hyperbolic tangent function

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21: 5.4 Special Values and Extrema

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5.4.17 $$\mathrm{\Im }\psi \left(\frac{1}{2}+\mathrm{i}y\right)=\frac{\pi }{2}\tanh \left(\pi y\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\tanh z$: hyperbolic tangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.3.12

Permalink:

http://dlmf.nist.gov/5.4.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(ii), §5.4 and Ch.5

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22: 10.24 Functions of Imaginary Order

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10.24.7 $${\tilde{J}}_{\nu }\left(x\right)={\left(\frac{2\tanh \left(\frac{1}{2}\pi \nu \right)}{\pi \nu }\right)}^{\frac{1}{2}}\cos \left(\nu \ln \left(\frac{1}{2}x\right)-{\gamma }_{\nu }\right)+O\left(x^2\right),$$

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Symbols:

${\tilde{J}}_{\nu }\left(x\right)$: Bessel function of imaginary order, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\tanh z$: hyperbolic tangent function, $\ln z$: principal branch of logarithm function, $x$: real variable, $\nu$: complex parameter and ${\gamma }_{\nu }$: parameter

Referenced by:

§10.24, §10.24

Permalink:

http://dlmf.nist.gov/10.24.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.24 and Ch.10

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23: 19.2 Definitions

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19.2.18 $$R_C(x,y)=\frac{1}{\sqrt{y-x}}\arctan \sqrt{\frac{y-x}{x}}=\frac{1}{\sqrt{y-x}}\arccos \sqrt{x/y},$$ ,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\arccos z$: arccosine function and $\arctan z$: arctangent function

Referenced by:

§19.2(iv)

Permalink:

http://dlmf.nist.gov/19.2.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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19.2.19 $$R_C(x,y)=\frac{1}{\sqrt{x-y}}\mathrm{arctanh}\sqrt{\frac{x-y}{x}}=\frac{1}{\sqrt{x-y}}\ln \frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{y}},$$ .

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{arctanh}z$: inverse hyperbolic tangent function and $\ln z$: principal branch of logarithm function

Referenced by:

§19.26(i), Figure 19.3.2, Figure 19.3.2, §19.36(ii)

Permalink:

http://dlmf.nist.gov/19.2.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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19.2.20 $$R_C(x,y)=\sqrt{\frac{x}{x-y}}{R}_C(x-y,-y)=\frac{1}{\sqrt{x-y}}\mathrm{arctanh}\sqrt{\frac{x}{x-y}}=\frac{1}{\sqrt{x-y}}\ln \frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{-y}},$$ .

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{arctanh}z$: inverse hyperbolic tangent function and $\ln z$: principal branch of logarithm function

Referenced by:

§19.2(iv), §19.20(iii), §19.25(i), Figure 19.3.2, Figure 19.3.2, §19.36(i)

Permalink:

http://dlmf.nist.gov/19.2.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iv), §19.2 and Ch.19

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24: 22.5 Special Values

§22.5 Special Values

… ►For the other nine functions ratios can be taken; compare (22.2.10). … ►

§22.5(ii) Limiting Values of $k$

… ►In these cases the elliptic functions degenerate into elementary trigonometric and hyperbolic functions, respectively. … ►

25: 15.12 Asymptotic Approximations

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15.12.5 $$𝐅(\genfrac{}{}{0pt}{}{a+\lambda ,b-\lambda }{c};\frac{1}{2}-\frac{1}{2}z)=2^{(a+b-1)/2}\frac{{(z+1)}^{(c-a-b-1)/2}}{{(z-1)}^{c/2}}\sqrt{\zeta \sinh \zeta }{\left(\lambda +\frac{1}{2}a-\frac{1}{2}b\right)}^{1-c}\left(I_{c-1}\left((\lambda +\frac{1}{2}a-\frac{1}{2}b)\zeta \right)(1+O({\lambda }^{-2}))+\frac{I_{c-2}\left((\lambda +\frac{1}{2}a-\frac{1}{2}b)\zeta \right)}{2\lambda +a-b}\left(\left(c-\frac{1}{2}\right)\left(c-\frac{3}{2}\right)\left(\frac{1}{\zeta }-\coth \zeta \right)+\frac{1}{2}(2c-a-b-1)(a+b-1)\tanh \left(\frac{1}{2}\zeta \right)+O({\lambda }^{-2})\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/15.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

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26: 4.23 Inverse Trigonometric Functions

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4.23.40 $$\mathrm{gd}\left(x\right)=2\arctan \left({\mathrm{e}}^x\right)-\frac{1}{2}\pi =\arcsin \left(\tanh x\right)=\mathrm{arccsc}\left(\coth x\right)=\arccos \left(\mathrm{sech}x\right)=\mathrm{arcsec}\left(\cosh x\right)=\arctan \left(\sinh x\right)=\mathrm{arccot}\left(\mathrm{csch}x\right).$$

