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41: 19.32 Conformal Map onto a Rectangle

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19.32.2 $$dz=-\frac{1}{2}\left(\prod _{j=1}^3{(p-x_j)}^{-1/2}\right)dp,$$ $\mathrm{\Im }p>0$; , $j=1,2,3$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase and $z(p)$: function

Referenced by:

§19.32

Permalink:

http://dlmf.nist.gov/19.32.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.32 and Ch.19

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$$z(\mathrm{\infty })=0,$$

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$$z(x_1)=R_F(0,x_1-x_2,x_1-x_3)\text{(}>0\text{)},$$

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$$z(x_3)=R_F(x_3-x_1,x_3-x_2,0)=-\mathrm{i}{R}_F(0,x_1-x_3,x_2-x_3).$$

…

42: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►With $\mathrm{e}$ denoting here the elementary charge, the Coulomb potential between two point particles with charges $Z_1\mathrm{e},Z_2\mathrm{e}$ and masses $m_1,m_2$ separated by a distance $s$ is $V(s)=Z_1{Z}_2{\mathrm{e}}^2/(4\pi {\epsilon }_{0}s)=Z_1{Z}_2\alpha \mathrm{\hslash }c/s$, where $Z_j$ are atomic numbers, ${\epsilon }_0$ is the electric constant, $\alpha$ is the fine structure constant, and $\mathrm{\hslash }$ is the reduced Planck’s constant. … ► ►►►

Attractive potentials:	, .
Zero potential ($V=0$):	$Z_1{Z}_2=0$, $\eta =0$.
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… ►For $Z_1{Z}_2=-1$ and $m=m_{\mathrm{e}}$, the electron mass, the scaling factors in (33.22.5) reduce to the Bohr radius, $a_0=\mathrm{\hslash }/(m_{\mathrm{e}}c\alpha )$, and to a multiple of the Rydberg constant, … ► ►►►

Attractive potentials:	, $r>0$.
Zero potential ($V=0$):	$Z_1{Z}_2=0$, $r=0$.
…

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Attractive potentials:	, .
Zero potential ($V=0$):	$Z_1{Z}_2=0$, $\kappa =0$.
…

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43: 19.27 Asymptotic Approximations and Expansions

… ►Assume $x$, $y$, and $z$ are real and nonnegative and at most one of them is 0. … ►Assume $x$, $y$, and $z$ are real and nonnegative and at most one of them is 0. … ►

19.27.4 $$R_G(x,y,z)=\frac{\sqrt{z}}{2}\left(1+O\left(\frac{a}{z}\ln \frac{z}{a}\right)\right),$$ $a/z\to 0$.

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Symbols:

$O\left(x\right)$: order not exceeding, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\ln z$: principal branch of logarithm function and $a$

Permalink:

http://dlmf.nist.gov/19.27.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.27(iii), §19.27 and Ch.19

… ►Assume $x$ and $y$ are real and nonnegative, at most one of them is 0, and $z>0$. … ►Assume $x$, $y$, and $z$ are real and nonnegative, at most one of them is 0, and $p>0$. …

44: 4.38 Inverse Hyperbolic Functions: Further Properties

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4.38.2 $$\mathrm{arcsinh}z=\ln \left(2z\right)+\frac{1}{2}\frac{1}{2z^2}-\frac{1\cdot 3}{2\cdot 4}\frac{1}{4z^4}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{1}{6z^6}-\mathrm{\cdots },$$ $\mathrm{\Re }z>0$, $|z|>1$.

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Symbols:

$\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.6.31 (misses a condition on $z$.)

Referenced by:

§4.38(i)

Permalink:

http://dlmf.nist.gov/4.38.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(i), §4.38 and Ch.4

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4.38.4 $$\mathrm{arccosh}z={(2(z-1))}^{1/2}\left(1+\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n\frac{1\cdot 3\cdot 5\mathrm{\cdots }(2n-1)}{2^{2n}n!(2n+1)}{(z-1)}^n\right),$$ $\mathrm{\Re }z>0$, $|z-1|\le 2$.

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Symbols:

$!$: factorial (as in $n!$), $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{\Re }$: real part, $n$: integer and $z$: complex variable

Referenced by:

§4.38(i)

Permalink:

http://dlmf.nist.gov/4.38.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(i), §4.38 and Ch.4

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4.38.6 $$\mathrm{arctanh}z=\pm \mathrm{i}\frac{\pi }{2}+\frac{1}{z}+\frac{1}{3z^3}+\frac{1}{5z^5}+\mathrm{\cdots },$$ $\mathrm{\Im }z\gtrless 0$, $\left|z\right|\ge 1$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.6.33

Permalink:

http://dlmf.nist.gov/4.38.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(i), §4.38 and Ch.4

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4.38.10 $$\frac{d}{dz}\mathrm{arccosh}z=\pm {(z^2-1)}^{-1/2},$$ $\mathrm{\Re }z\gtrless 0$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.6.38 (has an error.)

