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21: 7.5 Interrelations

… ►

7.5.6 $${\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}{z}^2}(\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right))=\frac{1}{2}(1\pm \mathrm{i})-(C\left(z\right)\pm \mathrm{i}S\left(z\right)).$$

ⓘ

Symbols:

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

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

… ►

7.5.7 $$\zeta =\frac{1}{2}\sqrt{\pi }(1\mp \mathrm{i})z,$$

ⓘ

Defines:

$\zeta$: change of variable (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§7.12(ii), §7.5

Permalink:

http://dlmf.nist.gov/7.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

… ►

7.5.8 $$C\left(z\right)\pm \mathrm{i}S\left(z\right)=\frac{1}{2}(1\pm \mathrm{i})\mathrm{erf}\zeta .$$

ⓘ

Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\mathrm{erf}z$: error function, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

Referenced by:

§7.5, §7.5, §7.6(i)

Permalink:

http://dlmf.nist.gov/7.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

►

7.5.9 $$C\left(z\right)\pm \mathrm{i}S\left(z\right)=\frac{1}{2}(1\pm \mathrm{i})\left(1-{\mathrm{e}}^{\pm \frac{1}{2}\pi \mathrm{i}{z}^2}w\left(\mathrm{i}\zeta \right)\right).$$

ⓘ

Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

►

7.5.10 $$\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right)=\frac{1}{2}(1\pm \mathrm{i}){\mathrm{e}}^{{\zeta }^2}\mathrm{erfc}\zeta .$$

ⓘ

Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

A&S Ref:

7.3.22 (in different form)

Referenced by:

§7.12(ii), §7.13(iv), §7.5, §7.5

Permalink:

http://dlmf.nist.gov/7.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

…

22: 32.10 Special Function Solutions

… ►For example, if $\alpha =\frac{1}{2}\epsilon$, with $\epsilon =\pm 1$, then the Riccati equation is … ►Solutions for other values of $\alpha$ are derived from $w(z;\pm \frac{1}{2})$ by application of the Bäcklund transformations (32.7.1) and (32.7.2). … ►with $n\in \mathbb{Z}$, and ${\epsilon }_1=\pm 1$, ${\epsilon }_2=\pm 1$, independently. … ►with $n\in \mathbb{Z}$ and $\epsilon =\pm 1$. In the case when $n=0$ in (32.10.15), the Riccati equation is …

23: 12.2 Differential Equations

… ►Standard solutions are $U(a,\pm z)$, $V(a,\pm z)$, $\overline{U}(a,\pm x)$ (not complex conjugate), $U(-a,\pm \mathrm{i}z)$ for (12.2.2); $W(a,\pm x)$ for (12.2.3); $D_{\nu }\left(\pm z\right)$ for (12.2.4), where … ►The solutions $W(a,\pm x)$ are treated in §12.14. ►In $ℂ$, for $j=0,1,2,3$, $U({(-1)}^{j-1}a,{(-\mathrm{i})}^{j-1}z)$ and $U({(-1)}^{j}a,{(-\mathrm{i})}^{j}z)$ comprise a numerically satisfactory pair of solutions in the half-plane $\frac{1}{4}(2j-3)\pi \le \mathrm{ph}z\le \frac{1}{4}(2j+1)\pi$. … ►

§12.2(vi) Solution $\overline{U}(a,x)$; Modulus and Phase Functions

►When $z$ $(=x)$ is real the solution $\overline{U}(a,x)$ is defined by …

24: 19.22 Quadratic Transformations

… ►

$4(p_{\pm }^2-a^2)={(\sqrt{p^2-x^2}\pm \sqrt{p^2-y^2})}^2.$

… ►Again, we assume that $a_0\ge g_0$ (except in (19.22.10)), and define $c_n=\sqrt{a_n^2-g_n^2}$. … ►

$4(z_{\pm }^2-a^2)={(\sqrt{z^2-x^2}\pm \sqrt{z^2-y^2})}^2.$

… ►However, if $x$ and $y$ are complex conjugates and $z$ and $p$ are real, then the right-hand sides of all transformations in §§19.22(i) and 19.22(iii)—except (19.22.3) and (19.22.22)—are free of complex numbers and $p_{\pm }^2-p_{\mp }^2=\pm |p^2-x^2|\ne 0$. … ►These relations need to be used with caution because $y$ is negative when . …

