confluent form

SearchAdvancedHelp

(0.002 seconds)

31—40 of 41 matching pages

31: Software Index

… ► ►►►►►

	Open Source													With Book						Commercial
…
13 Confluent Hypergeometric Functions
13.32(ii) $M(a,b,x)$, $U(a,b,x)$, $𝐌(a,b,x)$, $M_{\kappa ,\mu }\left(x\right)$, $W_{\kappa ,\mu }\left(x\right)$, $x,a,b\in ℝ$	✓		✓	✓	✓	✓						✓	✓	✓			✓		✓		✓	✓	a
13.32(iii) $M(a,b,z)$, $U(a,b,z)$, $𝐌(a,b,z)$, $M_{\kappa ,\mu }\left(z\right)$, $W_{\kappa ,\mu }\left(z\right)$, $z,a,b\in ℂ$	✓											✓	✓	✓					✓		✓	✓	a
…

… ►

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

…

32: 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 … ►Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being $L^2$ and forming a complete set. … ►The solutions (18.39.8) are called the stationary states as separation of variables in (18.39.9) yields solutions of form … ►The orthonormal stationary states and corresponding eigenvalues are then of the form … ►These, taken together with the infinite sets of bound states for each $l$, form complete sets. …

33: 8.19 Generalized Exponential Integral

… ►The right-hand sides are replaced by their limiting forms when $p=1,2,3,\mathrm{\dots }$. … ►

§8.19(vi) Relation to Confluent Hypergeometric Function

►

8.19.16 $$E_p\left(z\right)=z^{p-1}{\mathrm{e}}^{-z}U(p,p,z).$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $z$: complex variable and $p$: parameter

Permalink:

http://dlmf.nist.gov/8.19.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §8.19(vi), §8.19 and Ch.8

►For $U(a,b,z)$ see §13.2(i). …

34: 32.10 Special Function Solutions

… ►All solutions of ${\text{P}}_{\text{II}}$–${\text{P}}_{\text{VI}}$ that are expressible in terms of special functions satisfy a first-order equation of the form … ►

32.10.27 $$\varphi (z)=\frac{C_1{M}_{\kappa ,\mu }\left(\zeta \right)+C_2{W}_{\kappa ,\mu }\left(\zeta \right)}{{\zeta }^{(a-b+1)/2}}\exp \left(\frac{1}{2}\zeta \right),$$

ⓘ

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\exp z$: exponential function, $z$: real, $a$ , $b$ , $\varphi (z)$: function, $\kappa$ , $\mu$ and $C_j$: constants

Permalink:

http://dlmf.nist.gov/32.10.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §32.10(v), §32.10 and Ch.32

… ►The solution (32.10.34) is an essentially transcendental function of both constants of integration since ${\text{P}}_{\text{VI}}$ with $\alpha =\beta =\gamma =0$ and $\delta =\frac{1}{2}$ does not admit an algebraic first integral of the form $P(z,w,w^{\prime },C)=0$, with $C$ a constant. …

35: Bibliography S

… ►

Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

ⓘ

External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

… ►

Confluent hypergeometric functions on tube domains. Math. Ann. 260 (3), pp. 269–302.

ⓘ

External Links:

ISSN 0025-5831, Document, MathReview (A. B. Venkov), ZentralBlatt

Referenced by:

§35.6(ii).

Encodings:

BibTeX

… ►

Uniform Asymptotic Expansions of Confluent Hypergeometric Functions and Whittaker Functions. Doctoral dissertation, University of Copenhagen, Vol. 1965, Jul. Gjellerups Forlag, Copenhagen.

ⓘ

External Links:

MathReview (A. K. Agarwal), ZentralBlatt

Referenced by:

§13.21(iv).

Encodings:

BibTeX

… ►

Confluent Hypergeometric Functions. Cambridge University Press, Cambridge-New York.

ⓘ

Notes:

Table errata: Math. Comp. v. 30 (1976), no. 135, 677–678

External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§13.1, §13.10(ii), §13.10(iii), (13.11.3), §13.11, §13.13(i), §13.13(ii), §13.13(iii), §13.14(i), §13.14(vi), §13.15(i), §13.15(ii), §13.16(ii), §13.16(iii), §13.2(v), §13.2(vi), §13.21(i), §13.23(i), §13.24, §13.26(i), §13.26(ii), §13.26(iii), §13.3(i), §13.3(ii), 2nd item, §13.4(i), §13.4(ii), §13.8(i), §13.9(i), §13.9(ii), Ch.13, §8.5.

