partial differential equations

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31: 28.28 Integrals, Integral Representations, and Integral Equations

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28.28.4 $$\frac{\mathrm{i}h}{\pi }{\int }_0^{2\pi }\frac{\partial w}{\partial \alpha }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{ce}}_n(t,h^2)dt={\mathrm{i}}^n{\mathrm{ce}}_n^{\prime }(\alpha ,h^2){\mathrm{Mc}}_n^{(1)}(z,h),$$

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Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $h$: parameter, $n$: integer, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\alpha$: parameter

Permalink:

http://dlmf.nist.gov/28.28.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

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28.28.5 $$\frac{\mathrm{i}h}{\pi }{\int }_0^{2\pi }\frac{\partial w}{\partial \alpha }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{se}}_n(t,h^2)dt={\mathrm{i}}^n{\mathrm{se}}_n^{\prime }(\alpha ,h^2){\mathrm{Ms}}_n^{(1)}(z,h).$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $h$: parameter, $n$: integer, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\alpha$: parameter

Permalink:

http://dlmf.nist.gov/28.28.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

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28.28.8 $$\frac{1}{\pi }{\int }_{{ℒ}_j}2\mathrm{i}h\frac{\partial w}{\partial \alpha }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{me}}_{\nu }(t,h^2)dt={\mathrm{e}}^{\mathrm{i}\nu \pi /2}{\mathrm{me}}_{\nu }^{\prime }(\alpha ,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h),$$ $j=3,4$,

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $h$: parameter, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $w(z)$: Mathieu’s equation solution, $\alpha$: parameter and ${ℒ}_j$: integration paths

A&S Ref:

20.7.19 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.28.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

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32: 1.5 Calculus of Two or More Variables

… ►If $F(x,y)$ is continuously differentiable, $F(a,b)=0$, and $\partial F/\partial y\ne 0$ at $(a,b)$, then in a neighborhood of $(a,b)$, that is, an open disk centered at $a,b$, the equation $F(x,y)=0$ defines a continuously differentiable function $y=g(x)$ such that $F(x,g(x))=0$, $b=g(a)$, and $g^{\prime }(x)=-F_x/F_y$. …

33: 1.13 Differential Equations

§1.13 Differential Equations

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Fundamental Pair

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Wronskian

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§1.13(iii) Inhomogeneous Equations

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34: 18.39 Applications in the Physical Sciences

… ►Here the term $-\frac{{\mathrm{\hslash }}^2}{2m}\frac{{\partial }^2}{{\partial x}^2}$ represents the quantum kinetic energy of a single particle of mass $m$, and $V(x)$ its potential energy. … ►The finite system of functions ${\psi }_n$ is orthonormal in $L^2(ℝ,dx)$, see (18.34.7_3). … ►The Schrödinger equation with potential … ►

Other Analytically Solved Schrödinger Equations

… ►Substitution of (18.39.24) into (18.39.23) then gives the ordinary differential equation for the radial wave function $R_{n,l}(r)$, …

35: 13.8 Asymptotic Approximations for Large Parameters

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13.8.16 $$(k+1)c_{k+1}(z)+\sum _{s=0}^k\left(\frac{b{B}_{s+1}}{(s+1)!}+\frac{z(s+1)B_{s+2}}{(s+2)!}\right)c_{k-s}(z)=0,$$ $k=0,1,2,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $(a,b)$: open interval, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $s$: nonnegative integer and $z$: complex variable

Referenced by:

(13.8.16), §13.8(iii)

Permalink:

http://dlmf.nist.gov/13.8.E16

Encodings:

pMML, png, TeX

Addition (effective with 1.0.11):

The whole paragraph containing this equation has been added. In Temme (2015, Equation 10.3.28) the generating function $f(z,t)$ for $c_k(z)$ is given. It satisfies,

$$\frac{\partial f}{\partial t}=\left(b\left(\frac{1}{t}-\frac{1}{{\mathrm{e}}^t-1}\right)-z\left(\frac{1}{t^2}-\frac{{\mathrm{e}}^t}{{\left({\mathrm{e}}^t-1\right)}^2}\right)\right)f.$$

For (13.8.16) combine this differentialequation with $f(z,t)={\sum }_{k=0}^{\mathrm{\infty }}{c}_k(z)t^k$ and (24.4.1).

See also:

Annotations for §13.8(iii), §13.8 and Ch.13

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36: 21.9 Integrable Equations

§21.9 Integrable Equations

►Riemann theta functions arise in the study of integrable differential equations that have applications in many areas, including fluid mechanics (Ablowitz and Segur (1981, Chapter 4)), magnetic monopoles (Ercolani and Sinha (1989)), and string theory (Deligne et al. (1999, Part 3)). …Here, and in what follows, $x,y$, and $t$ suffixes indicate partial derivatives. … ► … ►

37: 32.6 Hamiltonian Structure

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§32.6(ii) First Painlevé Equation

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§32.6(iii) Second Painlevé Equation

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§32.6(iv) Third Painlevé Equation

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§32.6(v) Other Painlevé Equations

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38: 9.18 Tables

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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.

39: 30.14 Wave Equation in Oblate Spheroidal Coordinates

§30.14 Wave Equation in Oblate Spheroidal Coordinates

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§30.14(i) Oblate Spheroidal Coordinates

… ►The wave equation (30.13.7), transformed to oblate spheroidal coordinates $(\xi ,\eta ,\varphi )$, admits solutions of the form (30.13.8), where $w_1$ satisfies the differential equation …Equation (30.14.7) can be transformed to equation (30.2.1) by the substitution $z=\pm \mathrm{i}\xi$. … ►

40: 30.13 Wave Equation in Prolate Spheroidal Coordinates

§30.13 Wave Equation in Prolate Spheroidal Coordinates

… ►The wave equation …where $w_1$, $w_2$, $w_3$ satisfy the differential equations …Equations (30.13.9) and (30.13.10) agree with (30.2.1). … ►