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1: 14.26 Uniform Asymptotic Expansions

… ►The uniform asymptotic approximations given in §14.15 for $P_{\nu }^{-\mu }\left(x\right)$ and ${𝑸}_{\nu }^{\mu }\left(x\right)$ for are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). …

2: Sidebar 5.SB1: Gamma & Digamma Phase Plots

… ►The color encoded phases of $\mathrm{\Gamma }\left(z\right)$ (above) and $\psi \left(z\right)$ (below), are constrasted in the negative half of the complex plane. …

3: 35.5 Bessel Functions of Matrix Argument

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35.5.1 $$A_{\nu }\left(𝟎\right)=\frac{1}{{\mathrm{\Gamma }}_m\left(\nu +\frac{1}{2}(m+1)\right)},$$ $\nu \in ℂ$.

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Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $ℂ$: complex plane, $\in$: element of, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/35.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(i), §35.5 and Ch.35

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35.5.2 $$A_{\nu }\left(𝐓\right)=A_{\nu }\left(𝟎\right)\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k}{k!}\sum _{|\kappa |=k}\frac{1}{{\left[\nu +\frac{1}{2}(m+1)\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right),$$ $\nu \in ℂ$, $𝐓\in 𝓢$.

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Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $ℂ$: complex plane, $\in$: element of, $!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐓$: real symmetric $m\times m$ matrix, $k$: nonnegative integer, $m$: positive integer, $𝟎$: $m\times m$ matrix of zeros and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(i), §35.5 and Ch.35

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35.5.3 $$B_{\nu }\left(𝐓\right)={\int }_{𝛀}\mathrm{etr}\left(-(𝐓𝐗+{𝐗}^{-1})\right){\left|𝐗\right|}^{\nu -\frac{1}{2}(m+1)}d𝐗,$$ $\nu \in ℂ$, $𝐓\in 𝛀$.

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Defines:

$B_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (second kind)

Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices and $m$: positive integer

Referenced by:

§35.9

Permalink:

http://dlmf.nist.gov/35.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(i), §35.5 and Ch.35

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35.5.5 ${\nu }_j\in ℂ$, $\mathrm{\Re }\left({\nu }_j\right)>-1$, $j=1,2$; ${𝐒}_1,{𝐒}_2\in 𝓢$; $𝐓\in 𝛀$.

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Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $ℂ$: complex plane, $\in$: element of, $\int$: integral, $\mathrm{\Re }$: real part, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐒$: real symmetric $m\times m$ matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, : less-than for matrices, $j$: nonnegative integer, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/35.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(ii), §35.5 and Ch.35

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35.5.6 $$B_{\nu }\left(𝐓\right)={\left|𝐓\right|}^{-\nu }{B}_{-\nu }\left(𝐓\right),$$ $\nu \in ℂ$, $𝐓\in 𝛀$.

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Symbols:

$B_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (second kind), $ℂ$: complex plane, $\in$: element of, $𝐓$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ and $𝛀$: space of positive-definite real symmetric $m\times m$ matrices

Permalink:

http://dlmf.nist.gov/35.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(ii), §35.5 and Ch.35

…

4: 1.9 Calculus of a Complex Variable

… ►Also, the union of $S$ and its limit points is the closure of $S$. … ► … ► … ►

Jordan Curve Theorem

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§1.9(iv) Conformal Mapping

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5: 9.19 Approximations

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§9.19(iii) Approximations in the Complex Plane

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6: 4.37 Inverse Hyperbolic Functions

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4.37.16 $$\mathrm{arcsinh}z=\ln \left({(z^2+1)}^{1/2}+z\right),$$ $z/\mathrm{i}\in ℂ\setminus (-\mathrm{\infty },-1)\cup (1,\mathrm{\infty })$;

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Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $\setminus$: set subtraction, $\cup$: union and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

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4.37.19 $$\mathrm{arccosh}z=\ln \left(\pm {(z^2-1)}^{1/2}+z\right),$$ $z\in ℂ\setminus (-\mathrm{\infty },1)$,

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Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $\setminus$: set subtraction and $z$: complex variable

Referenced by:

§4.37(iv), §4.37(iii), §4.37(iv)

Permalink:

http://dlmf.nist.gov/4.37.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

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4.37.21 $$\mathrm{arccosh}z=2\ln \left({\left(\frac{z+1}{2}\right)}^{1/2}+{\left(\frac{z-1}{2}\right)}^{1/2}\right),$$ $z\in ℂ\setminus (-\mathrm{\infty },1)$;

