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11: 10.13 Other Differential Equations

\mathscr{C}_{\nu}\left(z\right)

\mathscr{D}_{\mu}(z)

are any cylinder functions of orders

\nu,\mu

, respectively, and

\vartheta=z(\!\ifrac{\mathrm{d}}{\mathrm{d}z})

. ►

10.13.9

{z^{2}w^{\prime\prime\prime}+3zw^{\prime\prime}+(4z^{2}+1-4\nu^{2})w^{\prime}+% 4zw=0},

w=\mathscr{C}_{\nu}\left(z\right)\mathscr{D}_{\nu}(z)

ⓘ

Symbols:: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $z$ : complex variable, $\nu$ : complex parameter and $\mathscr{D}_{\nu}(z)$ : cylinder function
A&S Ref:: 9.1.58 (modified)
Referenced by:: §10.13, §10.36
Permalink:: http://dlmf.nist.gov/10.13.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §10.13 and Ch.10

►

10.13.10

{z^{3}w^{\prime\prime\prime}+z(4z^{2}+1-4\nu^{2})w^{\prime}+(4\nu^{2}-1)w=0},

w=z\mathscr{C}_{\nu}\left(z\right)\mathscr{D}_{\nu}(z)

ⓘ

Symbols:: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $z$ : complex variable, $\nu$ : complex parameter and $\mathscr{D}_{\nu}(z)$ : cylinder function
A&S Ref:: 9.1.59
Permalink:: http://dlmf.nist.gov/10.13.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §10.13 and Ch.10

►

10.13.11

\left(\vartheta^{4}-2(\nu^{2}+\mu^{2})\vartheta^{2}+(\nu^{2}-\mu^{2})^{2}% \right)w+4z^{2}(\vartheta+1)(\vartheta+2)w=0,

w=\mathscr{C}_{\nu}\left(z\right)\mathscr{D}_{\mu}(z).

ⓘ

Symbols:: $\mathscr{C}_{\NVar{\nu}}\left(\NVar{z}\right)$ : cylinder function, $z$ : complex variable, $\nu$ : complex parameter, $\vartheta=z(\ifrac{\mathrm{d}}{\mathrm{d}z})$ and $\mathscr{D}_{\nu}(z)$ : cylinder function
A&S Ref:: 9.1.57
Referenced by:: §10.13, §10.36
Permalink:: http://dlmf.nist.gov/10.13.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §10.13 and Ch.10

… ►See also Watson (1944, pp. 95–100).

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…

14: Bibliography S

… ►

F. W. Schäfke and D. Schmidt (1966) Ein Verfahren zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung III. Numer. Math. 8 (1), pp. 68–71.

ⓘ

Notes:: Includes Algol program
External Links:: ISSN 0029-599X, Document, MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §28.36(ii).
Encodings:: BibTeX

… ►

B. I. Schneider, X. Guan, and K. Bartschat (2016) Time propagation of partial differential equations using the short iterative Lanczos method and finite-element discrete variable representation. Adv. Quantum Chem. 72, pp. 95–127.

ⓘ

Referenced by:: §18.38(i).
Encodings:: BibTeX

… ►

J. B. Seaborn (1991) Hypergeometric Functions and Their Applications. Texts in Applied Mathematics, Vol. 8, Springer-Verlag, New York.

ⓘ

External Links:: ISBN 0-387-97558-6, MathReview (Wolfgang Bühring), ZentralBlatt
Referenced by:: Erratum (V1.2.0) Section 18.39.
Encodings:: BibTeX

… ►

M. J. Seaton (1982) Coulomb functions analytic in the energy. Comput. Phys. Comm. 25 (1), pp. 87–95.

ⓘ

Notes:: Double-precision Fortran code.
External Links:: Document, Code
Referenced by:: §33.1, §33.14(iv), 9th item.
Encodings:: BibTeX

… ►

R. Sips (1949) Représentation asymptotique des fonctions de Mathieu et des fonctions d’onde sphéroidales. Trans. Amer. Math. Soc. 66 (1), pp. 93–134 (French).

