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11: 28.33 Physical Applications

… ►We shall derive solutions to the uniform, homogeneous, loss-free, and stretched elliptical ring membrane with mass $\rho$ per unit area, and radial tension $\tau$ per unit arc length. … ►

28.33.1 $$\frac{{\partial }^{2}W}{{\partial x}^2}+\frac{{\partial }^{2}W}{{\partial y}^2}-\frac{\rho }{\tau }\frac{{\partial }^{2}W}{{\partial t}^2}=0,$$

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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, $\rho$: mass, $\tau$: radial tension and $W(x,y,t)$: solution

Permalink:

http://dlmf.nist.gov/28.33.E1

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Annotations for §28.33(ii), §28.33 and Ch.28

►with $W(x,y,t)={\mathrm{e}}^{\mathrm{i}\omega t}V(x,y)$, reduces to (28.32.2) with $k^2={\omega }^2\rho /\tau$. …If we denote the positive solutions $q$ of (28.33.3) by $q_{n,m}$, then the vibration of the membrane is given by ${\omega }_{n,m}^2=4q_{n,m}\tau /(c^2\rho )$. … ►However, in response to a small perturbation at least one solution may become unbounded. …

12: 1.9 Calculus of a Complex Variable

… ►That is, given any positive number $ϵ$, however small, we can find a positive number $\delta$ such that for all $z$ in the open disk . … ►For $z$ in $\left|z-z_0\right|\le \rho$ (), the convergence is absolute and uniform. … ►Then the expansions (1.9.54), (1.9.57), and (1.9.60) hold for all sufficiently small $\left|z\right|$. …

13: 19.36 Methods of Computation

… ►When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. … ►This method loses significant figures in $\rho$ if ${\alpha }^2$ and $k^2$ are nearly equal unless they are given exact values—as they can be for tables. … ►When the values of complete integrals are known, addition theorems with $\psi =\pi /2$ (§19.11(ii)) ease the computation of functions such as $F(\varphi ,k)$ when $\frac{1}{2}\pi -\varphi$ is small and positive. Similarly, §19.26(ii) eases the computation of functions such as $R_F(x,y,z)$ when $x$ ($>0$) is small compared with $\min (y,z)$. …

14: 1.5 Calculus of Two or More Variables

… ►that is, for every arbitrarily small positive constant $ϵ$ there exists $\delta$ ($>0$) such that … ►With , $0\le \varphi \le 2\pi$, $0\le \theta \le \pi$, ►

$x=\rho \sin \theta \cos \varphi ,$

… ►

$z=\rho \cos \theta .$

… ►Suppose also that ${\int }_c^{d}f(x,y)dy$ converges and ${\int }_c^d(\partial f/\partial x)dy$ converges uniformly on $a\le x\le b$, that is, given any positive number $ϵ$, however small, we can find a number $c_0\in [c,d)$ that is independent of $x$ and is such that …

15: 10.73 Physical Applications

… ►Bessel functions first appear in the investigation of a physical problem in Daniel Bernoulli’s analysis of the small oscillations of a uniform heavy flexible chain. … ►The functions ${𝗃}_n\left(x\right)$, ${𝗒}_n\left(x\right)$, ${𝗁}_n^{(1)}\left(x\right)$, and ${𝗁}_n^{(2)}\left(x\right)$ arise in the solution (again by separation of variables) of the Helmholtz equation in spherical coordinates $\rho ,\theta ,\varphi$ (§1.5(ii)): ►

10.73.4 $$({\nabla }^2+k^2)f=\frac{1}{{\rho }^2}\frac{\partial }{\partial \rho }\left({\rho }^2\frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial f}{\partial \theta }\right)+\frac{1}{{\rho }^2{\sin }^2\theta }\frac{{\partial }^{2}f}{{\partial \varphi }^2}+k^{2}f.$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/10.73.E4

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Annotations for §10.73(ii), §10.73 and Ch.10

►With the spherical harmonic $Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ defined as in §14.30(i), the solutions are of the form $f=g_{\mathrm{\ell }}(k\rho )Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ with $g_{\mathrm{\ell }}={𝗃}_{\mathrm{\ell }}$, ${𝗒}_{\mathrm{\ell }}$, ${𝗁}_{\mathrm{\ell }}^{(1)}$, or ${𝗁}_{\mathrm{\ell }}^{(2)}$, depending on the boundary conditions. …

16: 2.6 Distributional Methods

… ►Let $\epsilon$ be a positive number, and ►

2.6.22 $${\varphi }_{\epsilon }(t)=\frac{{\mathrm{e}}^{-\epsilon t}}{t+z},$$ $t\in (0,\mathrm{\infty })$.

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Symbols:

$\in$: element of, $\mathrm{e}$: base of natural logarithm, $(a,b)$: open interval and $\epsilon$: small positive number

Permalink:

http://dlmf.nist.gov/2.6.E22

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Annotations for §2.6(ii), §2.6 and Ch.2

… ►

2.6.23 $$\underset{\epsilon \to 0}{lim}⟨t^{-s-\alpha },{\varphi }_{\epsilon }⟩=\frac{\pi }{\sin \left(\pi \alpha \right)}\frac{{(-1)}^s}{z^{s+\alpha }},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $\sin z$: sine function and $\epsilon$: small positive number

Permalink:

http://dlmf.nist.gov/2.6.E23

Encodings:

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Annotations for §2.6(ii), §2.6 and Ch.2

►

2.6.24 $$\underset{\epsilon \to 0}{lim}⟨t^{-s-1},{\varphi }_{\epsilon }⟩=\frac{{(-1)}^{s+1}}{z^{s+1}}\sum _{k=1}^s\frac{1}{k}+\frac{{(-1)}^s}{z^{s+1}}\ln z,$$

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Symbols:

$⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $\ln z$: principal branch of logarithm function and $\epsilon$: small positive number

Permalink:

http://dlmf.nist.gov/2.6.E24

Encodings:

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Annotations for §2.6(ii), §2.6 and Ch.2

… ►

2.6.32 $${\int }_0^{\mathrm{\infty }}\frac{f(t)}{{(t+z)}^{\rho }}dt,$$ $\rho >0$,

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Symbols:

$dx$: differential of $x$ , $\int$: integral and $f(t)$: locally integrable function

Permalink:

http://dlmf.nist.gov/2.6.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

…