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Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $\setminus$: set subtraction and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

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4.37.24 $$\mathrm{arctanh}z=\frac{1}{2}\ln \left(\frac{1+z}{1-z}\right),$$ $z\in ℂ\setminus (-\mathrm{\infty },-1]\cup [1,\mathrm{\infty })$;

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Symbols:

$[a,b)$: half-closed interval, $ℂ$: complex plane, $\in$: element of, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, $\ln z$: principal branch of logarithm function, $(a,b]$: half-closed interval, $\setminus$: set subtraction, $\cup$: union and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.37.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iv), §4.37(iv), §4.37 and Ch.4

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7: 4.5 Inequalities

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4.5.16 $$|{\mathrm{e}}^z-1|\le {\mathrm{e}}^{|z|}-1\le |z|{\mathrm{e}}^{|z|},$$ $z\in ℂ$.

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Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

4.2.39

Referenced by:

§4.5(ii)

Permalink:

http://dlmf.nist.gov/4.5.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §4.5(ii), §4.5 and Ch.4

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8: 5.23 Approximations

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§5.23(iii) Approximations in the Complex Plane

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9: 18.24 Hahn Class: Asymptotic Approximations

… ►When the parameters $\alpha$ and $\beta$ are fixed and the ratio $n/N=c$ is a constant in the interval (0,1), uniform asymptotic formulas (as $n\to \mathrm{\infty }$ ) of the Hahn polynomials $Q_n(z;\alpha ,\beta ,N)$ can be found in Lin and Wong (2013) for $z$ in three overlapping regions, which together cover the entire complex plane. …

10: 4.23 Inverse Trigonometric Functions

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4.23.19 $$\arcsin z=-\mathrm{i}\ln \left({(1-z^2)}^{1/2}+\mathrm{i}z\right),$$ $z\in ℂ\setminus (-\mathrm{\infty },-1)\cup (1,\mathrm{\infty })$;

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Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{i}$: imaginary unit, $\arcsin z$: arcsine function, $\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $\setminus$: set subtraction, $\cup$: union and $z$: complex variable

Referenced by:

§4.23(iv)

Permalink:

http://dlmf.nist.gov/4.23.E19

Encodings:

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See also:

Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

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4.23.22 $$\arccos z=\frac{1}{2}\pi +\mathrm{i}\ln \left({(1-z^2)}^{1/2}+\mathrm{i}z\right),$$ $z\in ℂ\setminus (-\mathrm{\infty },-1)\cup (1,\mathrm{\infty })$;

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $ℂ$: complex plane, $\in$: element of, $\mathrm{i}$: imaginary unit, $\arccos z$: arccosine function, $\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $\setminus$: set subtraction, $\cup$: union and $z$: complex variable

Referenced by:

§4.23(iv)

Permalink:

http://dlmf.nist.gov/4.23.E22

Encodings:

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See also:

Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

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4.23.23 $$\arccos z=-2\mathrm{i}\ln \left({\left(\frac{1+z}{2}\right)}^{1/2}+\mathrm{i}{\left(\frac{1-z}{2}\right)}^{1/2}\right),$$ $z\in ℂ\setminus (-\mathrm{\infty },-1)\cup (1,\mathrm{\infty })$;

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Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{i}$: imaginary unit, $\arccos z$: arccosine function, $\ln z$: principal branch of logarithm function, $(a,b)$: open interval, $\setminus$: set subtraction, $\cup$: union and $z$: complex variable

Referenced by:

§4.23(iv)

Permalink:

http://dlmf.nist.gov/4.23.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

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4.23.26 $$\arctan z=\frac{\mathrm{i}}{2}\ln \left(\frac{\mathrm{i}+z}{\mathrm{i}-z}\right),$$ $z/\mathrm{i}\in ℂ\setminus (-\mathrm{\infty },-1]\cup [1,\mathrm{\infty })$;

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Symbols:

$[a,b)$: half-closed interval, $ℂ$: complex plane, $\in$: element of, $\mathrm{i}$: imaginary unit, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $(a,b]$: half-closed interval, $\setminus$: set subtraction, $\cup$: union and $z$: complex variable

Referenced by:

§4.23(iv)

Permalink:

http://dlmf.nist.gov/4.23.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §4.23(iv), §4.23(iv), §4.23 and Ch.4

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