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21: 13.7 Asymptotic Expansions for Large Argument

… ►For the special case $\mathrm{ph}z=\pm \pi$ see Paris (2013). …

22: 23.5 Special Lattices

… ►For the case ${\omega }_3={\mathrm{e}}^{\pi \mathrm{i}/3}{\omega }_1$ see §23.5(v). … ►Note also that in this case $\tau ={\mathrm{e}}^{\mathrm{i}\pi /3}$. …

23: 1.8 Fourier Series

… ►Formally, if $f(x)$ is a real- or complex-valued $2\pi$-periodic function, … ►where $f(x)$ is square-integrable on $[-\pi ,\pi ]$ and $a_n,b_n,c_n$ are given by (1.8.2), (1.8.4). … ►If $f(x)$ is of period $2\pi$, and $f^{(m)}(x)$ is piecewise continuous, then … ►It follows from definition (1.14.1) that the integral in (1.8.14) is equal to $\sqrt{2\pi }ℱ\left(f\right)\left(-2\pi n\right)$. … ►As a special case …

24: 14.20 Conical (or Mehler) Functions

… ►

14.20.3 $${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)=\frac{\pi {\mathrm{e}}^{-\tau \pi }\sin \left(\mu \pi \right)\sinh \left(\tau \pi \right)}{2({\cosh }^2\left(\tau \pi \right)-{\sin }^2\left(\mu \pi \right))}{𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)+\frac{\pi ({\mathrm{e}}^{-\tau \pi }{\cos }^2\left(\mu \pi \right)+\sinh \left(\tau \pi \right))}{2({\cosh }^2\left(\tau \pi \right)-{\sin }^2\left(\mu \pi \right))}{𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(-x\right).$$

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Symbols:

: Ferrers function of the first kind, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$: conical function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $\tau$: real variable and $\mu$: general order

Referenced by:

§14.20(i), §14.20(iii), §14.20(ix)

Permalink:

http://dlmf.nist.gov/14.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(i), §14.20 and Ch.14

… ►When , … ►Special cases: … ►uniformly for $\theta \in (0,\pi -\delta ]$, where $I$ and $K$ are the modified Bessel functions (§10.25(ii)) and $\delta$ is an arbitrary constant such that . … ►For the case of purely imaginary order and argument see Dunster (2013). …

25: 15.4 Special Cases

… ►

15.4.33 $$F(3a,\frac{1}{3}+a;\frac{2}{3}+2a;{\mathrm{e}}^{\mathrm{i}\pi /3})=\sqrt{\pi }{\mathrm{e}}^{\mathrm{i}\pi a/2}{\left(\frac{16}{27}\right)}^{(3a+1)/6}\frac{\mathrm{\Gamma }\left(\frac{5}{6}+a\right)}{\mathrm{\Gamma }\left(\frac{2}{3}+a\right)\mathrm{\Gamma }\left(\frac{2}{3}\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $a$: real or complex parameter

Notes:

See §15.2(ii) for treatment of the case when the third parameter is a nonpositive integer.

Referenced by:

§15.4(iii), §15.4(iii)

Permalink:

http://dlmf.nist.gov/15.4.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(iii), §15.4 and Ch.15

►

15.4.34 $$F(3a,a;2a;{\mathrm{e}}^{\mathrm{i}\pi /3})=\sqrt{\pi }{\mathrm{e}}^{\mathrm{i}\pi a/2}\frac{2^{2a}\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{3^{(3a+1)/2}}\left(\frac{1}{\mathrm{\Gamma }\left(\frac{1}{3}+a\right)\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}+a\right)\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $a$: real or complex parameter

Proof sketch:

From §15.5(ii) one obtains that $w(a)={\mathrm{e}}^{-\mathrm{i}\pi a/2}{2}^{-2a}{3}^{(3a+1)/2}F(3a,a;2a;{\mathrm{e}}^{\mathrm{i}\pi /3})\mathrm{\Gamma }\left(\frac{1}{3}+a\right)/\mathrm{\Gamma }\left(\frac{1}{2}+a\right)$ is a solution of the difference equation $(3a+5)w(a+2)-6(a+1)w(a+1)+(3a+1)w(a)=0$. Two linearly independent solutions of this difference equation are $w_1(a)=1$ and $w_2(a)=\mathrm{\Gamma }\left(\frac{1}{3}+a\right)/\mathrm{\Gamma }\left(\frac{2}{3}+a\right)$. Combining (15.6.1) with Laplace’s method (see §2.4(iii)) we obtain that $w(a)\sim \sqrt{\pi }\left(\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{a^{-1/3}}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right)$, as $a\to \mathrm{\infty }$. Hence, $w(a)=\sqrt{\pi }\left(\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{w_2(a)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right)$.

