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31: 10.12 Generating Function and Associated Series

… ►For $z\in ℂ$ and $t\in ℂ\setminus \{0\}$, … ►Jacobi–Anger expansions: for $z,\theta \in ℂ$, ►

$\cos \left(z\sin \theta \right)=J_0\left(z\right)+2{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{J}_{2k}\left(z\right)\cos \left(2k\theta \right),$

►

$\sin \left(z\sin \theta \right)=2{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{J}_{2k+1}\left(z\right)\sin \left((2k+1)\theta \right),$

►

$\cos \left(z\cos \theta \right)=J_0\left(z\right)+2{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{(-1)}^k{J}_{2k}\left(z\right)\cos \left(2k\theta \right),$

…

32: 14.13 Trigonometric Expansions

… ►When , and $\nu +\mu$ is not a negative integer, …These Fourier series converge absolutely when . If then they converge, but, if $\theta \ne \frac{1}{2}\pi$, they do not converge absolutely. … ►

14.13.3 $${𝖯}_n\left(\cos \theta \right)=\frac{2^{2n+2}{(n!)}^2}{\pi (2n+1)!}\sum _{k=0}^{\mathrm{\infty }}\frac{1\cdot 3\mathrm{\cdots }(2k-1)}{k!}\frac{(n+1)(n+2)\mathrm{\cdots }(n+k)}{(2n+3)(2n+5)\mathrm{\cdots }(2n+2k+1)}\sin \left((n+2k+1)\theta \right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function, $n$: nonnegative integer and : variable

A&S Ref:

8.7.3

Permalink:

http://dlmf.nist.gov/14.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

►

14.13.4 $${𝖰}_n\left(\cos \theta \right)=\frac{2^{2n+1}{(n!)}^2}{(2n+1)!}\sum _{k=0}^{\mathrm{\infty }}\frac{1\cdot 3\mathrm{\cdots }(2k-1)}{k!}\frac{(n+1)(n+2)\mathrm{\cdots }(n+k)}{(2n+3)(2n+5)\mathrm{\cdots }(2n+2k+1)}\cos \left((n+2k+1)\theta \right),$$

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Symbols:

$\cos z$: cosine function, $!$: factorial (as in $n!$), ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $n$: nonnegative integer and : variable

A&S Ref:

8.7.4

Permalink:

http://dlmf.nist.gov/14.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

…

33: 14.30 Spherical and Spheroidal Harmonics

… ►With $l$ and $m$ integers such that $|m|\le l$, and $\theta$ and $\varphi$ angles such that $0\le \theta \le \pi$, $0\le \varphi \le 2\pi$, …$Y_{l,m}(\theta ,\varphi )$ are known as spherical harmonics. …Sometimes $Y_{l,m}(\theta ,\varphi )$ is denoted by ${\mathrm{i}}^{-l}{𝔇}_{lm}(\theta ,\varphi )$; also the definition of $Y_{l,m}(\theta ,\varphi )$ can differ from (14.30.1), for example, by inclusion of a factor ${(-1)}^m$. … ►where $𝐚=(\frac{1}{2\lambda }-\frac{\lambda }{2},-\frac{\mathrm{i}}{2\lambda }-\frac{\mathrm{i}\lambda }{2},1)$ and $𝐱=(r\sin \theta \cos \varphi ,r\sin \theta \sin \varphi ,r\cos \theta )$. … ►has solutions $W(\rho ,\theta ,\varphi )={\rho }^l{Y}_{l,m}(\theta ,\varphi )$, which are everywhere one-valued and continuous. …

34: 18.15 Asymptotic Approximations

… ►as $n\to \mathrm{\infty }$, uniformly with respect to $\theta \in [\delta ,\pi -\delta ]$. … ►as $n\to \mathrm{\infty }$, uniformly with respect to $\theta \in [\delta ,\pi -\delta ]$. … ►as $n\to \mathrm{\infty }$ uniformly with respect to $\theta \in [\delta ,\pi -\delta ]$, where … ►Also, when , the right-hand side of (18.15.12) with $M=\mathrm{\infty }$ converges; paradoxically, however, the sum is $2P_n\left(\cos \theta \right)$ and not $P_n\left(\cos \theta \right)$ as stated erroneously in Szegő (1975, §8.4(3)). … ►For asymptotic expansions of $P_n\left(\cos \theta \right)$ and $P_n\left(\cosh \xi \right)$ that are uniformly valid when $0\le \theta \le \pi -\delta$ and see §14.15(iii) with $\mu =0$ and $\nu =n$. …

35: 4.17 Special Values and Limits

… ►

Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

► ►►►

$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
0	0	1	0	$\mathrm{\infty }$	1	$\mathrm{\infty }$
…

►

►

4.17.1 $$\underset{z\to 0}{lim}\frac{\sin z}{z}=1,$$

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Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.74

Permalink:

http://dlmf.nist.gov/4.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

►

4.17.2 $$\underset{z\to 0}{lim}\frac{\tan z}{z}=1.$$

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Symbols:

$\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.75

Permalink:

http://dlmf.nist.gov/4.17.E2

Encodings:

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See also:

Annotations for §4.17 and Ch.4

►

4.17.3 $$\underset{z\to 0}{lim}\frac{1-\cos z}{z^2}=\frac{1}{2}.$$

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Symbols:

