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1: 28.10 Integral Equations

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§28.10(iii) Further Equations

… ►For relations with variable boundaries see Volkmer (1983).

2: 1.4 Calculus of One Variable

… ►For the functions discussed in the following DLMF chapters these two integration measures are adequate, as these special functions are analytic functions of their variables, and thus $C^{\mathrm{\infty }}$, and well defined for all values of these variables; possible exceptions being at boundary points. …

3: 12.17 Physical Applications

… ►By using instead coordinates of the parabolic cylinder $\xi ,\eta ,\zeta$, defined by …Setting $w=U(\xi )V(\eta )W(\zeta )$ and separating variables, we obtain … ►Buchholz (1969) collects many results on boundary-value problems involving PCFs. Miller (1974) treats separation of variables by group theoretic methods. … ►For this topic and other boundary-value problems see Boyd (1973), Hillion (1997), Magnus (1941), Morse and Feshbach (1953a, b), Müller (1988), Ott (1985), Rice (1954), and Shanmugam (1978). …

4: 1.13 Differential Equations

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§1.13(iv) Change of Variables

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Elimination of First Derivative by Change of Independent Variable

… ►Assuming that $u(x)$ satisfies un-mixed boundary conditions of the form …or periodic boundary conditions … ►For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, $\lambda$; (ii) the corresponding (real) eigenfunctions, $u(x)$ and $w(t)$, have the same number of zeros, also called nodes, for $t\in (0,c)$ as for $x\in (a,b)$; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …

5: 28.32 Mathematical Applications

… ►If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient. These are given by … ►

28.32.2 $$\frac{{\partial }^{2}V}{{\partial x}^2}+\frac{{\partial }^{2}V}{{\partial y}^2}+k^{2}V=0$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable, $y$: real variable, $k$: parameter and $V(\xi ,\eta )$: solution

Referenced by:

§28.33(ii)

Permalink:

http://dlmf.nist.gov/28.32.E2

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Annotations for §28.32(i), §28.32 and Ch.28

… ►This leads to integral equations and an integral relation between the solutions of Mathieu’s equation (setting $\zeta =\mathrm{i}\xi$, $z=\eta$ in (28.32.3)). … ►

28.32.6 $$w(z)={\int }_{ℒ}K(z,\zeta )u(\zeta )d\zeta$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $z$: complex variable, $\zeta$: variable, $u(\zeta )$: solution, $K(z,\zeta )$: solution and $ℒ$: curve

Permalink:

http://dlmf.nist.gov/28.32.E6

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Annotations for §28.32(i), §28.32 and Ch.28

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6: 10.73 Physical Applications

… ►and on separation of variables we obtain solutions of the form ${\mathrm{e}}^{\pm \mathrm{i}n\varphi }{\mathrm{e}}^{\pm \kappa z}{J}_n\left(\kappa r\right)$, from which a solution satisfying prescribed boundary conditions may be constructed. …

7: 12.15 Generalized Parabolic Cylinder Functions

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12.15.1 $$\frac{d^{2}w}{{dz}^2}+\left(\nu +{\lambda }^{-1}-{\lambda }^{-2}{z}^{\lambda }\right)w=0$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $\nu$: real or complex parameter

Referenced by:

§12.15

Permalink:

http://dlmf.nist.gov/12.15.E1

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Annotations for §12.15 and Ch.12

… ►This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. …

8: 20.13 Physical Applications

… ►For $\tau =\mathrm{i}t$, with $\alpha ,t,z$ real, (20.13.1) takes the form of a real-time $t$ diffusion equation ►

20.13.2 $$\partial \theta /\partial t=\alpha {\partial }^2\theta /{\partial z}^2,$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $t$: time, $\alpha$: real parameter and $z$: real variable

Referenced by:

§20.13

Permalink:

http://dlmf.nist.gov/20.13.E2

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Annotations for §20.13 and Ch.20

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20.13.3 $$g(z,t)=\sqrt{\frac{\pi }{4\alpha t}}\exp \left(-\frac{z^2}{4\alpha t}\right)$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $t$: time, $\alpha$: real parameter and $z$: real variable

Permalink:

http://dlmf.nist.gov/20.13.E3

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Annotations for §20.13 and Ch.20

… ►These two apparently different solutions differ only in their normalization and boundary conditions. …Theta-function solutions to the heat diffusion equation with simple boundary conditions are discussed in Lawden (1989, pp. 1–3), and with more general boundary conditions in Körner (1989, pp. 274–281). …

9: 28.33 Physical Applications

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Boundary-values problems arising from solution of the two-dimensional wave equation in elliptical coordinates. This yields a pair of equations of the form (28.2.1) and (28.20.1), and the appropriate solution of (28.2.1) is usually a periodic solution of integer order. See §28.33(ii).

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§28.33(ii) Boundary-Value Problems

… ►with $W(x,y,t)={\mathrm{e}}^{\mathrm{i}\omega t}V(x,y)$, reduces to (28.32.2) with $k^2={\omega }^2\rho /\tau$. …The boundary conditions for $\xi ={\xi }_0$ (outer clamp) and $\xi ={\xi }_1$ (inner clamp) yield the following equation for $q$: … ►For a visualization see Gutiérrez-Vega et al. (2003), and for references to other boundary-value problems see: …

10: 29.3 Definitions and Basic Properties

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Table 29.3.2: Lamé functions.

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boundary conditions	eigenvalue $h$	eigenfunction $w(z)$	parity of $w(z)$	parity of $w(z-K)$	period of $w(z)$
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