Encodings:

BibTeX

… ►

Tables of Airy Functions and Special Confluent Hypergeometric Functions. Pergamon Press, New York.

ⓘ

Notes:

Translated from the Russian by D. G. Fry

External Links:

MathReview (W. F. Freiberger), ZentralBlatt

Referenced by:

(9.13.19), (9.13.20), (9.13.21), §9.13(i), 1st item.

Encodings:

BibTeX

…

36: 9.6 Relations to Other Functions

… ►

§9.6(iii) Airy Functions as Confluent Hypergeometric Functions

… ►

9.6.21 $$\mathrm{Ai}\left(z\right)=\frac{1}{2}{\pi }^{-1/2}{z}^{-1/4}{W}_{0,1/3}\left(2\zeta \right)=3^{-1/6}{\pi }^{-1/2}{\zeta }^{2/3}{\mathrm{e}}^{-\zeta }U(\frac{5}{6},\frac{5}{3},2\zeta ),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $U(a,b,z)$: Kummer confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Combine (9.6.2) with (10.39.8) and (10.39.6).

Referenced by:

(9.5.8)

Permalink:

http://dlmf.nist.gov/9.6.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(iii), §9.6 and Ch.9

►

9.6.22 $${\mathrm{Ai}}^{\prime }\left(z\right)=-\frac{1}{2}{\pi }^{-1/2}{z}^{1/4}{W}_{0,2/3}\left(2\zeta \right)=-3^{1/6}{\pi }^{-1/2}{\zeta }^{4/3}{\mathrm{e}}^{-\zeta }U(\frac{7}{6},\frac{7}{3},2\zeta ),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $U(a,b,z)$: Kummer confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Combine (9.6.3) with (10.39.8) and (10.39.6).

Permalink:

http://dlmf.nist.gov/9.6.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(iii), §9.6 and Ch.9

►

9.6.23 $$\mathrm{Bi}\left(z\right)=\frac{1}{2^{1/3}\mathrm{\Gamma }\left(\frac{2}{3}\right)}{z}^{-1/4}{M}_{0,-1/3}\left(2\zeta \right)+\frac{3}{2^{5/3}\mathrm{\Gamma }\left(\frac{1}{3}\right)}{z}^{-1/4}{M}_{0,1/3}\left(2\zeta \right),$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Combine (9.6.4) with (10.39.7).

Referenced by:

(9.6.25)

Permalink:

http://dlmf.nist.gov/9.6.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(iii), §9.6 and Ch.9

►

9.6.24 $${\mathrm{Bi}}^{\prime }\left(z\right)=\frac{2^{1/3}}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}{z}^{1/4}{M}_{0,-2/3}\left(2\zeta \right)+\frac{3}{2^{10/3}\mathrm{\Gamma }\left(\frac{2}{3}\right)}{z}^{1/4}{M}_{0,2/3}\left(2\zeta \right),$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $z$: complex variable and $\zeta$: change of variable

Proof sketch:

Combine (9.6.5) with (10.39.7).

Referenced by:

(9.6.26)

Permalink:

http://dlmf.nist.gov/9.6.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(iii), §9.6 and Ch.9

…

37: 7.18 Repeated Integrals of the Complementary Error Function

… ►

7.18.2 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)={\int }_z^{\mathrm{\infty }}{\mathrm{i}}^{n-1}\mathrm{erfc}\left(t\right)dt=\frac{2}{\sqrt{\pi }}{\int }_z^{\mathrm{\infty }}\frac{{(t-z)}^n}{n!}{\mathrm{e}}^{-t^2}dt.$$

ⓘ

Defines:

${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.1 (in modified form) 7.2.3

Permalink:

http://dlmf.nist.gov/7.18.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(i), §7.18 and Ch.7

