hyperbolic tangent function

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11: 4.40 Integrals

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4.40.3 $$\int \tanh xdx=\ln \left(\cosh x\right).$$

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Symbols:

$dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.5.79

Permalink:

http://dlmf.nist.gov/4.40.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

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4.40.4 $$\int \mathrm{csch}xdx=\ln \left(\tanh \left(\frac{1}{2}x\right)\right),$$ .

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Symbols:

$dx$: differential of $x$ , $\mathrm{csch}z$: hyperbolic cosecant function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.5.80

Permalink:

http://dlmf.nist.gov/4.40.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(ii), §4.40 and Ch.4

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4.40.8 $${\int }_0^{\mathrm{\infty }}\frac{\sinh \left(ax\right)}{\sinh \left(\pi x\right)}dx=\frac{1}{2}\tan \left(\frac{1}{2}a\right),$$ ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, $\tan z$: tangent function, $a$: real or complex constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.40.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iii), §4.40 and Ch.4

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4.40.10 $${\int }_0^{\mathrm{\infty }}\frac{\tanh \left(ax\right)-\tanh \left(bx\right)}{x}dx=\ln \left(\frac{a}{b}\right),$$ $a>0$, $b>0$.

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Symbols:

$dx$: differential of $x$ , $\tanh z$: hyperbolic tangent function, $\int$: integral, $\ln z$: principal branch of logarithm function, $a$: real or complex constant and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.40.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iii), §4.40 and Ch.4

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4.40.13 $$\int \mathrm{arctanh}xdx=x\mathrm{arctanh}x+\frac{1}{2}\ln \left(1-x^2\right),$$ ,

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Symbols:

$dx$: differential of $x$ , $\mathrm{arctanh}z$: inverse hyperbolic tangent function, $\int$: integral, $\ln z$: principal branch of logarithm function and $x$: real variable

A&S Ref:

4.6.45 (modified)

Permalink:

http://dlmf.nist.gov/4.40.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §4.40(iv), §4.40 and Ch.4

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12: 4.39 Continued Fractions

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4.39.1 $$\tanh z=\frac{z}{1+}\frac{z^2}{3+}\frac{z^2}{5+}\frac{z^2}{7+}\mathrm{\cdots },$$ $z\ne \pm \frac{1}{2}\pi \mathrm{i},\pm \frac{3}{2}\pi \mathrm{i},\mathrm{\dots }$.

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Symbols:

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

A&S Ref:

4.5.70

Permalink:

http://dlmf.nist.gov/4.39.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.39 and Ch.4

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4.39.3 $$\mathrm{arctanh}z=\frac{z}{1-}\frac{z^2}{3-}\frac{4z^2}{5-}\frac{9z^2}{7-}\mathrm{\cdots },$$

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Symbols:

$\mathrm{arctanh}z$: inverse hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.6.35

Permalink:

http://dlmf.nist.gov/4.39.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.39 and Ch.4

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13: 29.5 Special Cases and Limiting Forms

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29.5.5 $$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{\mathrm{𝐸𝑐}}_{\nu }^m(0,k^2)}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(0,k^2)}=\frac{1}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu ,\frac{1}{2}\mu +\frac{1}{2}\nu +\frac{1}{2}}{\frac{1}{2}};{\tanh }^{2}z),$$ $m$ even,

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.5 and Ch.29

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29.5.6 $$\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)}{{d{\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)/dz|}_{z=0}}=\underset{k\to 1-}{lim}\frac{{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)}{{d{\mathrm{𝐸𝑠}}_{\nu }^{m+1}(z,k^2)/dz|}_{z=0}}=\frac{\tanh z}{{(\cosh z)}^{\mu }}F(\genfrac{}{}{0pt}{}{\frac{1}{2}\mu -\frac{1}{2}\nu +\frac{1}{2},\frac{1}{2}\mu +\frac{1}{2}\nu +1}{\frac{3}{2}};{\tanh }^{2}z),$$ $m$ odd,

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $m$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $\mu$: parameter

Permalink:

http://dlmf.nist.gov/29.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.5 and Ch.29

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14: 28.23 Expansions in Series of Bessel Functions

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28.23.3 $${\mathrm{me}}_{\nu }^{\prime }(0,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h)=\mathrm{i}\tanh z\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n(\nu +2n)c_{2n}^{\nu }(h^2){𝒞}_{\nu +2n}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $c_{2m}(q)$: coefficients and ${𝒞}_{\mu }^{(j)}$: cylinder functions

Permalink:

http://dlmf.nist.gov/28.23.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.10 $${\mathrm{Ms}}_{2m+1}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{se}}_{2m+1}^{\prime }(0,h^2)\right)}^{-1}\tanh z\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}(2\mathrm{\ell }+1)B_{2\mathrm{\ell }+1}^{2m+1}(h^2){𝒞}_{2\mathrm{\ell }+1}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.12 $${\mathrm{Ms}}_{2m+2}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{se}}_{2m+2}^{\prime }(0,h^2)\right)}^{-1}\tanh z\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}(2\mathrm{\ell }+2)B_{2\mathrm{\ell }+2}^{2m+2}(h^2){𝒞}_{2\mathrm{\ell }+2}^{(j)}(2h\cosh z),$$

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Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.23.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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15: 19.10 Relations to Other Functions

§19.10 Relations to Other Functions

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§19.10(i) Theta and Elliptic Functions

►For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. … ►

§19.10(ii) Elementary Functions

… ►In each case when $y=1$, the quantity multiplying $R_C$ supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0. …

16: 4.33 Maclaurin Series and Laurent Series

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4.33.3 $$\tanh z=z-\frac{z^3}{3}+\frac{2}{15}{z}^5-\frac{17}{315}{z}^7+\mathrm{\cdots }+\frac{2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}{z}^{2n-1}+\mathrm{\cdots },$$ .