25: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►

8.12.5 $$\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}}{2\mathrm{i}\sin \left(\pi a\right)}Q(-a,z{\mathrm{e}}^{\pm \pi \mathrm{i}})=\pm \frac{1}{2}\mathrm{erfc}\left(\pm \mathrm{i}\eta \sqrt{a/2}\right)-\mathrm{i}T(a,\eta ),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $Q(a,z)$: normalized incomplete gamma function, $\sin z$: sine function, $z$: complex variable, $a$: parameter, $\eta (\lambda )$ and $T(a,\eta )$: function

Referenced by:

Erratum (V1.0.16) for Equation (8.12.5), Erratum (V1.0.16) for Equation (8.12.5)

Permalink:

http://dlmf.nist.gov/8.12.E5

Encodings:

pMML, png, TeX

Change (effective with 1.0.16):

To be consistent with the notation used in (8.12.16), the original equation,

$$\mathrm{\Gamma }\left(a+1\right)\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}}{2\pi \mathrm{i}}\mathrm{\Gamma }(-a,z{\mathrm{e}}^{\pm \pi \mathrm{i}})=\mp \frac{1}{2}\mathrm{erfc}\left(\pm \mathrm{i}\eta \sqrt{a/2}\right)+\mathrm{i}T(a,\eta ),$$

was rewritten.

See also:

Annotations for §8.12 and Ch.8

… ►

8.12.16 $$\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}}{2\mathrm{i}\sin \left(\pi a\right)}Q(-a,a{\mathrm{e}}^{\pm \pi \mathrm{i}})\sim \pm \frac{1}{2}-\frac{\mathrm{i}}{\sqrt{2\pi a}}\sum _{k=0}^{\mathrm{\infty }}{c}_k(0){(-a)}^{-k},$$ $|\mathrm{ph}a|\le \pi -\delta$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $Q(a,z)$: normalized incomplete gamma function, $\mathrm{ph}$: phase, $\sin z$: sine function, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant and $c_k(\eta )$: coefficients

Referenced by:

(8.12.5), Erratum (V1.0.16) for Equation (8.12.5)

Permalink:

http://dlmf.nist.gov/8.12.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §8.12 and Ch.8

… ►

8.12.18 $$\begin{array}{c}Q(a,z)\\ P(a,z)\end{array}\}\sim \frac{z^{a-\frac{1}{2}}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a\right)}\left(d(\pm \chi )\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\chi )}{z^{k/2}}\mp \sum _{k=1}^{\mathrm{\infty }}\frac{B_k(\chi )}{z^{k/2}}\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $\chi$ , $d(\pm \chi )$ , $A_k(\chi )$ and $B_k(\chi )$

Referenced by:

§8.12, Erratum (V1.0.16) for Equation (8.12.18)

Permalink:

http://dlmf.nist.gov/8.12.E18

Encodings:

pMML, png, TeX

Errata (effective with 1.0.16):

The original $\pm$ in front of the second summation was replaced by $\mp$ to correct an error in Paris (2002b); for details see https://arxiv.org/abs/1611.00548.

Reported 2017-01-28 by Richard Paris

See also:

Annotations for §8.12 and Ch.8

►for $z\to \mathrm{\infty }$ in , with $\mathrm{\Re }\left(z-a\right)\le 0$ for $P(a,z)$ and $\mathrm{\Re }\left(z-a\right)\ge 0$ for $Q(a,z)$. … ►

$d(\pm \chi )=\sqrt{\frac{1}{2}\pi }{\mathrm{e}}^{{\chi }^2/2}\mathrm{erfc}\left(\pm \chi /\sqrt{2}\right),$