… ►

Confluent Hypergeometric Functions

►

7.18.9 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)={\mathrm{e}}^{-z^2}\left(\frac{1}{2^n\mathrm{\Gamma }\left(\frac{1}{2}n+1\right)}M(\frac{1}{2}n+\frac{1}{2},\frac{1}{2},z^2)-\frac{z}{2^{n-1}\mathrm{\Gamma }\left(\frac{1}{2}n+\frac{1}{2}\right)}M(\frac{1}{2}n+1,\frac{3}{2},z^2)\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

7.2.12

Permalink:

http://dlmf.nist.gov/7.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iv), §7.18(iv), §7.18 and Ch.7

►

7.18.10 $${\mathrm{i}}^n\mathrm{erfc}\left(z\right)=\frac{{\mathrm{e}}^{-z^2}}{2^n\sqrt{\pi }}U(\frac{1}{2}n+\frac{1}{2},\frac{1}{2},z^2).$$

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$: repeated integrals of the complementary error function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§7.18(iv), §7.18(iv), Erratum (V1.0.21) for Paragraph Confluent Hypergeometric Functions (in §7.18(iv))

Permalink:

http://dlmf.nist.gov/7.18.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.18(iv), §7.18(iv), §7.18 and Ch.7

►The confluent hypergeometric function on the right-hand side of (7.18.10) is multivalued and in the sectors one has to use the analytic continuation formula (13.2.12). …

38: Errata

… ►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. … ►

A note about the multivalued nature of the Kummer confluent hypergeometric function of the second kind $U$ on the right-hand side of (7.18.10) was inserted.

… ►

Confluent hypergeometric functions were incorrectly linked to the definitions of the Kummer confluent hypergeometric and parabolic cylinder functions. However, to the eye, the functions appeared correct. The links were corrected.

… ►

There were clarifications made in the conditions on the parameter $a$ in $U(a,b,z)$ of those equations.

… ►

A new Subsection Continued Fractions, has been added to cover computation of confluent hypergeometric functions by continued fractions.

…

39: Bibliography B

… ►

The Confluent Hypergeometric Function with Special Emphasis on Its Applications. Springer-Verlag, New York.

ⓘ

Notes:

Translated from the German by H. Lichtbau and K. Wetzel

External Links:

MathReview, ZentralBlatt

Referenced by:

§12.17, §12.17, Ch.12, §13.1, §13.10(v), §13.14(i), §13.16(ii), §13.16(iii), §13.23(i), §13.24, §13.4(i), §13.4(ii), §13.4(iii), §13.6(iv), §13.9(i), §13.9(ii), Ch.13, §33.6, Notations M.

Encodings:

BibTeX

… ►

The double confluent Heun equation: Characteristic exponent and connection formulae. Methods Appl. Anal. 1 (3), pp. 348–370.

ⓘ

External Links:

ISSN 1073-2772, FullText, MathReview (W. Balser), ZentralBlatt

Referenced by:

§31.12.

Encodings:

BibTeX

… ►

The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms. Dover Publications, New York.

ⓘ

Notes:

Two volumes. A reproduction of the seventh edition [vol. 1, Longmans, Green, London 1912; vol. 2, Longmans, Green, London 1928].

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.11(i), §1.11(ii), §1.11(iv).

Encodings:

BibTeX

…

40: Bibliography O

… ►

Uniform, exponentially improved, asymptotic expansions for the confluent hypergeometric function and other integral transforms. SIAM J. Math. Anal. 22 (5), pp. 1475–1489.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (Hans-Jürgen Glaeske), ZentralBlatt

Referenced by:

§10.17(v), §10.40(iv), §13.7(iii), (9.7.18), (9.7.19), (9.7.20), (9.7.21), (9.7.22), (9.7.23), §9.7(v).

Encodings:

BibTeX

►

Exponentially-improved asymptotic solutions of ordinary differential equations I: The confluent hypergeometric function. SIAM J. Math. Anal. 24 (3), pp. 756–767.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (Hussain S. Nur), ZentralBlatt

Referenced by:

§10.17(v), §13.7(iii), (9.7.18), (9.7.19), (9.7.20), (9.7.21), (9.7.22), (9.7.23), §9.7(v).

Encodings:

BibTeX

… ►

A closed form solution of the $s$-wave Bethe-Goldstone equation with an infinite repulsive core. J. Math. Phys. 27 (4), pp. 1154–1158.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§30.13(v).

Encodings:

BibTeX

…