ${𝒞}_{\nu }\left(z\right)$ denotes $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu$. …

24: Bibliography F

… ►

Multivariate Calculation. Use of the Continuous Groups. Springer Series in Statistics, Springer-Verlag, New York.

ⓘ

External Links:

ISBN 0-387-96049-X, MathReview (Yasuko Chikuse), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

►

Theta Functions on Riemann Surfaces. Springer-Verlag, Berlin.

ⓘ

Notes:

Lecture Notes in Mathematics, Vol. 352

External Links:

MathReview (H. M. Farkas), ZentralBlatt

Referenced by:

§21.1, §21.7(ii), Ch.21.

Encodings:

BibTeX

… ►

On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.

ⓘ

External Links:

Document, MathReview (B. C. Carlson), ZentralBlatt

Referenced by:

§19.7(i).

Encodings:

BibTeX

… ►

Tables of the line broadening function $H(a,v)$ . Monthly Notices Roy. Astronom. Soc. 129, pp. 221–235.

ⓘ

External Links:

FullText

Referenced by:

§7.19(i), 3rd item.

Encodings:

BibTeX

… ►

Numerical evaluation of the elliptic integral of the third kind. Math. Comp. 19 (91), pp. 494–496.

ⓘ

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§19.36(iv).

Encodings:

BibTeX

…

25: 28.26 Asymptotic Approximations for Large $q$

… ►

28.26.1 $${\mathrm{Mc}}_m^{(3)}(z,h)=\frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}\left({\mathrm{Fc}}_m(z,h)-\mathrm{i}{\mathrm{Gc}}_m(z,h)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

►

28.26.2 $$\mathrm{i}{\mathrm{Ms}}_{m+1}^{(3)}(z,h)=\frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}\left({\mathrm{Fs}}_m(z,h)-\mathrm{i}{\mathrm{Gs}}_m(z,h)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

… ►Then as $h\to +\mathrm{\infty }$ with fixed $z$ in $\mathrm{\Re }z>0$ and fixed $s=2m+1$, … ►The asymptotic expansions of ${\mathrm{Fs}}_m(z,h)$ and ${\mathrm{Gs}}_m(z,h)$ in the same circumstances are also given by the right-hand sides of (28.26.4) and (28.26.5), respectively. … ►For asymptotic approximations for ${\mathrm{M}}_{\nu }^{(3,4)}(z,h)$ see also Naylor (1984, 1987, 1989).

26: 28.1 Special Notation

… ► ►►►

$m,n$	integers.
…
$a,q,h$	real or complex parameters of Mathieu’s equation with $q=h^2$.
…

… ►The functions ${\mathrm{Mc}}_n^{(j)}(z,h)$ and ${\mathrm{Ms}}_n^{(j)}(z,h)$ are also known as the radial Mathieu functions. … ►

$g_{e,n}(h),$

►

$g_{o,n}(h),$

… ►The radial functions ${\mathrm{Mc}}_n^{(j)}(z,h)$ and ${\mathrm{Ms}}_n^{(j)}(z,h)$ are denoted by ${\mathrm{Mc}}_n^{(j)}(z,q)$ and ${\mathrm{Ms}}_n^{(j)}(z,q)$, respectively.

27: 29.9 Stability

… ►The Lamé equation (29.2.1) with specified values of $k,h,\nu$ is called stable if all of its solutions are bounded on $ℝ$; otherwise the equation is called unstable. If $\nu$ is not an integer, then (29.2.1) is unstable iff $h\le a_{\nu }^0\left(k^2\right)$ or $h$ lies in one of the closed intervals with endpoints $a_{\nu }^m\left(k^2\right)$ and $b_{\nu }^m\left(k^2\right)$, $m=1,2,\mathrm{\dots }$. If $\nu$ is a nonnegative integer, then (29.2.1) is unstable iff $h\le a_{\nu }^0\left(k^2\right)$ or $h\in [b_{\nu }^m\left(k^2\right),a_{\nu }^m\left(k^2\right)]$ for some $m=1,2,\mathrm{\dots },\nu$.