…

26: 10.38 Derivatives with Respect to Order

… ►

10.38.1 $$\frac{\partial I_{\pm \nu }\left(z\right)}{\partial \nu }=\pm I_{\pm \nu }\left(z\right)\ln \left(\frac{1}{2}z\right)\mp {(\frac{1}{2}z)}^{\pm \nu }\sum _{k=0}^{\mathrm{\infty }}\frac{\psi \left(k+1\pm \nu \right)}{\mathrm{\Gamma }\left(k+1\pm \nu \right)}\frac{{(\frac{1}{4}{z}^2)}^k}{k!},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\ln z$: principal branch of logarithm function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.42

Referenced by:

§10.38, §10.38, Erratum (V1.0.17) for Equations (10.15.1), (10.38.1)

Permalink:

http://dlmf.nist.gov/10.38.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.17):

This equation has been generalized to include the additional case of $\partial I_{-\nu }\left(z\right)/\partial \nu$ because it will help the user who wants to combine it with (10.38.2).

See also:

Annotations for §10.38 and Ch.10

… ►

10.38.3 $${{(-1)}^n\frac{\partial I_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=-K_n\left(z\right)+\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}{(-1)}^k\frac{{(\frac{1}{2}z)}^k{I}_k\left(z\right)}{k!(n-k)},$$

ⓘ

Symbols:

$!$: 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, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.44

Referenced by:

§10.38

Permalink:

http://dlmf.nist.gov/10.38.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

… ►

10.38.4 $${\frac{\partial K_{\nu }\left(z\right)}{\partial \nu }|}_{\nu =n}=\frac{n!}{2{(\frac{1}{2}z)}^n}\sum _{k=0}^{n-1}\frac{{(\frac{1}{2}z)}^k{K}_k\left(z\right)}{k!(n-k)}.$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.45

Referenced by:

§10.38

Permalink:

http://dlmf.nist.gov/10.38.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

… ►

10.38.6 $${\frac{\partial I_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\pm \frac{1}{2}}=-\frac{1}{\sqrt{2\pi x}}\left(E_1\left(2x\right){\mathrm{e}}^x\pm \mathrm{Ei}\left(2x\right){\mathrm{e}}^{-x}\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.2.32 and 10.2.33 (both corrected)

Referenced by:

§10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.38.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

►

10.38.7 $${\frac{\partial K_{\nu }\left(x\right)}{\partial \nu }|}_{\nu =\pm \frac{1}{2}}=\pm \sqrt{\frac{\pi }{2x}}{E}_1\left(2x\right){\mathrm{e}}^x.$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable and $\nu$: complex parameter

A&S Ref:

10.2.34 (corrected)

Referenced by:

§10.38, Ch.10

Permalink:

http://dlmf.nist.gov/10.38.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.38, §10.38 and Ch.10

…

27: 10.59 Integrals

… ►

10.59.1

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind and $n$: integer

A&S Ref:

11.4.26

Referenced by:

§10.59

Permalink:

http://dlmf.nist.gov/10.59.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.59 and Ch.10

►where $P_n$ is the Legendre polynomial (§18.3). …

28: 30.6 Functions of Complex Argument

… ►The solutions …of (30.2.1) with $\mu =m$ and $\lambda ={\lambda }_n^m\left({\gamma }^2\right)$ are real when $z\in (1,\mathrm{\infty })$, and their principal values (§4.2(i)) are obtained by analytic continuation to $ℂ\setminus (-\mathrm{\infty },1]$. … ►with $A_n^{\pm m}({\gamma }^2)$ as in (30.11.4). … ►

30.6.4 $${\mathrm{𝑃𝑠}}_n^m(x\pm \mathrm{i}0,{\gamma }^2)={(\mp \mathrm{i})}^m{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),$$

ⓘ

Symbols:

$\mathrm{i}$: imaginary unit, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.6, §30.6 and Ch.30

►

30.6.5 $${\mathrm{𝑄𝑠}}_n^m(x\pm \mathrm{i}0,{\gamma }^2)={(\mp \mathrm{i})}^m\left({\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)\mp \frac{1}{2}\mathrm{i}\pi {\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §30.6, §30.6 and Ch.30

…

29: 4.25 Continued Fractions

… ►

4.25.1 $$\tan z=\frac{z}{1-}\frac{z^2}{3-}\frac{z^2}{5-}\frac{z^2}{7-}\mathrm{\cdots },$$ $z\ne \pm \frac{1}{2}\pi$, $\pm \frac{3}{2}\pi$, $\mathrm{\dots }$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.94

