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1: 1.9 Calculus of a Complex Variable

… ►and when $z\ne 0$, … ►

Conformal Transformation

… ►The angle between $C_1$ and $C_2$ at $z_0$ is the angle between the tangents to the two arcs at $z_0$, that is, the difference of the signed angles that the tangents make with the positive direction of the real axis. If $f^{\prime }(z_0)\ne 0$, then the angle between $C_1$ and $C_2$ equals the angle between $C_1^{\prime }$ and $C_2^{\prime }$ both in magnitude and sense. We then say that the mapping $w=f(z)$ is conformal (angle-preserving) at $z_0$. …

2: 4.42 Solution of Triangles

… ►

4.42.4 $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C},$$

ⓘ

Symbols:

$\sin z$: sine function, $a$: length, $b$: base, $c$: length, $A$: angle, $B$: angle and $C$: angle

A&S Ref:

4.3.148

Permalink:

http://dlmf.nist.gov/4.42.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(ii), §4.42 and Ch.4

… ►

4.42.6 $$a=b\cos C+c\cos B$$

ⓘ

Symbols:

$\cos z$: cosine function, $a$: length, $b$: base, $c$: length, $B$: angle and $C$: angle

A&S Ref:

4.3.148

Permalink:

http://dlmf.nist.gov/4.42.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(ii), §4.42 and Ch.4

… ►

4.42.9 $$\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c},$$

ⓘ

Symbols:

$\sin z$: sine function, $A$: angle, $B$: angle, $C$: angle, $a$: arc length, $b$: arc length and $c$: arc length

A&S Ref:

4.3.149

Permalink:

http://dlmf.nist.gov/4.42.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

►

4.42.10 $$\sin a\cos B=\cos b\sin c-\sin b\cos c\cos A,$$

ⓘ

Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $A$: angle, $B$: angle, $a$: arc length, $b$: arc length and $c$: arc length

Permalink:

http://dlmf.nist.gov/4.42.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

… ►

4.42.12 $$\cos A=-\cos B\cos C+\sin B\sin C\cos a.$$

ⓘ

Symbols:

$\cos z$: cosine function, $\sin z$: sine function, $A$: angle, $B$: angle, $C$: angle and $a$: arc length

A&S Ref:

4.3.149

Permalink:

http://dlmf.nist.gov/4.42.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.42(iii), §4.42 and Ch.4

…

3: 31.16 Mathematical Applications

… ►Expansions of Heun polynomial products in terms of Jacobi polynomial (§18.3) products are derived in Kalnins and Miller (1991a, b, 1993) from the viewpoint of interrelation between two bases in a Hilbert space: ►

31.16.1 $${\mathrm{𝐻𝑝}}_{n,m}\left(x\right){\mathrm{𝐻𝑝}}_{n,m}\left(y\right)=\sum _{j=0}^n{A}_j{\sin }^{2j}\theta P_{n-j}^{(\gamma +\delta +2j-1,ϵ-1)}\left(\cos \left(2\theta \right)\right)P_j^{(\delta -1,\gamma -1)}\left(\cos \left(2\varphi \right)\right),$$

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Symbols:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\cos z$: cosine function, $\sin z$: sine function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\beta$: real or complex parameter, $\theta$: angle, $\varphi$: angle, $x$: variable, $y$: variable and $A_j$: coefficients

Referenced by:

§31.16(ii), §31.16(ii)

Permalink:

http://dlmf.nist.gov/31.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.16(ii), §31.16 and Ch.31

… ►

4: 7.20 Mathematical Applications

… ►Then the arc length between the origin and $P(t)$ equals $t$, and is directly proportional to the curvature at $P(t)$, which equals $\pi t$. Furthermore, because $dy/dx=\tan \left(\frac{1}{2}\pi t^2\right)$, the angle between the $x$-axis and the tangent to the spiral at $P(t)$ is given by $\frac{1}{2}\pi t^2$. …

5: 1.6 Vectors and Vector-Valued Functions

… ►

Magnitude and Angle of Vector $𝐚$

… ►

1.6.4 $$\cos \theta =\frac{𝐚\cdot 𝐛}{\Vert 𝐚\Vert \Vert 𝐛\Vert };$$

ⓘ

Symbols:

$\cos z$: cosine function, $\Vert 𝐯\Vert$: vector norm ($l$²) and $\theta$: angle between $𝐚$ and $𝐛$

Permalink:

http://dlmf.nist.gov/1.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(i), §1.6(i), §1.6 and Ch.1

► $\theta$ is the angle between $𝐚$ and $𝐛$. … ►

1.6.9

ⓘ

Symbols:

$det$: determinant, $\sin z$: sine function, $\Vert 𝐯\Vert$: vector norm ($l$²), $\theta$: angle between $𝐚$ and $𝐛$ , $𝐢$: unit vector, $𝐣$: unit vector and $𝐤$: unit vector

Referenced by:

§1.6(i)

Permalink:

http://dlmf.nist.gov/1.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(i), §1.6(i), §1.6 and Ch.1

… ►For a sphere $x=\rho \sin \theta \cos \varphi$, $y=\rho \sin \theta \sin \varphi$, $z=\rho \cos \theta$, …

6: 19.11 Addition Theorems

… ►

19.11.1 $$F(\theta ,k)+F(\varphi ,k)=F(\psi ,k),$$

ⓘ

Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\varphi$: real or complex argument, $k$: real or complex modulus, $\theta$: amplitude and $\psi$: angle