$\cos z$: cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

36: 1.15 Summability Methods

… ►If $f(\theta )$ is periodic and integrable on $[0,2\pi ]$, then as $n\to \mathrm{\infty }$ the Abel means $A(r,\theta )$ and the (C,1) means ${\sigma }_n(\theta )$ converge to …at every point $\theta$ where both limits exist. If $f(\theta )$ is also continuous, then the convergence is uniform for all $\theta$. ►For real-valued $f(\theta )$, if …is the Fourier series of $f(\theta )$, then the series …

37: 27.13 Functions

… ►If $3^k=q{2}^k+r$ with , then equality holds in (27.13.2) provided $r+q\le 2^k$, a condition that is satisfied with at most a finite number of exceptions. … ►Jacobi (1829) notes that $r_2\left(n\right)$ is the coefficient of $x^n$ in the square of the theta function $\vartheta \left(x\right)$: ►

27.13.4 $$\vartheta \left(x\right)=1+2\sum _{m=1}^{\mathrm{\infty }}{x}^{m^2},$$ .

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Defines:

$\vartheta \left(x\right)={\theta }_3(0,x)$: alternative notation

Symbols:

${\theta }_j(z,q)$: theta function, $m$: positive integer and $x$: real number

A&S Ref:

16.27.3 (with $z=0$, $q=x$)

Referenced by:

§27.13(iv)

Permalink:

http://dlmf.nist.gov/27.13.E4

Encodings:

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See also:

Annotations for §27.13(iv), §27.13 and Ch.27

►(In §20.2(i), $\vartheta \left(x\right)$ is denoted by ${\theta }_3(0,x)$.) … ►Mordell (1917) notes that $r_k\left(n\right)$ is the coefficient of $x^n$ in the power-series expansion of the $k$th power of the series for $\vartheta \left(x\right)$. …

38: 20.6 Power Series

§20.6 Power Series

… ►where $z_{m,n}$ is given by (20.2.5) and the minimum is for $m,n\in \mathbb{Z}$, except $m=n=0$. … ►

20.6.2 $${\theta }_1\left(\pi z\right|\tau )=\pi z{\theta }_1^{\prime }\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\delta }_{2j}(\tau )z^{2j}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $z$: complex, $\tau$: lattice parameter and ${\delta }_{2j}(\tau )$: coefficient

Referenced by:

§20.6

Permalink:

http://dlmf.nist.gov/20.6.E2

Encodings:

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See also:

Annotations for §20.6 and Ch.20

►

20.6.3 $${\theta }_2\left(\pi z\right|\tau )={\theta }_2\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\alpha }_{2j}(\tau )z^{2j}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $z$: complex, $\tau$: lattice parameter and ${\alpha }_{2j}(\tau )$: coefficient

Permalink:

http://dlmf.nist.gov/20.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

►

20.6.4 $${\theta }_3\left(\pi z\right|\tau )={\theta }_3\left(0\right|\tau )\exp \left(-\sum _{j=1}^{\mathrm{\infty }}\frac{1}{2j}{\beta }_{2j}(\tau )z^{2j}\right),$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $z$: complex, $\tau$: lattice parameter and ${\beta }_{2j}(\tau )$: coefficient

Permalink:

http://dlmf.nist.gov/20.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.6 and Ch.20

…

39: 10.68 Modulus and Phase Functions

… ►where $M_{\nu }\left(x\right)(>0)$, $N_{\nu }\left(x\right)(>0)$, ${\theta }_{\nu }\left(x\right)$, and ${\varphi }_{\nu }\left(x\right)$ are continuous real functions of $x$ and $\nu$, with the branches of ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ chosen to satisfy (10.68.18) and (10.68.21) as $x\to \mathrm{\infty }$. … ►

${\theta }_0^{\prime }=(M_1/M_0)\sin \left({\theta }_1-{\theta }_0-\frac{1}{4}\pi \right).$

… ►Equations (10.68.8)–(10.68.14) also hold with the symbols $\mathrm{ber}$, $\mathrm{bei}$, $M$, and $\theta$ replaced throughout by $\ker$, $\mathrm{kei}$, $N$, and $\varphi$, respectively. … ►Thus this reference gives ${\varphi }_1\left(0\right)=\frac{5}{4}\pi$ (Eq. …However, numerical tabulations show that if the second of these equations applies and ${\varphi }_1\left(x\right)$ is continuous, then ${\varphi }_1\left(0\right)=-\frac{3}{4}\pi$; compare Abramowitz and Stegun (1964, p. 433).

40: 2.11 Remainder Terms; Stokes Phenomenon

… ►uniformly when $\theta \in [-\pi +\delta ,\pi -\delta ]$ ($\delta >0$) and $|\alpha |$ is bounded. The coefficients are rational functions of $\alpha$ and $1+{\mathrm{e}}^{\mathrm{i}\theta }$, for example, $a_0(\theta ,\alpha )=1$, and … ►Here $\mathrm{erfc}$ is the complementary error function (§7.2(i)), and …Also, … ►Thus if $0\le \theta \le \pi -\delta$ (), then $c(\theta )$ lies in the right half-plane. …