Permalink:

http://dlmf.nist.gov/4.38.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(ii), §4.38 and Ch.4

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4.38.12 $$\frac{d}{dz}\mathrm{arccsch}z=\mp \frac{1}{z{(1+z^2)}^{1/2}},$$ $\mathrm{\Re }z\gtrless 0$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arccsch}z$: inverse hyperbolic cosecant function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.6.40

Permalink:

http://dlmf.nist.gov/4.38.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.38(ii), §4.38 and Ch.4

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45: 6.11 Relations to Other Functions

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6.11.1 $$E_1\left(z\right)=\mathrm{\Gamma }(0,z).$$

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Symbols:

$E_1\left(z\right)$: exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable

A&S Ref:

5.1.45 (special case of)

Referenced by:

§6.11

Permalink:

http://dlmf.nist.gov/6.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

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46: 10.25 Definitions

… ►In particular, the principal branch of $I_{\nu }\left(z\right)$ is defined in a similar way: it corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu }$, is analytic in $ℂ\setminus (-\mathrm{\infty },0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. … ►It has a branch point at $z=0$ for all $\nu \in ℂ$. … ►Both $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ are real when $\nu$ is real and $\mathrm{ph}z=0$. ►For fixed $z$ $(\ne 0)$ each branch of $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ is entire in $\nu$. … ►When , $I_{\nu }\left(z\right)$ is replaced by $I_{-\nu }\left(z\right)$. …

47: 10.31 Power Series

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10.31.1 $$K_n\left(z\right)=\frac{1}{2}{(\frac{1}{2}z)}^{-n}\sum _{k=0}^{n-1}\frac{(n-k-1)!}{k!}{(-\frac{1}{4}{z}^2)}^k+{(-1)}^{n+1}\ln \left(\frac{1}{2}z\right)I_n\left(z\right)+{(-1)}^n\frac{1}{2}{(\frac{1}{2}z)}^n\sum _{k=0}^{\mathrm{\infty }}\left(\psi \left(k+1\right)+\psi \left(n+k+1\right)\right)\frac{{(\frac{1}{4}{z}^2)}^k}{k!(n+k)!},$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.6.11

Referenced by:

§10.30(i), §10.65(ii)

Permalink:

http://dlmf.nist.gov/10.31.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.31 and Ch.10

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10.31.2 $$K_0\left(z\right)=-\left(\ln \left(\frac{1}{2}z\right)+\gamma \right)I_0\left(z\right)+\frac{\frac{1}{4}{z}^2}{{(1!)}^2}+(1+\frac{1}{2})\frac{{(\frac{1}{4}{z}^2)}^2}{{(2!)}^2}+(1+\frac{1}{2}+\frac{1}{3})\frac{{(\frac{1}{4}{z}^2)}^3}{{(3!)}^2}+\mathrm{\cdots }.$$

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Symbols:

$\gamma$: Euler’s constant, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

9.6.13

Referenced by:

§10.45

Permalink:

http://dlmf.nist.gov/10.31.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.31 and Ch.10

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10.31.3 $$I_{\nu }\left(z\right)I_{\mu }\left(z\right)={(\frac{1}{2}z)}^{\nu +\mu }\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\nu +\mu +k+1\right)}_k{(\frac{1}{4}{z}^2)}^k}{k!\mathrm{\Gamma }\left(\nu +k+1\right)\mathrm{\Gamma }\left(\mu +k+1\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.31

Permalink:

http://dlmf.nist.gov/10.31.E3

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §10.31 and Ch.10

48: 25.3 Graphics

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Figure 25.3.4: $Z(t)$, $0\le t\le 50$. … Magnify

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49: 16.21 Differential Equation

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16.21.1 $$\left({(-1)}^{p-m-n}z(\mathit{\vartheta }-a_1+1)\mathrm{\cdots }(\mathit{\vartheta }-a_p+1)-(\mathit{\vartheta }-b_1)\mathrm{\cdots }(\mathit{\vartheta }-b_q)\right)w=0,$$

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Symbols:

$p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $w$: solution and $\mathit{\vartheta }$: differential operator

Referenced by:

§16.21, §16.21

Permalink:

http://dlmf.nist.gov/16.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.21 and Ch.16

… ►With the classification of §16.8(i), when the only singularities of (16.21.1) are a regular singularity at $z=0$ and an irregular singularity at $z=\mathrm{\infty }$. When $p=q$ the only singularities of (16.21.1) are regular singularities at $z=0$, ${(-1)}^{p-m-n}$, and $\mathrm{\infty }$. …

50: 28.9 Zeros

… ►For real $q$ each of the functions ${\mathrm{ce}}_{2n}(z,q)$, ${\mathrm{se}}_{2n+1}(z,q)$, ${\mathrm{ce}}_{2n+1}(z,q)$, and ${\mathrm{se}}_{2n+2}(z,q)$ has exactly $n$ zeros in . …Furthermore, for $q>0$ ${\mathrm{ce}}_m(z,q)$ and ${\mathrm{se}}_m(z,q)$ also have purely imaginary zeros that correspond uniquely to the purely imaginary $z$-zeros of $J_m\left(2\sqrt{q}\cos z\right)$ (§10.21(i)), and they are asymptotically equal as $q\to 0$ and $\left|\mathrm{\Im }z\right|\to \mathrm{\infty }$. …