Permalink:

http://dlmf.nist.gov/4.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

►

4.25.2 $$\tan \left(az\right)=\frac{a\tan z}{1+}\frac{(1-a^2){\tan }^{2}z}{3+}\frac{(4-a^2){\tan }^{2}z}{5+}\frac{(9-a^2){\tan }^{2}z}{7+}\mathrm{\cdots },$$ , $az\ne \pm \frac{1}{2}\pi ,\pm \frac{3}{2}\pi ,\mathrm{\dots }$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Re }$: real part, $\tan z$: tangent function, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.3.95

Permalink:

http://dlmf.nist.gov/4.25.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

►

4.25.3 $$\frac{\arcsin z}{\sqrt{1-z^2}}=\frac{z}{1-}\frac{1\cdot 2z^2}{3-}\frac{1\cdot 2z^2}{5-}\frac{3\cdot 4z^2}{7-}\frac{3\cdot 4z^2}{9-}\mathrm{\cdots },$$

ⓘ

Symbols:

$\arcsin z$: arcsine function and $z$: complex variable

A&S Ref:

4.4.44

Permalink:

http://dlmf.nist.gov/4.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

… ►

4.25.4 $$\arctan z=\frac{z}{1+}\frac{z^2}{3+}\frac{4z^2}{5+}\frac{9z^2}{7+}\frac{16z^2}{9+}\mathrm{\cdots },$$

ⓘ

Symbols:

$\arctan z$: arctangent function and $z$: complex variable

A&S Ref:

4.4.43

Referenced by:

§3.10(ii)

Permalink:

http://dlmf.nist.gov/4.25.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

… ►

4.25.5 $${\mathrm{e}}^{2a\arctan \left(1/z\right)}=1+\frac{2a}{z-a+}\frac{a^2+1}{3z+}\frac{a^2+4}{5z+}\frac{a^2+9}{7z+}\mathrm{\cdots },$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $\arctan z$: arctangent function, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.2.42

Permalink:

http://dlmf.nist.gov/4.25.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

…

30: 4.35 Identities

… ►

4.35.1 $$\sinh \left(u\pm v\right)=\sinh u\cosh v\pm \cosh u\sinh v,$$

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function and $\sinh z$: hyperbolic sine function

A&S Ref:

4.5.24 (modified)

Permalink:

http://dlmf.nist.gov/4.35.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.35(i), §4.35 and Ch.4

►

4.35.2 $$\cosh \left(u\pm v\right)=\cosh u\cosh v\pm \sinh u\sinh v,$$

ⓘ

Symbols:

$\cosh z$: hyperbolic cosine function and $\sinh z$: hyperbolic sine function

A&S Ref:

4.5.25 (modified)

Permalink:

http://dlmf.nist.gov/4.35.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.35(i), §4.35 and Ch.4

►

4.35.3 $$\tanh \left(u\pm v\right)=\frac{\tanh u\pm \tanh v}{1\pm \tanh u\tanh v},$$

ⓘ

Symbols:

$\tanh z$: hyperbolic tangent function

A&S Ref:

4.5.26 (modified)

Permalink:

http://dlmf.nist.gov/4.35.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.35(i), §4.35 and Ch.4

►

4.35.4 $$\coth \left(u\pm v\right)=\frac{\pm \coth u\coth v+1}{\coth u\pm \coth v}.$$

ⓘ

Symbols:

$\coth z$: hyperbolic cotangent function

A&S Ref:

4.5.27 (modified)

Permalink:

http://dlmf.nist.gov/4.35.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.35(i), §4.35 and Ch.4

… ►

4.35.33 $$\cosh \left(nz\right)\pm \sinh \left(nz\right)={(\cosh z\pm \sinh z)}^n,$$ $n\in \mathbb{Z}$.

ⓘ

Symbols:

$\in$: element of, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathbb{Z}$: set of all integers, $n$: integer and $z$: complex variable

A&S Ref:

4.5.53

Permalink:

http://dlmf.nist.gov/4.35.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §4.35(iii), §4.35 and Ch.4

…