Notes:

See §15.2(ii) for treatment of the case when the third parameter is a nonpositive integer.

Referenced by:

§15.4(iii), §15.4(iii), Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/15.4.E34

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §15.4(iii), §15.4 and Ch.15

… ►

26: 20.1 Special Notation

… ► ►►►►

$m$, $n$	integers.
…
$q$ $(\in ℂ)$	the nome, $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$, . Since $\tau$ is not a single-valued function of $q$, it is assumed that $\tau$ is known, even when $q$ is specified. Most applications concern the rectangular case $\mathrm{\Re }\tau =0$, $\mathrm{\Im }\tau >0$, so that and $\tau$ and $q$ are uniquely related.
$q^{\alpha }$	${\mathrm{e}}^{\mathrm{i}\alpha \pi \tau }$ for $\alpha \in ℝ$ (resolving issues of choice of branch).
…

…

27: 6.12 Asymptotic Expansions

… ►

6.12.1 $$E_1\left(z\right)\sim \frac{{\mathrm{e}}^{-z}}{z}\left(1-\frac{1!}{z}+\frac{2!}{z^2}-\frac{3!}{z^3}+\mathrm{\cdots }\right),$$ $z\to \mathrm{\infty }$, .

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

5.1.51 (special case of)

Permalink:

http://dlmf.nist.gov/6.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.12(i), §6.12 and Ch.6

…

28: 14.16 Zeros

… ►For all cases concerning ${𝖯}_{\nu }^{\mu }\left(x\right)$ and $P_{\nu }^{\mu }\left(x\right)$ we assume that $\nu \ge -\frac{1}{2}$ without loss of generality (see (14.9.5) and (14.9.11)). … ► ${𝖰}_{\nu }^{\mu }\left(x\right)$ has $\max (\lceil \nu -|\mu |\rceil ,0)+k$ zeros in the interval $(-1,1)$, where $k$ can take one of the values $-1$, $0$, $1$, $2$, subject to $\max (\lceil \nu -|\mu |\rceil ,0)+k$ being even or odd according as $\cos \left(\nu \pi \right)$ and $\cos \left(\mu \pi \right)$ have opposite signs or the same sign. In the special case $\mu =0$ and $\nu =n=0,1,2,3,\mathrm{\dots }$, ${𝖰}_n\left(x\right)$ has $n+1$ zeros in the interval . … ►

$\mu >0$, $\mu >\nu$, $\mu \notin \mathbb{Z}$, and $\sin \left((\mu -\nu )\pi \right)$ and $\sin \left(\mu \pi \right)$ have opposite signs.

…

29: 32.11 Asymptotic Approximations for Real Variables

… ►Consider the special case of ${\text{P}}_{\text{II}}$ with $\alpha =0$: … ►In the case when … ►In the generic case … ►where $\lambda$ is an arbitrary constant such that , and … ►

32.11.27 $$\sigma =(2/\pi )\arcsin \left(\pi \lambda \right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\arcsin z$: arcsine function and $\sigma$: constant

Permalink:

http://dlmf.nist.gov/32.11.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §32.11(iv), §32.11 and Ch.32

…

30: 7.8 Inequalities

… ►

7.8.8 $x>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm and $x$: real variable

Source:

Pólya (1949, (1.5))

Referenced by:

§7.8, Erratum (V1.0.17) for Equation (7.8.8)

Permalink:

http://dlmf.nist.gov/7.8.E8

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This inequality was added. It is Pólya (1949, (1.5)). Note that it is a special case of (8.10.11).

Suggested by Roberto Iacono

See also:

Annotations for §7.8 and Ch.7