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11: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

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28.25.1 $${\mathrm{M}}_{\nu }^{(3,4)}(z,h)\sim \frac{{\mathrm{e}}^{\pm \mathrm{i}\left(2h\cosh z-\left(\frac{1}{2}\nu +\frac{1}{4}\right)\pi \right)}}{{\left(\pi h(\cosh z+1)\right)}^{\frac{1}{2}}}\sum _{m=0}^{\mathrm{\infty }}\frac{D_m^{\pm }}{{\left(\mp 4\mathrm{i}h(\cosh z+1)\right)}^m},$$

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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, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.1 (for 3 and for integer $\nu$) 20.9.3 (for 4 and for integer $\nu$)

Referenced by:

§28.25

Permalink:

http://dlmf.nist.gov/28.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

… ►

$D_{-1}^{\pm }=0,$

►

$D_0^{\pm }=1,$

… ►

28.25.3 $$(m+1)D_{m+1}^{\pm }+\left({(m+\frac{1}{2})}^2\pm (m+\frac{1}{4})8\mathrm{i}h+2h^2-a\right)D_m^{\pm }\pm (m-\frac{1}{2})\left(8\mathrm{i}hm\right)D_{m-1}^{\pm }=0,$$ $m\ge 0$.

ⓘ

Symbols:

$\mathrm{i}$: imaginary unit, $m$: integer, $h$: parameter, $a$: parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.2 (for $+$) 20.9.4 (for $-$)

Permalink:

http://dlmf.nist.gov/28.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

…

12: 4.13 Lambert $W$-Function

… ►The decreasing solution can be identified as $W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$. …$W_0\left(z\right)$ is a single-valued analytic function on $ℂ\setminus (-\mathrm{\infty },-{\mathrm{e}}^{-1}]$, real-valued when $z>-{\mathrm{e}}^{-1}$, and has a square root branch point at $z=-{\mathrm{e}}^{-1}$. …The other branches $W_k\left(z\right)$ are single-valued analytic functions on $ℂ\setminus (-\mathrm{\infty },0]$, have a logarithmic branch point at $z=0$, and, in the case $k=\pm 1$, have a square root branch point at $z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}$ respectively. … ►and has several advantages over the Lambert $W$-function (see Lawrence et al. (2012)), and the tree $T$-function $T\left(z\right)=-W\left(-z\right)$, which is a solution of … ►where $t\ge 0$ for $W_0$, $t\le 0$ for $W_{\pm 1}$ on the relevant branch cuts, …

13: 36.7 Zeros

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$$x_{m,n}^{\pm }=\sqrt{\frac{2}{-y_m}}\left(2n+\frac{1}{2}+{(-1)}^m\frac{1}{2}\pm \frac{1}{4}\right)\pi ,$$ $m=1,2,3,\mathrm{\dots }$, $n=0,\pm 1,\pm 2,\mathrm{\dots }$.

►

Table 36.7.1: Zeros of cusp diffraction catastrophe to 5D. …

► ►►►►►

Zeros $\left\{\genfrac{}{}{0.0pt}{}{x}{y}\right\}$ inside, and zeros $\left[\genfrac{}{}{0.0pt}{}{x}{y}\right]$ outside, the cusp $x^2=\frac{8}{27}{|y|}^3$.
$\left\{{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 0.52768}{-4.37804}}\right\}$	$\left[{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 2.35218}{-1.74360}}\right]$
$\left\{{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 1.41101}{-5.55470}}\right\}$	$\left\{{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 2.36094}{-5.52321}}\right\}$	$\left[{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 4.42707}{-3.05791}}\right]$
…
$\left\{{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 0.38488}{-8.31916}}\right\}$	$\left\{{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 2.71193}{-8.22315}}\right\}$	$\left\{{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 3.49286}{-8.20326}}\right\}$	$\left\{{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 5.96669}{-7.85723}}\right\}$	$\left\{{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 6.79538}{-7.80456}}\right\}$	$\left[{\displaystyle \genfrac{}{}{0.0pt}{}{\pm 9.17308}{-5.55831}}\right]$

►

… ►

$x_n=\pm {\left({\displaystyle \frac{8}{27}}\right)}^{1/2}{|y_n|}^{3/2}(1+{\xi }_n),$

…

14: 22.9 Cyclic Identities

§22.9 Cyclic Identities

… ►

§22.9(ii) Typical Identities of Rank 2

… ► ►

§22.9(iii) Typical Identities of Rank 3

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15: 6.4 Analytic Continuation

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6.4.3 $$E_1\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\mathrm{Ein}\left(-z\right)-\ln z-\gamma \mp \pi \mathrm{i},$$ $|\mathrm{ph}z|\le \pi$.