ⓘ

External Links:: FullText, MathReview (M. Strutt), ZentralBlatt
Referenced by:: §28.8(ii).
Encodings:: BibTeX

…

15: Bibliography C

… ►

L. G. Cabral-Rosetti and M. A. Sanchis-Lozano (2000) Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams. J. Comput. Appl. Math. 115 (1-2), pp. 93–99.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (Lorenzo L. Salcedo), ZentralBlatt
Referenced by:: §16.24(ii).
Encodings:: BibTeX

… ►

B. C. Carlson (1977a) Elliptic integrals of the first kind. SIAM J. Math. Anal. 8 (2), pp. 231–242.

ⓘ

External Links:: Document, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §19.29(iii).
Encodings:: BibTeX

… ►

C. W. Clark (1979) Coulomb phase shift. American Journal of Physics 47 (8), pp. 683–684.

ⓘ

External Links:: Document
Referenced by:: §5.20.
Encodings:: BibTeX

… ►

C. W. Clenshaw, F. W. J. Olver, and P. R. Turner (1989) Level-Index Arithmetic: An Introductory Survey. In Numerical Analysis and Parallel Processing (Lancaster, 1987), P. R. Turner (Ed.), Lecture Notes in Math., Vol. 1397, pp. 95–168.

ⓘ

External Links:: MathReview Entry, ZentralBlatt
Referenced by:: §3.1(iv).
Encodings:: BibTeX

… ►

D. Colton and R. Kress (1998) Inverse Acoustic and Electromagnetic Scattering Theory. 2nd edition, Applied Mathematical Sciences, Vol. 93, Springer-Verlag, Berlin.

ⓘ

External Links:: ISBN 3-540-62838-X, MathReview, ZentralBlatt
Referenced by:: §10.73(i).
Encodings:: BibTeX

…

16: 24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of

N_{n}

and

E_{n}

for

n=20(2)60

and

n=8(2)42

, respectively. In Wagstaff (2002) these results are extended to

n=60(2)152

and

n=40(2)88

, respectively, with further complete and partial factorizations listed up to

n=300

and

n=200

, respectively. …

17: Bibliography N

… ►

G. Nemes (2013c) Generalization of Binet’s Gamma function formulas. Integral Transforms Spec. Funct. 24 (8), pp. 597–606.

ⓘ

External Links:: ISSN 1065-2469, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt
Referenced by:: §5.11(i), §5.11(i).
Encodings:: BibTeX

… ►

N. Nielsen (1909) Der Eulersche Dilogarithmus und seine Verallgemeinerungen. Nova Acta Leopoldina 90, pp. 123–212.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §25.12(i).
Encodings:: BibTeX

… ►

V. Yu. Novokshënov (1985) The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).

ⓘ

Notes:: In Russian. English translation: Sov. Math., Dokl. 30(1988), no. 8, pp. 666–668.
External Links:: ISSN 0002-3264, MathReview (Vojislav Marić), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

… ►

Numerical Recipes (commercial C, C++, Fortran 77, and Fortran 90 libraries)

ⓘ

Notes:: A large general purpose mathematical software collection containing some special function codes. Implementation is in single precision.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

►

H. M. Nussenzveig (1965) High-frequency scattering by an impenetrable sphere. Ann. Physics 34 (1), pp. 23–95.

ⓘ

External Links:: Document, MathReview (E. Ar)
Referenced by:: §14.31(iii).
Encodings:: BibTeX

…

18: Bibliography B

… ►

W. N. Bailey (1938) The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.

ⓘ

External Links:: ISSN 0024-6107; 1469-7750/e, Document, ZentralBlatt
Referenced by:: §18.18(vii).
Encodings:: BibTeX

… ►

A. P. Bassom, P. A. Clarkson, and A. C. Hicks (1995) Bäcklund transformations and solution hierarchies for the fourth Painlevé equation. Stud. Appl. Math. 95 (1), pp. 1–71.

