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31: 32.3 Graphics

§32.3 Graphics

… ►Plots of solutions $w_k(x)$ of ${\text{P}}_{\text{I}}$ with $w_k(0)=0$ and $w_k^{\prime }(0)=k$ for various values of $k$, and the parabola $6w^2+x=0$. … ►Here $w_k(x)$ is the solution of ${\text{P}}_{\text{II}}$ with $\alpha =0$ and such that … ►Here $u=u_k(x;\nu )$ is the solution of …The corresponding solution of ${\text{P}}_{\text{IV}}$ is given by …

32: 15.10 Hypergeometric Differential Equation

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§15.10(i) Fundamental Solutions

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§15.10(ii) Kummer’s 24 Solutions and Connection Formulas

►The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions. … ►The $\left(\genfrac{}{}{0.0pt}{}{6}{3}\right)=20$ connection formulas for the principal branches of Kummer’s solutions are: …

33: 15.11 Riemann’s Differential Equation

… ►The most general form is given by … ►The complete set of solutions of (15.11.1) is denoted by Riemann’s $P$-symbol: … ►

15.11.4

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Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $w$: solution

Permalink:

http://dlmf.nist.gov/15.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(i), §15.11 and Ch.15

►denotes the set of solutions of (15.10.1). ►

§15.11(ii) Transformation Formulas

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34: Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate

… ►Jacobian elliptic functions arise as solutions to certain nonlinear Schrödinger equations, which model many types of wave propagation phenomena. …

35: 30.12 Generalized and Coulomb Spheroidal Functions

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

30.12.1 $$\frac{d}{dz}\left((1-z^2)\frac{dw}{dz}\right)+\left(\lambda +\alpha z+{\gamma }^2(1-z^2)-\frac{{\mu }^2}{1-z^2}\right)w=0,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $w$: solution to DE, $\lambda$: real parameter, ${\gamma }^2$: real parameter and $\mu$: real parameter

Referenced by:

§30.12

Permalink:

http://dlmf.nist.gov/30.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.12 and Ch.30

… ►For the theory and computation of solutions of (30.12.1) see Falloon (2001), Judd (1975), Leaver (1986), and Komarov et al. (1976). … ►

30.12.2 $$\frac{d}{dz}\left((1-z^2)\frac{dw}{dz}\right)+\left(\lambda +{\gamma }^2(1-z^2)-\frac{\alpha (\alpha +1)}{z^2}-\frac{{\mu }^2}{1-z^2}\right)w=0,$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $w$: solution to DE, $\lambda$: real parameter, ${\gamma }^2$: real parameter and $\mu$: real parameter

Permalink:

http://dlmf.nist.gov/30.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.12 and Ch.30

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36: 31.1 Special Notation

… ►Sometimes the parameters are suppressed.

37: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

§31.5 Solutions Analytic at Three Singularities: Heun Polynomials

… ►is a polynomial of degree $n$, and hence a solution of (31.2.1) that is analytic at all three finite singularities $0,1,a$. These solutions are the Heun polynomials. …

38: T. Mark Dunster

… ►He has received a number of National Science Foundation grants, and has published numerous papers in the areas of uniform asymptotic solutions of differential equations, convergent WKB methods, special functions, quantum mechanics, and scattering theory. …

39: 4.42 Solution of Triangles

§4.42 Solution of Triangles

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§4.42(iii) Spherical Triangles

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40: 13.27 Mathematical Applications

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