Tanh

Hyperbolic tangent: $\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaHyperbolic functions↗MathWorldHyperbolicTangent↗HyperbolicFunctions↗WikidataQ204034↗Britannicatopic/hyperbolic-functions↗DLMF4.28.E4↗FungrimTanh✓↗Fungrim entry6f3fec✓↗Wolfram LanguageTanh✓↗

Tanh(x)the hyperbolic tangent of x.

SageMathtanh(x)SymPytanh(x)mpmathtanh(x)Juliaenumeratio_ball(tanh, x)Rusttanh(x)

Typed signatures

Type	Package	Applies
(complex | quantity | signed_infinity) -> number	compute-engine	always
(complex | complex | infinity | quantity | signed_infinity) -> number	analyticreplaces compute-engine	always

Domain: Elementary functions

Details

• $\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$, ranging over $(-1, 1)$.
• Odd function: $\tanh(-x) = -\tanh(x)$.
• Same fold-only-with-N(...) behavior as Sinh and Cosh.

Examples

2 more kept as data, checked against the oracles

Basic #

N(...) forces the numeric result that plain evaluation withholds

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In$N(\tanh(0))$Out$0.0$

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In$N(\tanh(1))$Out$0.761\,594\,155\,955\,764\,888\,119$

Exact at 0, as Sinh and Cosh are

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In$\tanh(0)$Out$0$

A float argument evaluates directly, no N(...) needed

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In$\tanh(2.2)$Out$0.975\,743\,130\,031\,451\,520\,414$

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In$\operatorname{Series}(\tanh(x), x, 0, 10)$Out$x-\frac{x^3}{3}+\frac{2x^5}{15}-\frac{17x^7}{315}+\frac{62x^9}{2\,835}+O\left(x^{11}\right)$

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In$\operatorname{Series}(\tanh(x), x, \frac{1}{2}i π, 3)$Out$\frac{1}{x+(-\frac{1}{2}i)π}+\frac{1}{3}(x+(-\frac{1}{2}i)π)-\frac{1}{45}(x+(-\frac{1}{2}i)π)^3+O\left((x+(-\frac{1}{2}i)π)^5\right)$

Scope #

Arbitrary precision via N(...)

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In$N(\tanh(\frac{11}{5}))$Out$0.975\,743\,130\,031\,451\,520\,414$

Evaluates at a complex argument

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In$\tanh(2.5+i)$Out$1.005\,548\,011\,895\,06+0.012\,322\,138\,255\,828\,353i$

Interval arithmetic: $\tanh$ is increasing, so $\tanh[0, \ln 2] = [0, \tfrac35]$

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In$\tanh(\mathrm{Interval}(0, \ln(2)))$Out$\mathrm{Interval}(0, \frac{3}{5})$

Uncertainty propagation: $\tanh(2/\sqrt3 \pm 0.1) \approx 0.819 \pm 0.033$

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In$\tanh(\mathrm{Around}(\frac{2}{\sqrt{3}}, 0.1))$Out$\mathrm{Around}(0.819\,305\,290\,078\,300\,8, 0.032\,873\,884\,164\,971\,15)$

The horizontal asymptote at $+\infty$

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In$\tanh(∞)$Out$1$

$\tanh(i\pi/4) = i\tan(\pi/4) = i$

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In$\tanh(\frac{i π}{4})$Out$i$

Oddness pulls the sign out of a symbolic argument: $\tanh(-x) = -\tanh x$

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In$\tanh(-x)$Out$-\tanh(x)$

$\tanh(\operatorname{artanh} x) = x$ for every $x$

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In$\tanh(\mathrm{artanh}(x))$Out$x$

$\tanh(\operatorname{artanh}(2)) = 2$, even past $(-1, 1)$

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In$\tanh(\mathrm{artanh}(2))$Out$2$

Derivative: $\operatorname{sech}^2 x$

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In$\frac{\mathrm{d}\tanh(x)}{\mathrm{d}x}$Out$\mathrm{sech}(x)^2$

Antiderivative: $\ln\cosh x$

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In$\int\!\tanh(x)\, \mathrm{d}x$Out$\ln(\cosh(x))$

$\tanh$ has no limit at complex infinity, undefined from any direction: NaN

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In$\tanh(\tilde∞)$Out$\operatorname{NaN}$

To 30 significant digits, every one correctly rounded

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In$N(\tanh(1), 30)$Out$0.761\,594\,155\,955\,764\,888\,119\,458\,282\,605$

A 50-digit value of $\tanh\frac{11}5$

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In$N(\tanh(\frac{11}{5}), 50)$Out$0.975\,743\,130\,031\,451\,520\,414\,306\,668\,030\,114\,652\,781\,305\,034\,057\,99$