Referenced by:

§19.11(i)

Permalink:

http://dlmf.nist.gov/19.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(i), §19.11 and Ch.19

►

19.11.2 $$E(\theta ,k)+E(\varphi ,k)=E(\psi ,k)+k^2\sin \theta \sin \varphi \sin \psi .$$

ⓘ

Symbols:

$E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $\theta$: amplitude and $\psi$: angle

Permalink:

http://dlmf.nist.gov/19.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(i), §19.11 and Ch.19

… ►

$\sin \psi ={\displaystyle \frac{(\sin \theta \cos \varphi )\mathrm{\Delta }(\varphi )+(\sin \varphi \cos \theta )\mathrm{\Delta }(\theta )}{1-k^2{\sin }^2\theta {\sin }^2\varphi }},$

… ►

19.11.12 $$F(\psi ,k)=2F(\theta ,k),$$

ⓘ

Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $k$: real or complex modulus, $\theta$: amplitude and $\psi$: angle

Permalink:

http://dlmf.nist.gov/19.11.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(iii), §19.11 and Ch.19

… ►

19.11.14 $$\sin \psi =(\sin 2\theta )\mathrm{\Delta }(\theta )/(1-k^2{\sin }^4\theta ),$$

ⓘ

Defines:

$\psi$: angle (locally)

Symbols:

$\sin z$: sine function, $k$: real or complex modulus, $\theta$: amplitude and $\mathrm{\Delta }(\theta )$

Permalink:

http://dlmf.nist.gov/19.11.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(iii), §19.11 and Ch.19

…

7: 20.15 Tables

… ►This reference gives ${\theta }_j(x,q)$, $j=1,2,3,4$, and their logarithmic $x$-derivatives to 4D for $x/\pi =0(.1)1$, $\alpha =0(9^{\circ }){90}^{\circ }$, where $\alpha$ is the modular angle given by ►

20.15.1 $$\sin \alpha ={\theta }_2^2(0,q)/{\theta }_3^2(0,q)=k.$$

ⓘ

Defines:

$\alpha$: modular angle (locally)

Symbols:

${\theta }_j(z,q)$: theta function, $\sin z$: sine function and $q$: nome

Referenced by:

§20.15, §20.15

Permalink:

http://dlmf.nist.gov/20.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.15 and Ch.20

…

8: 28.27 Addition Theorems

… ►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. …

9: 31.10 Integral Equations and Representations

… ►

31.10.8 $${\sin }^2\theta \left(\frac{{\partial }^2𝒦}{{\partial \theta }^2}+\left((1-2\gamma )\tan \theta +2(\delta +ϵ-\frac{1}{2})\cot \theta \right)\frac{\partial 𝒦}{\partial \theta }-4\alpha \beta 𝒦\right)+\frac{{\partial }^2𝒦}{{\partial \varphi }^2}+\left((1-2\delta )\cot \varphi -(1-2ϵ)\tan \varphi \right)\frac{\partial 𝒦}{\partial \varphi }=0.$$

ⓘ

Symbols:

$\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\tan z$: tangent function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $𝒦(z,t)$: kernel, $\theta$: angle and $\varphi$: angle

Referenced by:

§31.10(i)

Permalink:

http://dlmf.nist.gov/31.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(i), §31.10(i), §31.10 and Ch.31

… ►

31.10.9

ⓘ

Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $\cos z$: cosine function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $𝒦(z,t)$: kernel, $\theta$: angle, $\varphi$: angle and $\sigma$: separation constant

Referenced by:

§31.10(i)

Permalink:

http://dlmf.nist.gov/31.10.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(i), §31.10(i), §31.10 and Ch.31

… ►A further change of variables, to spherical coordinates, … ►

31.10.21 $$\frac{{\partial }^2𝒦}{{\partial r}^2}+\frac{2(\gamma +\delta +ϵ)-1}{r}\frac{\partial 𝒦}{\partial r}+\frac{1}{r^2}\frac{{\partial }^2𝒦}{{\partial \theta }^2}+\frac{(2(\delta +ϵ)-1)\cot \theta -(2\gamma -1)\tan \theta }{r^2}\frac{\partial 𝒦}{\partial \theta }+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^2𝒦}{{\partial \varphi }^2}+\frac{(2\delta -1)\cot \varphi -(2ϵ-1)\tan \varphi }{r^2{\sin }^2\theta }\frac{\partial 𝒦}{\partial \varphi }=0.$$

ⓘ

Symbols:

$\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\tan z$: tangent function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $𝒦(z;s,t)$: kernel, $r$: radius, $\theta$: angle and $\varphi$: angle

Permalink:

http://dlmf.nist.gov/31.10.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(ii), §31.10(ii), §31.10 and Ch.31

… ►

31.10.22

ⓘ

Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $\cos z$: cosine function, $\sin z$: sine function, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $𝒦(z;s,t)$: kernel, $r$: radius, $\theta$: angle, $\varphi$: angle, $b$ and $c$

Permalink:

http://dlmf.nist.gov/31.10.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §31.10(ii), §31.10(ii), §31.10 and Ch.31

…

10: Simon Ruijsenaars

… ►His main research interests cover integrable systems, special functions, analytic difference equations, classical and quantum mechanics, and the relations between these areas. …