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

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6.4.4 $$\mathrm{Ci}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pm \pi \mathrm{i}+\mathrm{Ci}\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

►

6.4.5 $$\mathrm{Chi}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pm \pi \mathrm{i}+\mathrm{Chi}\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{Chi}\left(z\right)$: hyperbolic cosine integral, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

►

6.4.6 $$\mathrm{f}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pi {\mathrm{e}}^{\mp \mathrm{i}z}-\mathrm{f}\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.12(ii), §6.4

Permalink:

http://dlmf.nist.gov/6.4.E6

Encodings:

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See also:

Annotations for §6.4 and Ch.6

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6.4.7 $$\mathrm{g}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\mp \pi \mathrm{i}{\mathrm{e}}^{\mp \mathrm{i}z}+\mathrm{g}\left(z\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.12(ii), §6.4

Permalink:

http://dlmf.nist.gov/6.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

…

16: 26.15 Permutations: Matrix Notation

… ►where the sum is over and $n\ge h>\mathrm{\ell }\ge 1$. … ►For $(j,k)\in B$, $B\setminus [j,k]$ denotes $B$ after removal of all elements of the form $(j,t)$ or $(t,k)$, $t=1,2,\mathrm{\dots },n$. $B\setminus (j,k)$ denotes $B$ with the element $(j,k)$ removed. ►

26.15.5 $$R(x,B)=xR(x,B\setminus [j,k])+R(x,B\setminus (j,k)).$$

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Symbols:

$[a,b]$: closed interval, $(a,b)$: open interval, $\setminus$: set subtraction, $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $B$: subset and $R(x,B)$: rook polynomial

Permalink:

http://dlmf.nist.gov/26.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §26.15 and Ch.26

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26.15.8 $$N_0(B)\equiv N(0,B)=\sum _{k=0}^n{(-1)}^k{r}_k(B)(n-k)!.$$

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Symbols:

$\equiv$: equals by definition, $!$: factorial (as in $n!$), $k$: nonnegative integer, $n$: nonnegative integer, $B$: subset, $r_j(B)$: number of rook positions and $N(x,B)$: generating function

Permalink:

http://dlmf.nist.gov/26.15.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §26.15 and Ch.26

…

17: 14.2 Differential Equations

… ►Standard solutions: ${𝖯}_{\nu }\left(\pm x\right)$, ${𝖰}_{\nu }\left(\pm x\right)$, ${𝖰}_{-\nu -1}\left(\pm x\right)$, $P_{\nu }\left(\pm x\right)$, $Q_{\nu }\left(\pm x\right)$, $Q_{-\nu -1}\left(\pm x\right)$. … ►Standard solutions: ${𝖯}_{\nu }^{\mu }\left(\pm x\right)$, ${𝖯}_{\nu }^{-\mu }\left(\pm x\right)$, ${𝖰}_{\nu }^{\mu }\left(\pm x\right)$, ${𝖰}_{-\nu -1}^{\mu }\left(\pm x\right)$, $P_{\nu }^{\mu }\left(\pm x\right)$, $P_{\nu }^{-\mu }\left(\pm x\right)$, ${𝑸}_{\nu }^{\mu }\left(\pm x\right)$, ${𝑸}_{-\nu -1}^{\mu }\left(\pm x\right)$. …

18: 33.8 Continued Fractions

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33.8.2 $$\frac{H_{\mathrm{\ell }}^{\pm }{}_{}{}^{\prime }}{H_{\mathrm{\ell }}^{\pm }}=c\pm \frac{\mathrm{i}}{\rho }\frac{ab}{2(\rho -\eta \pm \mathrm{i})+}\frac{(a+1)(b+1)}{2(\rho -\eta \pm 2\mathrm{i})+}\mathrm{\cdots },$$