Threads over a matrix of imaginary angles, giving $i\tan\theta$

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In$\tanh(\bigl\lbrack\bigl\lbrack(1/3)i π, (-(1/3)i)π\bigr\rbrack, \bigl\lbrack0, (-(1/6)i)π\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\sqrt{3}i, -\sqrt{3}i\bigr\rbrack, \bigl\lbrack0, -\frac{3^{1/2}}{3}i\bigr\rbrack\bigr\rbrack$

On the imaginary axis, $\tanh(i\theta)=i\tan\theta$, a radical for $\pi/5$

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In$\tanh(\frac{1}{5}i π)$Out$i\sqrt{5-2\sqrt{5}}$

Left as a tangent when no radical form is known

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In$\tanh(\frac{1}{24}i π)$Out$i\tan(\frac{π}{24})$

A matrix function of a triangular matrix of imaginary angles

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In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\tanh(\operatorname{\_1}), \bigl\lbrack\bigl\lbrack(1/3)i π, (-(1/3)i)π\bigr\rbrack, \bigl\lbrack0, (-(1/6)i)π\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\sqrt{3}i, -\frac{8×3^{1/2}}{9}i\bigr\rbrack, \bigl\lbrack0, -\frac{3^{1/2}}{3}i\bigr\rbrack\bigr\rbrack$

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In$\mathrm{FunctionMeromorphic}(\tanh(x), x)$Out$\mathrm{True}$

Properties #

$\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$

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In$\tanh(1)=\frac{\sinh(1)}{\cosh(1)}$Out$\mathrm{True}$

In exponentials: $\tanh x = \frac{e^x - e^{-x}}{e^x + e^{-x}}$

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In$\mathrm{TrigToExp}(\tanh(x))$Out$\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$

$\tanh(i\pi/12) = i(2-\sqrt3)$

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In$\tanh(\frac{1}{12}i π)$Out$i(2-\sqrt{3})$

Known value · Wolfram MathematicalFunctionData, Tanh

A tenth of $\pi$ gives nested radicals

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In$\tanh(\frac{1}{10}i π)$Out$\frac{\sqrt{5}}{5}i\sqrt{5-2\sqrt{5}}$

Known value · Wolfram MathematicalFunctionData, Tanh

A ninth of $\pi$ stays as a tangent

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In$\tanh(\frac{1}{9}i π)$Out$i\tan(\frac{π}{9})$

Known value · Wolfram MathematicalFunctionData, Tanh

$\tanh(i\pi/8)=i(\sqrt2-1)$

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In$\tanh(\frac{1}{8}i π)$Out$i(\sqrt{2}-1)$

Known value · Wolfram MathematicalFunctionData, Tanh

A seventh of $\pi$ stays as a tangent

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In$\tanh(\frac{1}{7}i π)$Out$i\tan(\frac{π}{7})$

Known value · Wolfram MathematicalFunctionData, Tanh

$\tanh(i\pi/6)=\frac{i}{\sqrt3}$

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In$\tanh(\frac{1}{6}i π)$Out$\frac{\sqrt{3}}{3}i$

Known value · Wolfram MathematicalFunctionData, Tanh

Another non-constructible angle, $2\pi/9$

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In$\tanh(\frac{2}{9}i π)$Out$i\tan(\frac{2π}{9})$

Known value · Wolfram MathematicalFunctionData, Tanh

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In$N((1+\tanh(1/2)^2)\tanh(1))$Out$0.924\,234\,314\,520\,019\,517\,003\,851\,527\,548\,909\,824\,992\,15$

Known value · Wolfram MathematicalFunctionData, Tanh, FunctionalEquations

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In$N(\tanh(\frac{1}{2}+\frac{1}{2}i π))$Out$2.163\,953\,413\,738\,653$

Known value · Wolfram MathematicalFunctionData, Tanh, FunctionalEquations

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In$N((1+\tanh(2)^2)\tanh(4))$Out$1.928\,055\,160\,151\,633\,767\,891\,606\,112\,455\,327\,689\,634\,896$

Known value · Wolfram MathematicalFunctionData, Tanh, FunctionalEquations

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In$N(\tanh(2+\frac{1}{2}i π))$Out$1.037\,314\,720\,727\,548\,2$

Known value · Wolfram MathematicalFunctionData, Tanh, FunctionalEquations

About

Examples	40
Checked against	Wolfram Language 39 agreeSageMath 33 agree1 errorSymPy 35 agree2 differmpmath 27 agreeJulia 25 agree2 errorRust 10 agree1 error
Crosswalk	WikipediaMathWorld ×2WikidataBritannicaDLMFFungrimFungrim entryWolfram LanguageSageMathSymPympmathJuliaRust

Implementation

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

See also: Sinh, Cosh