from differential equation

SearchAdvancedHelp

(0.006 seconds)

1—10 of 200 matching pages

1: 9.13 Generalized Airy Functions

§9.13 Generalized Airy Functions

►

§9.13(i) Generalizations from the Differential Equation

… ►and ${𝒵}_p$ is any linear combination of the modified Bessel functions $I_p$ and ${\mathrm{e}}^{p\pi \mathrm{i}}{K}_p$ (§10.25(ii)). … ►As $z\to \mathrm{\infty }$ … ► …

2: Bibliography J

… ►

Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions ${\overline{ps}}_n^r(\eta ,h)$ and ${\overline{qs}}_n^r(\eta ,h)$ for large $h$ . Proc. Roy. Soc. London Ser. A 321, pp. 545–555.

ⓘ

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§30.9(ii).

Encodings:

BibTeX

…

3: 9.18 Tables

… ►

Smirnov (1960) tabulates $U_1(x,\alpha )$, $U_2(x,\alpha )$, defined by (9.13.20), (9.13.21), and also $\partial U_1(x,\alpha )/\partial x$, $\partial U_2(x,\alpha )/\partial x$, for $\alpha =1$, $x=-6(.01)10$ to 5D or 5S, and also for $\alpha =\pm \frac{1}{4}$, $\pm \frac{1}{3}$, $\pm \frac{1}{2}$, $\pm \frac{2}{3}$, $\pm \frac{3}{4}$, $\frac{5}{4}$, $\frac{4}{3}$, $\frac{3}{2}$, $\frac{5}{3}$, $\frac{7}{4}$, 2, $x=0(.01)6$; 4D.

4: 22.10 Maclaurin Series

… ►Further terms may be derived from the differential equations (22.13.13), (22.13.14), (22.13.15), or from the integral representations of the inverse functions in §22.15(ii). …

5: 2.7 Differential Equations

… ►The radii of convergence of the series (2.7.4), (2.7.6) are not less than the distance of the next nearest singularity of the differential equation from $z_0$. …

6: 10.36 Other Differential Equations

… ►Differential equations for products can be obtained from (10.13.9)–(10.13.11) by replacing $z$ by $\mathrm{i}z$.

7: 19.18 Derivatives and Differential Equations

§19.18 Derivatives and Differential Equations

… ►

§19.18(ii) Differential Equations

… ►and also a system of $n(n-1)/2$ Euler–Poisson differential equations (of which only $n-1$ are independent): …If $n=2$, then elimination of ${\partial }_{2}v$ between (19.18.11) and (19.18.12), followed by the substitution $(b_1,b_2,z_1,z_2)=(b,c-b,1-z,1)$, produces the Gauss hypergeometric equation (15.10.1). ►The next four differential equations apply to the complete case of $R_F$ and $R_G$ in the form $R_{-a}(\frac{1}{2},\frac{1}{2};z_1,z_2)$ (see (19.16.20) and (19.16.23)). …

8: 10.13 Other Differential Equations

§10.13 Other Differential Equations

►In the following equations $\nu ,\lambda ,p,q$, and $r$ are real or complex constants with $\lambda \ne 0$, $p\ne 0$, and $q\ne 0$. ►

10.13.1 $$w^{\prime \prime }+\left({\lambda }^2-\frac{{\nu }^2-\frac{1}{4}}{z^2}\right)w=0,$$ $w=z^{\frac{1}{2}}{𝒞}_{\nu }\left(\lambda z\right)$,

ⓘ

Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.49

Referenced by:

§1.18(vi), §10.36

Permalink:

http://dlmf.nist.gov/10.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

… ►

10.13.4 $$w^{\prime \prime }+\frac{1\mp 2\nu }{z}{w}^{\prime }+{\lambda }^{2}w=0,$$ $w=z^{\pm \nu }{𝒞}_{\nu }\left(\lambda z\right)$,

ⓘ

Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.52

Referenced by:

Erratum (V1.0.8) for Equation (10.13.4)

Permalink:

http://dlmf.nist.gov/10.13.E4

Encodings:

pMML, png, TeX

Generalization (effective with 1.0.8):

This equation has been generalized to allow an additional case. Originally only the case $w=z^{\nu }{𝒞}_{\nu }\left(\lambda z\right)$ was covered.

Suggested 2014-01-27 by Tom Koornwinder

See also:

Annotations for §10.13 and Ch.10

… ►For further differential equations see Kamke (1977, pp. 440–451). …

9: 3.11 Approximation Techniques

… ►

3.11.26 $$F(s)=ℒf\left(s\right)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}f(t)dt$$

ⓘ

Defines:

$F(s)$: Laplace transform of $f(t)$ (locally)

Symbols:

$ℒ\left(f\right)\left(s\right)$: Laplace transform, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm and $\int$: integral

Keywords:

Laplace transform

A&S Ref:

29.1.1

Permalink:

http://dlmf.nist.gov/3.11.E26

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Laplace transform was changed to $ℒf\left(z\right)$ from $ℒ(f;z)$.

See also:

Annotations for §3.11(iv), §3.11(iv), §3.11 and Ch.3

… ►From the equations $\partial S/\partial a_k=0$, $k=0,1,\mathrm{\dots },n$, we derive the normal equations …

10: Bibliography G

… ►

Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.

ⓘ

External Links:

ISBN 0-8176-3837-7, MathReview, ZentralBlatt

Referenced by:

§15.17(v).

Encodings:

BibTeX

…