ⓘ

External Links:: ISSN 0022-2526, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.10(iv), §32.2(iii), §32.7(iv), §32.8(iv).
Encodings:: BibTeX

… ►

M. V. Berry (1975) Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces. J. Phys. A 8 (4), pp. 566–584.

ⓘ

External Links:: Document, MathReview
Referenced by:: §36.14(iii).
Encodings:: BibTeX

… ►

P. L. Butzer, M. Hauss, and M. Leclerc (1992) Bernoulli numbers and polynomials of arbitrary complex indices. Appl. Math. Lett. 5 (6), pp. 83–88.

ⓘ

External Links:: ISSN 0893-9659, MathReview, ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

►

P. L. Butzer and M. Hauss (1992) Riemann zeta function: Rapidly converging series and integral representations. Appl. Math. Lett. 5 (2), pp. 83–88.

ⓘ

External Links:: ISSN 0893-9659, Document, MathReview (Jacek Fabrykowski), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

…

19: Bibliography F

… ►

H. E. Fettis (1965) Calculation of elliptic integrals of the third kind by means of Gauss’ transformation. Math. Comp. 19 (89), pp. 97–104.

ⓘ

External Links:: FullText, MathReview (J. E. L. Peck), ZentralBlatt
Referenced by:: §19.36(ii).
Encodings:: BibTeX

… ►

A. M. S. Filho and G. Schwachheim (1967) Algorithm 309. Gamma function with arbitrary precision. Comm. ACM 10 (8), pp. 511–512.

ⓘ

External Links:: ISSN 0001-0782, Document
Referenced by:: §5.24(ii).
Encodings:: BibTeX

… ►

V. Fock (1945) Diffraction of radio waves around the earth’s surface. Acad. Sci. USSR. J. Phys. 9, pp. 255–266.

ⓘ

External Links:: MathReview, ZentralBlatt
Referenced by:: §9.1, Notations U, Notations V.
Encodings:: BibTeX

… ►

C. K. Frederickson and P. L. Marston (1994) Travel time surface of a transverse cusp caustic produced by reflection of acoustical transients from a curved metal surface. J. Acoust. Soc. Amer. 95 (2), pp. 650–660.

ⓘ

External Links:: Document
Referenced by:: §36.14(iv).
Encodings:: BibTeX

… ►

T. Fukushima (2012) Series expansions of symmetric elliptic integrals. Math. Comp. 81 (278), pp. 957–990.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §20.7(ii), §20.7(ii).
Encodings:: BibTeX

…

20: 19.2 Definitions

… ►The principal values of

K\left(k\right)

and

E\left(k\right)

are even functions. … ►

§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$

… ►Formulas involving

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

that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using

R_{C}\left(x,y\right)

. … ►When

x

and

y

are positive,

R_{C}\left(x,y\right)

is an inverse circular function if

x<y

and an inverse hyperbolic function (or logarithm) if

x>y

: …For the special cases of

R_{C}\left(x,x\right)

and

R_{C}\left(0,y\right)

see (19.6.15). …

11—20 of 662 matching pages

11: 10.13 Other Differential Equations

12: 27.2 Functions

13: DLMF Project News

14: Bibliography S

15: Bibliography C

16: 24.20 Tables

17: Bibliography N

18: Bibliography B

19: Bibliography F

20: 19.2 Definitions

§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$

11—20 of 662 matching pages

11: 10.13 Other Differential Equations

12: 27.2 Functions

13: DLMF Project News

14: Bibliography S

15: Bibliography C

16: 24.20 Tables

17: Bibliography N

18: Bibliography B

19: Bibliography F

20: 19.2 Definitions

§19.2(iv) A Related Function: R C ⁡ ( x , y )

§19.2(iv) A Related Function: $R_{C}\left(x,y\right)$