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Symbols:

$\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $a$: coefficient, $b$: coefficient and $c$: coefficient

Referenced by:

§33.8

Permalink:

http://dlmf.nist.gov/33.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.8 and Ch.33

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$a=1+\mathrm{\ell }\pm \mathrm{i}\eta ,$

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$b=-\mathrm{\ell }\pm \mathrm{i}\eta ,$

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$c=\pm \mathrm{i}(1-(\eta /\rho )).$

… ►

$F_{\mathrm{\ell }}=\pm {(q^{-1}{(u-p)}^2+q)}^{-1/2},$

…

19: 28.20 Definitions and Basic Properties

… ►When $z$ is replaced by $\pm \mathrm{i}z$, (28.2.1) becomes the modified Mathieu’s equation: … ►Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to ${\zeta }^{1/2}{\mathrm{e}}^{\pm 2\mathrm{i}h\zeta }$ as $\zeta \to \mathrm{\infty }$ in the respective sectors $|\mathrm{ph}\left(\mp \mathrm{i}\zeta \right)|\le \frac{3}{2}\pi -\delta$, $\delta$ being an arbitrary small positive constant. … ►

$${\mathrm{Mc}}_{2n}^{(j)}(z\pm \frac{1}{2}\pi \mathrm{i},h)={\mathrm{Mc}}_{2n}^{(j)}(z,\pm \mathrm{i}h),$$

►

$${\mathrm{Ms}}_{2n+1}^{(j)}(z\pm \frac{1}{2}\pi \mathrm{i},h)={\mathrm{Mc}}_{2n+1}^{(j)}(z,\pm \mathrm{i}h),$$

… ►And for the corresponding identities for the radial functions use (28.20.15) and (28.20.16).

20: 4.24 Inverse Trigonometric Functions: Further Properties

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4.24.13 $$\mathrm{Arcsin}u\pm \mathrm{Arcsin}v=\mathrm{Arcsin}\left(u{(1-v^2)}^{1/2}\pm v{(1-u^2)}^{1/2}\right),$$

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Symbols:

$\mathrm{Arcsin}z$: general arcsine function

A&S Ref:

4.4.32

Permalink:

http://dlmf.nist.gov/4.24.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.14 $$\mathrm{Arccos}u\pm \mathrm{Arccos}v=\mathrm{Arccos}\left(uv\mp {((1-u^2)(1-v^2))}^{1/2}\right),$$

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Symbols:

$\mathrm{Arccos}z$: general arccosine function

A&S Ref:

4.4.33

Permalink:

http://dlmf.nist.gov/4.24.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.15 $$\mathrm{Arctan}u\pm \mathrm{Arctan}v=\mathrm{Arctan}\left(\frac{u\pm v}{1\mp uv}\right),$$

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Symbols:

$\mathrm{Arctan}z$: general arctangent function

A&S Ref:

4.4.34

Referenced by:

§4.45(i)

Permalink:

http://dlmf.nist.gov/4.24.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.16 $$\mathrm{Arcsin}u\pm \mathrm{Arccos}v=\mathrm{Arcsin}\left(uv\pm {((1-u^2)(1-v^2))}^{1/2}\right)=\mathrm{Arccos}\left(v{(1-u^2)}^{1/2}\mp u{(1-v^2)}^{1/2}\right),$$

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Symbols:

$\mathrm{Arccos}z$: general arccosine function and $\mathrm{Arcsin}z$: general arcsine function

A&S Ref:

4.4.35

Permalink:

http://dlmf.nist.gov/4.24.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

►

4.24.17 $$\mathrm{Arctan}u\pm \mathrm{Arccot}v=\mathrm{Arctan}\left(\frac{uv\pm 1}{v\mp u}\right)=\mathrm{Arccot}\left(\frac{v\mp u}{uv\pm 1}\right).$$

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Symbols:

$\mathrm{Arccot}z$: general arccotangent function and $\mathrm{Arctan}z$: general arctangent function

A&S Ref:

4.4.36

Permalink:

http://dlmf.nist.gov/4.24.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(iii), §4.24 and Ch.4

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