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1: 19.30 Lengths of Plane Curves

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§19.30(i) Ellipse

►The arclength $s$ of the ellipse … ►The length of the ellipse is … ►Let $a^2$ and $b^2$ be replaced respectively by $a^2+\lambda$ and $b^2+\lambda$, where $\lambda \in (-b^2,\mathrm{\infty })$, to produce a family of confocal ellipses. … ►See Carlson (1977b, Ex. 9.4-1 and (9.4-4)) for arclengths of hyperbolas and ellipses in terms of $R_{-a}$ that differ only in the sign of $b^2$. …

2: 14.28 Sums

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14.28.2 $$\sum _{n=0}^{\mathrm{\infty }}(2n+1)Q_n\left(z_1\right)P_n\left(z_2\right)=\frac{1}{z_1-z_2},$$ $z_1\in {ℰ}_1$, $z_2\in {ℰ}_2$,

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Symbols:

$\in$: element of, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $z$: complex variable, $n$: nonnegative integer and ${ℰ}_1$, ${ℰ}_2$: ellipses

Permalink:

http://dlmf.nist.gov/14.28.E2

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See also:

Annotations for §14.28(ii), §14.28 and Ch.14

►where ${ℰ}_1$ and ${ℰ}_2$ are ellipses with foci at $\pm 1$, ${ℰ}_2$ being properly interior to ${ℰ}_1$. …

3: 19.15 Advantages of Symmetry

… ►For the many properties of ellipses and triaxial ellipsoids that can be represented by elliptic integrals, any symmetry in the semiaxes remains obvious when symmetric integrals are used (see (19.30.5) and §19.33). …

4: 22.18 Mathematical Applications

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Ellipse

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5: 31.11 Expansions in Series of Hypergeometric Functions

… ►and (31.11.1) converges to (31.3.10) outside the ellipse $ℰ$ in the $z$-plane with foci at 0, 1, and passing through the third finite singularity at $z=a$. … ►The expansion (31.11.1) for a Heun function that is associated with any branch of (31.11.2)—other than a multiple of the right-hand side of (31.11.12)—is convergent inside the ellipse $ℰ$. … ►For Heun functions (§31.4) they are convergent inside the ellipse $ℰ$. …

6: 4.15 Graphics

… ►Lines parallel to the real axis in the $z$-plane map onto ellipses in the $w$-plane with foci at $w=\pm 1$, and lines parallel to the imaginary axis in the $z$-plane map onto rectangular hyperbolas confocal with the ellipses. …

7: 14.24 Analytic Continuation

… ►Let $s$ be an arbitrary integer, and $P_{\nu }^{-\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)$ and ${𝑸}_{\nu }^{\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)$ denote the branches obtained from the principal branches by making $\frac{1}{2}s$ circuits, in the positive sense, of the ellipse having $\pm 1$ as foci and passing through $z$. …

8: 19.33 Triaxial Ellipsoids

… ►For additional geometrical properties of ellipsoids (and ellipses), see Carlson (1964, p. 417). …

9: 28.32 Mathematical Applications

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

10: 19.9 Inequalities

… ►The perimeter $L(a,b)$ of an ellipse with semiaxes $a,b$ is given by …Even for the extremely eccentric ellipse with $a=99$ and $b=1$, this is correct within 0. …