solution of triangles and spherical triangles

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1: 4.42 Solution of Triangles

§4.42 Solution of Triangles

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§4.42(i) Planar Right Triangles

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§4.42(ii) Planar Triangles

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§4.42(iii) Spherical Triangles

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Figure 4.42.3: Spherical triangle. Magnify

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2: 15.17 Mathematical Applications

… ►The quotient of two solutions of (15.10.1) maps the closed upper half-plane $\mathrm{\Im }z\ge 0$ conformally onto a curvilinear triangle. …

3: Sidebar 9.SB2: Interference Patterns in Caustics

… ►The bright sharp-edged triangle is a caustic, that is a line of focused light. …

4: 34.10 Zeros

… ►In a $\mathit{3}j$ symbol, if the three angular momenta $j_1,j_2,j_3$ do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the $\mathit{3}j$ symbol is zero. Similarly the $\mathit{6}j$ symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four $\mathit{3}j$ symbols in the summation. …However, the $\mathit{3}j$ and $\mathit{6}j$ symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. …

5: 28.29 Definitions and Basic Properties

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28.29.15 $$△(\lambda )=w_{\text{I}}(\pi ,\lambda )+w_{\text{II}}^{\prime }(\pi ,\lambda )$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $w_{\text{I}}(z;a,q)$: fundamental solution, $w_{\text{II}}(z;a,q)$: fundamental solution and $\lambda$: parameter

Permalink:

http://dlmf.nist.gov/28.29.E15

Encodings:

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See also:

Annotations for §28.29(iii), §28.29 and Ch.28

… ►For a given $\nu$, the characteristic equation $△(\lambda )-2\cos \left(\pi \nu \right)=0$ has infinitely many roots $\lambda$. Conversely, for a given $\lambda$, the value of $△(\lambda )$ is needed for the computation of $\nu$. … ►

28.29.16 $${\lambda }_n,n=0,1,2,\mathrm{\dots },\text{ with }△({\lambda }_n)=2,$$

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Defines:

${\lambda }_n$: eigenvalues (locally)

Symbols:

$n$: integer

Referenced by:

§28.30(i)

Permalink:

http://dlmf.nist.gov/28.29.E16

Encodings:

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See also:

Annotations for §28.29(iii), §28.29 and Ch.28

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28.29.17 $${\mu }_n,n=1,2,3,\mathrm{\dots },\text{ with }△({\mu }_n)=-2.$$

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Defines:

${\mu }_n$: eigenvalues (locally)

Symbols:

$n$: integer

Referenced by:

§28.30(i)

Permalink:

http://dlmf.nist.gov/28.29.E17

Encodings:

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See also:

Annotations for §28.29(iii), §28.29 and Ch.28

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6: 18.37 Classical OP’s in Two or More Variables

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§18.37(ii) OP’s on the Triangle

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Definition in Terms of Jacobi Polynomials

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18.37.7 $$P_{m,n}^{\alpha ,\beta ,\gamma }(x,y)=P_{m-n}^{(\alpha ,\beta +\gamma +2n+1)}\left(2x-1\right)x^n{P}_n^{(\beta ,\gamma )}\left(2x^{-1}y-1\right),$$ $m\ge n\ge 0$, $\alpha ,\beta ,\gamma >-1$.

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Defines:

$P_{m,n}^{\alpha ,\beta ,\gamma }(x,y)$: triangle polynomial

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $y$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.37(ii)

Permalink:

http://dlmf.nist.gov/18.37.E7

Encodings:

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See also:

Annotations for §18.37(ii), §18.37(ii), §18.37 and Ch.18

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18.37.8 $m\ne j$ and/or $n\ne \mathrm{\ell }$.

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Symbols:

$dx$: differential of $x$ , $P_{m,n}^{\alpha ,\beta ,\gamma }(x,y)$: triangle polynomial, $y$: real variable, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.37.E8

Encodings:

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See also:

Annotations for §18.37(ii), §18.37(ii), §18.37 and Ch.18

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7: 18.1 Notation

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Triangle: $P_{m,n}^{\alpha ,\beta ,\gamma }(x,y)$.

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8: 34.3 Basic Properties: $\mathit{3}j$ Symbol

… ►Then assuming the triangle conditions are satisfied … ►Again it is assumed that in (34.3.7) the triangle conditions are satisfied. … ►In the following three equations it is assumed that the triangle conditions are satisfied by each $\mathit{3}j$ symbol. …

9: 10.47 Definitions and Basic Properties

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§10.47(ii) Standard Solutions

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§10.47(iii) Numerically Satisfactory Pairs of Solutions

►For (10.47.1) numerically satisfactory pairs of solutions are given by Table 10.2.1 with the symbols $J$, $Y$, $H$, and $\nu$ replaced by $𝗃$, $𝗒$, $𝗁$, and $n$, respectively. ►For (10.47.2) numerically satisfactory pairs of solutions are ${𝗂}_n^{(1)}\left(z\right)$ and ${𝗄}_n\left(z\right)$ in the right half of the $z$-plane, and ${𝗂}_n^{(1)}\left(z\right)$ and ${𝗄}_n\left(-z\right)$ in the left half of the $z$-plane. …

10: 34.2 Definition: $\mathit{3}j$ Symbol

… ►They must form the sides of a triangle (possibly degenerate). They therefore satisfy the triangle conditions …