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Symbols:

$\mathrm{gd}x$: Gudermannian function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\coth z$: hyperbolic cotangent function, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $\mathrm{arccsc}z$: arccosecant function, $\arccos z$: arccosine function, $\mathrm{arccot}z$: arccotangent function, $\mathrm{arcsec}z$: arcsecant function, $\arcsin z$: arcsine function, $\arctan z$: arctangent function and $x$: real variable

Referenced by:

§4.23(viii), §4.40(ii)

Permalink:

http://dlmf.nist.gov/4.23.E40

Encodings:

pMML, png, TeX

See also:

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

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4.23.42 $${\mathrm{gd}}^{-1}\left(x\right)=\ln \tan \left(\frac{1}{2}x+\frac{1}{4}\pi \right)=\ln \left(\sec x+\tan x\right)=\mathrm{arcsinh}\left(\tan x\right)=\mathrm{arccsch}\left(\cot x\right)=\mathrm{arccosh}\left(\sec x\right)=\mathrm{arcsech}\left(\cos x\right)=\mathrm{arctanh}\left(\sin x\right)=\mathrm{arccoth}\left(\csc x\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $\cot z$: cotangent function, $\mathrm{arccsch}z$: inverse hyperbolic cosecant function, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{arccoth}z$: inverse hyperbolic cotangent function, $\mathrm{arcsech}z$: inverse hyperbolic secant function, $\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\ln z$: principal branch of logarithm function, $\sec z$: secant function, $\sin z$: sine function, $\tan z$: tangent function and $x$: real variable

Referenced by:

§19.9(ii), §4.23(viii), §4.26(ii)

Permalink:

http://dlmf.nist.gov/4.23.E42

Encodings:

pMML, png, TeX

See also:

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

27: 10.20 Uniform Asymptotic Expansions for Large Order

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10.20.17 $$z=\pm {(\tau \coth \tau -{\tau }^2)}^{\frac{1}{2}}\pm \mathrm{i}{({\tau }^2-\tau \tanh \tau )}^{\frac{1}{2}},$$ $0\le \tau \le {\tau }_0$,

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Symbols:

$\coth z$: hyperbolic cotangent function, $\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.20.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(ii), §10.20 and Ch.10

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28: 14.5 Special Values

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14.5.25 $$P_{-\frac{1}{2}}\left(\cosh \xi \right)=\frac{2}{\pi \cosh \left(\frac{1}{2}\xi \right)}K\left(\tanh \left(\frac{1}{2}\xi \right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind and $\xi >0$: variable

A&S Ref:

8.13.2

Permalink:

http://dlmf.nist.gov/14.5.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(v), §14.5 and Ch.14

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29: 14.15 Uniform Asymptotic Approximations

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14.15.27 $$\frac{1}{2}\zeta {\left({\zeta }^2-{\alpha }^2\right)}^{1/2}-\frac{1}{2}{\alpha }^2\mathrm{arccosh}\left(\frac{\zeta }{\alpha }\right)={\left(1-a^2\right)}^{1/2}\mathrm{arctanh}\left(\frac{1}{x}{\left(\frac{x^2-a^2}{1-a^2}\right)}^{1/2}\right)-\mathrm{arccosh}\left(\frac{x}{a}\right),$$ , ,

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Symbols:

$\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, $x$: real variable, $a$ , $\zeta$ and $\alpha$

Permalink:

http://dlmf.nist.gov/14.15.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(v), §14.15 and Ch.14

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14.15.31 $$\frac{1}{2}\zeta {\left({\zeta }^2+{\alpha }^2\right)}^{1/2}+\frac{1}{2}{\alpha }^2\mathrm{arcsinh}\left(\frac{\zeta }{\alpha }\right)={\left(1+a^2\right)}^{1/2}\mathrm{arctanh}\left(x{\left(\frac{1+a^2}{x^2+a^2}\right)}^{1/2}\right)-\mathrm{arcsinh}\left(\frac{x}{a}\right),$$ , ,

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Symbols:

$\mathrm{arcsinh}z$: inverse hyperbolic sine function, $\mathrm{arctanh}z$: inverse hyperbolic tangent function, $x$: real variable, $a$ , $\zeta$ and $\alpha$

Permalink:

http://dlmf.nist.gov/14.15.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(v), §14.15 and Ch.14

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30: 14.20 Conical (or Mehler) Functions

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14.20.14 $$\pi {\int }_0^{\mathrm{\infty }}\frac{\tau \tanh \left(\tau \pi \right)}{\cosh \left(\tau \pi \right)}{P}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)P_{-\frac{1}{2}+\mathrm{i}\tau }\left(y\right)d\tau =\frac{1}{y+x}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, $\int$: integral, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $x$: real variable and $\tau$: real variable

Permalink:

http://dlmf.nist.gov/14.20.E14

Encodings:

pMML, png, TeX

See also:

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

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