28: Bibliography M

… ►

Siegel’s modular forms and Dirichlet series. Lecture Notes in Mathematics, Vol. 216, Springer-Verlag, Berlin.

ⓘ

External Links:

Document, MathReview (K.-B. Gundlach), ZentralBlatt

Referenced by:

§35.4(ii).

Encodings:

BibTeX

… ►

Complex-energy method for elastic $e$-H scattering above the ionization threshold. Phys. Rev. Lett. 23 (7), pp. 361–363.

ⓘ

External Links:

Document

Referenced by:

1st item.

Encodings:

BibTeX

… ►

The exponential integral distribution. Statist. Probab. Lett. 5 (3), pp. 209–211.

ⓘ

External Links:

ISSN 0167-7152, Document, MathReview Entry, ZentralBlatt

Referenced by:

§8.19(xi).

Encodings:

BibTeX

… ►

Zeros of first derivatives of Bessel functions of the first kind, $J_n^{\prime }(x)$, $21\le n\le 51$, $0\le x\le 100$ . J. Res. Nat. Bur. Standards Sect. B 67B (3), pp. 181–183.

ⓘ

External Links:

ISSN 0160-1741, MathReview, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

… ►

On asymptotic expansions of ellipsoidal wave functions. Math. Nachr. 32, pp. 157–172.

ⓘ

External Links:

Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.11, §29.7(ii).

Encodings:

BibTeX

…

29: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

… ►For fixed $h(\ne 0)$ and fixed $\nu$, … ►

28.25.3 $$(m+1)D_{m+1}^{\pm }+\left({(m+\frac{1}{2})}^2\pm (m+\frac{1}{4})8\mathrm{i}h+2h^2-a\right)D_m^{\pm }\pm (m-\frac{1}{2})\left(8\mathrm{i}hm\right)D_{m-1}^{\pm }=0,$$ $m\ge 0$.

ⓘ

Symbols:

$\mathrm{i}$: imaginary unit, $m$: integer, $h$: parameter, $a$: parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.2 (for $+$) 20.9.4 (for $-$)

Permalink:

http://dlmf.nist.gov/28.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

►The upper signs correspond to ${\mathrm{M}}_{\nu }^{(3)}(z,h)$ and the lower signs to ${\mathrm{M}}_{\nu }^{(4)}(z,h)$. The expansion (28.25.1) is valid for ${\mathrm{M}}_{\nu }^{(3)}(z,h)$ when …and for ${\mathrm{M}}_{\nu }^{(4)}(z,h)$ when …

30: Bibliography S

… ►

Note on the complex zeros of $H_{\nu }^{\prime }(x)+i\zeta H_{\nu }(x)=0$ . J. Comput. Appl. Math. 201 (1), pp. 3–7.

ⓘ

External Links:

ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§10.21(xiv), §10.21(xiv).

Encodings:

BibTeX

… ►

Numerical evaluation of spherical Bessel transforms via fast Fourier transforms. J. Comput. Phys. 100 (2), pp. 294–296.

ⓘ

External Links:

ISSN 0021-9991, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

… ►

Principles of Atomic Spectra. John Wiley & Sons Ltd., New York.

ⓘ

External Links:

ISBN 0-471-78835-X

Referenced by:

§34.12.

Encodings:

BibTeX

… ►

On inequalities of the Turán type. Math. Scand. 2, pp. 65–73.

ⓘ

External Links:

ISSN 0025-5521, FullText, MathReview (G. Szegő), ZentralBlatt

Referenced by:

§18.14(ii).

Encodings:

BibTeX

… ►

Perturbations of Kerr-de Sitter black holes and Heun’s equations. Progr. Theoret. Phys. 100 (3), pp. 491–505.

ⓘ

External Links:

ISSN 0033-068X, FullText, MathReview (Volker Perlick)

Referenced by:

§31.17(ii).

Encodings:

BibTeX

…