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\tanh z$: hyperbolic tangent function, $n$: integer and $z$: complex variable

A&S Ref:

4.5.64

Permalink:

http://dlmf.nist.gov/4.33.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.33 and Ch.4

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17: 33.7 Integral Representations

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33.7.3 $$H_{\mathrm{\ell }}^{-}(\eta ,\rho )=\frac{-\mathrm{i}{\mathrm{e}}^{-\pi \eta }{\rho }^{\mathrm{\ell }+1}}{(2\mathrm{\ell }+1)!C_{\mathrm{\ell }}\left(\eta \right)}{\int }_0^{\mathrm{\infty }}\left(\frac{\exp \left(-\mathrm{i}(\rho \tanh t-2\eta t)\right)}{{(\cosh t)}^{2\mathrm{\ell }+2}}+\mathrm{i}{(1+t^2)}^{\mathrm{\ell }}\exp \left(-\rho t+2\eta \arctan t\right)\right)dt,$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\arctan z$: arctangent function, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

A&S Ref:

14.3.3

Referenced by:

§33.7

Permalink:

http://dlmf.nist.gov/33.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.7 and Ch.33

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18: 12.12 Integrals

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12.12.4 $${(U(a,z))}^2+{(\overline{U}(a,z))}^2=\frac{2^{\frac{3}{2}}}{\pi }\mathrm{\Gamma }\left(\frac{1}{2}-a\right){\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2at+\frac{1}{2}{z}^2\tanh t}}{\sqrt{\sinh \left(2t\right)}}dt,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\overline{U}(a,x)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: real or complex parameter

Notes:

See Durand (1975).

Referenced by:

§12.12

Permalink:

http://dlmf.nist.gov/12.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12, §12.12 and Ch.12

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19: 4.38 Inverse Hyperbolic Functions: Further Properties

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4.38.5 $$\mathrm{arctanh}z=z+\frac{z^3}{3}+\frac{z^5}{5}+\frac{z^7}{7}+\mathrm{\cdots },$$ $\left|z\right|\le 1$, $z\ne \pm 1$.

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Symbols:

$\mathrm{arctanh}z$: inverse hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.6.33

Permalink:

http://dlmf.nist.gov/4.38.E5

Encodings:

pMML, png, TeX

See also:

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

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4.38.7 $$\mathrm{arctanh}z=\frac{z}{1-z^2}\left(1+\frac{2}{3}\frac{z^2}{z^2-1}+\frac{2\cdot 4}{3\cdot 5}{\left(\frac{z^2}{z^2-1}\right)}^2+\mathrm{\cdots }\right),$$ ,

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Symbols:

$\mathrm{arctanh}z$: inverse hyperbolic tangent function, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§4.38(i)

Permalink:

http://dlmf.nist.gov/4.38.E7

Encodings:

pMML, png, TeX

See also:

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

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4.38.11 $$\frac{d}{dz}\mathrm{arctanh}z=\frac{1}{1-z^2}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arctanh}z$: inverse hyperbolic tangent function and $z$: complex variable

A&S Ref:

4.6.39

Permalink:

http://dlmf.nist.gov/4.38.E11

Encodings:

pMML, png, TeX

See also:

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

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4.38.17 $$\mathrm{Arctanh}u\pm \mathrm{Arctanh}v=\mathrm{Arctanh}\left(\frac{u\pm v}{1\pm uv}\right),$$

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Symbols:

$\mathrm{Arctanh}z$: general inverse hyperbolic tangent function

A&S Ref:

4.6.28

Permalink:

http://dlmf.nist.gov/4.38.E17

Encodings:

pMML, png, TeX

See also:

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

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4.38.19 $$\mathrm{Arctanh}u\pm \mathrm{Arccoth}v=\mathrm{Arctanh}\left(\frac{uv\pm 1}{v\pm u}\right)=\mathrm{Arccoth}\left(\frac{v\pm u}{uv\pm 1}\right).$$

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Symbols:

$\mathrm{Arccoth}z$: general inverse hyperbolic cotangent function and $\mathrm{Arctanh}z$: general inverse hyperbolic tangent function

A&S Ref:

4.6.30

Permalink:

http://dlmf.nist.gov/4.38.E19

Encodings:

pMML, png, TeX

See also:

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

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20: 10.19 Asymptotic Expansions for Large Order

§10.19 Asymptotic Expansions for Large Order

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§10.19(i) Asymptotic Forms

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§10.19(ii) Debye’s Expansions

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§10.19(iii) Transition Region

… ►See also §10.20(i).