solutions of differential equations

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21: 36.15 Methods of Computation

… ►For numerical solution of partial differential equations satisfied by the canonical integrals see Connor et al. (1983).

22: 9.15 Mathematical Applications

… ►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …

23: 30.2 Differential Equations

§30.2 Differential Equations

►

§30.2(i) Spheroidal Differential Equation

… ► … ►The Liouville normal form of equation (30.2.1) is … ►

§30.2(iii) Special Cases

…

24: 16.21 Differential Equation

§16.21 Differential Equation

► $w=G_{p,q}^{m,n}(z;𝐚;𝐛)$ satisfies the differential equation ►

16.21.1 $$\left({(-1)}^{p-m-n}z(\mathit{\vartheta }-a_1+1)\mathrm{\cdots }(\mathit{\vartheta }-a_p+1)-(\mathit{\vartheta }-b_1)\mathrm{\cdots }(\mathit{\vartheta }-b_q)\right)w=0,$$

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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, $w$: solution and $\mathit{\vartheta }$: differential operator

Referenced by:

§16.21, §16.21

Permalink:

http://dlmf.nist.gov/16.21.E1

Encodings:

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See also:

Annotations for §16.21 and Ch.16

… ►This equation is of order $\max (p,q)$. … ►A fundamental set of solutions of (16.21.1) is given by …

25: 4.7 Derivatives and Differential Equations

… ►For a nonvanishing analytic function $f(z)$, the general solution of the differential equation … ►The general solution of the differential equation … ►The general solution of the differential equation …

26: 30.12 Generalized and Coulomb Spheroidal Functions

… ►Generalized spheroidal wave functions and Coulomb spheroidal functions are solutions of the differential equation …

27: 12.2 Differential Equations

§12.2 Differential Equations

… ►PCFs are solutions of the differential equation … ►All solutions are entire functions of $z$ and entire functions of $a$ or $\nu$. …

28: 16.8 Differential Equations

… ►When no $b_j$ is an integer, and no two $b_j$ differ by an integer, a fundamental set of solutions of (16.8.3) is given by …For other values of the $b_j$, series solutions in powers of $z$ (possibly involving also $\ln z$) can be constructed via a limiting process; compare §2.7(i) in the case of second-order differential equations. …

29: 31.1 Special Notation

… ►Sometimes the parameters are suppressed.

30: 10.47 Definitions and Basic Properties

… ►

§10.47(ii) Standard Solutions

… ►

§10.47(iii) Numerically Satisfactory Pairs of Solutions

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