Sinh

Hyperbolic sine: $\sinh(x) = \frac{e^x - e^{-x}}{2}$.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaHyperbolic functions↗MathWorldHyperbolicSine↗HyperbolicFunctions↗WikidataQ204034↗Britannicatopic/hyperbolic-functions↗DLMF4.28.E1↗FungrimSinh✓↗Fungrim entryWolfram LanguageSinh!↗

Sinh(x)the hyperbolic sine of x.

SageMathsinh(x)SymPysinh(x)mpmathsinh(x)Juliaenumeratio_ball(sinh, x)Rustsinh(x)

Typed signatures

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

Domain: Elementary functions

Details

• Defined in terms of Exp: $\sinh(x) = \frac{e^x - e^{-x}}{2}$.
• Odd function: $\sinh(-x) = -\sinh(x)$.
• Exact at $x = 0$ as Sin is: $\sinh(0) = 0$; a floating-point argument, or N(...), gives a numeric result.

Examples

Basic #

N(...) gives the numeric value

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

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In$N(\sinh(1))$Out$1.175\,201\,193\,643\,801\,456\,88$

Exact at 0, as Sin is

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In$\sinh(0)$Out$0$

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

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In$\sinh(1.4)$Out$1.904\,301\,501\,451\,534\,055\,14$

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In$\operatorname{Series}(\sinh(x), x, 0, 10)$Out$x+\frac{x^3}{6}+\frac{x^5}{120}+\frac{x^7}{5\,040}+\frac{x^9}{362\,880}+O\left(x^{11}\right)$

Scope #

Arbitrary precision via N(...)

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In$N(\sinh(\frac{7}{5}))$Out$1.904\,301\,501\,451\,534\,055\,14$

Evaluates at a complex argument

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In$\sinh(2.5+i)$Out$3.268\,939\,432\,079\,549+5.160\,143\,667\,579\,705i$

$\sinh(i\pi/2) = i\sin(\pi/2) = i$

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In$\sinh(\frac{i π}{2})$Out$i$

Element-wise over a matrix of imaginary multiples of $\pi$, folding through $\sinh(ix) = i\sin x$

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

Interval arithmetic: $\sinh$ is increasing, so $\sinh[0, \ln 3] = [0, \tfrac43]$, the interval of the endpoint values

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In$\sinh(\mathrm{Interval}(0, \ln(3)))$Out$\mathrm{Interval}(0, \frac{4}{3})$

Uncertainty propagation: $\sinh(2/\sqrt3 \pm 0.1) \approx 1.429 \pm 0.174$

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In$\sinh(\mathrm{Around}(\frac{2}{\sqrt{3}}, 0.1))$Out$\mathrm{Around}(1.428\,960\,581\,481\,575\,8, 0.174\,411\,248\,015\,377\,8)$

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In$\sinh(∞)$Out$∞$

$\sinh(\ln 2) = \frac{2 - 1/2}{2} = \tfrac34$

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In$\sinh(\ln(2))$Out$\frac{3}{4}$

A real symbolic argument is untouched

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In$\sinh(x)$Out$\sinh(x)$

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

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

$\sinh(\operatorname{arsinh} x) = x$ for every $x$

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In$\sinh(\mathrm{arsinh}(x))$Out$x$

$\sinh(\operatorname{arsinh}(2)) = 2$

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In$\sinh(\mathrm{arsinh}(2))$Out$2$

Derivative: $\cosh x$

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In$\frac{\mathrm{d}\sinh(x)}{\mathrm{d}x}$Out$\cosh(x)$

Antiderivative

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

Taylor series about 0

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In$\operatorname{Series}(\sinh(x), x, 0, 5)$Out$x+\frac{x^3}{6}+\frac{x^5}{120}+O\left(x^7\right)$

To 30 significant digits, the last one correctly rounded

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In$N(\sinh(1), 30)$Out$1.175\,201\,193\,643\,801\,456\,882\,381\,850\,6$

N(...) decimalizes $\sinh(i\pi/2)$, not just the exact $i$ form

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In$N(\sinh(\frac{i π}{2}))$Out$1.0i$

A 50-digit value of $\sinh\frac75$

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In$N(\sinh(\frac{7}{5}), 50)$Out$1.904\,301\,501\,451\,534\,055\,142\,123\,827\,697\,426\,309\,551\,638\,683\,721\,3$

On the imaginary axis, $\sinh(i\theta)=i\sin\theta$, with a radical for $\theta=\pi/5$

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In$\sinh(\frac{1}{5}i π)$Out$\frac{\sqrt{2}}{4}i\sqrt{5-\sqrt{5}}$

Left in terms of sine when no radical form is known

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In$\sinh(\frac{1}{24}i π)$Out$i\sin(\frac{π}{24})$

A matrix function of an upper triangular imaginary matrix

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

Properties #

Odd function: $\sinh(-x) = -\sinh(x)$

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In$\sinh(-1)=-\sinh(1)$Out$\mathrm{True}$

Definition in terms of Exp: $\sinh(x) = \frac{e^x - e^{-x}}{2}$

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In$\sinh(1)=\frac{e^1-1/e}{2}$Out$\mathrm{True}$

The definition $\sinh x = \frac{e^x - e^{-x}}{2}$

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In$\mathrm{TrigToExp}(\sinh(x))$Out$\frac{1}{2}(e^{x}-e^{-x})$

$\sinh(i\pi/12)$ is $i\sin 15^\circ$ in radicals

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In$\sinh(\frac{1}{12}i π)$Out$\frac{\sqrt{2}}{4}i(\sqrt{3}-1)$

Known value · Wolfram MathematicalFunctionData, Sinh

$\sinh(i\pi/10) = i\,\frac{\sqrt5-1}{4}$

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In$\sinh(\frac{1}{10}i π)$Out$\frac{1}{4}i(\sqrt{5}-1)$

Known value · Wolfram MathematicalFunctionData, Sinh

A ninth of $\pi$ has no radical form, so it stays as sine

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In$\sinh(\frac{1}{9}i π)$Out$i\sin(\frac{π}{9})$

Known value · Wolfram MathematicalFunctionData, Sinh

$\sinh(i\pi/8)=i\sin(\pi/8)$ in nested radicals

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

Known value · Wolfram MathematicalFunctionData, Sinh

A seventh of $\pi$ is also left as sine

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In$\sinh(\frac{1}{7}i π)$Out$i\sin(\frac{π}{7})$

Known value · Wolfram MathematicalFunctionData, Sinh

$\sinh(i\pi/6)=\frac i2$

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In$\sinh(\frac{1}{6}i π)$Out$\frac{1}{2}i$

Known value · Wolfram MathematicalFunctionData, Sinh

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

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In$\sinh(\frac{2}{9}i π)$Out$i\sin(\frac{2π}{9})$

Known value · Wolfram MathematicalFunctionData, Sinh

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In$\sinh(1)^4-2(2\sinh(\frac{1}{2})^2+2\sinh(\frac{1}{2})^4)\sinh(1)^2$Out$\sinh(1)^4-4(\sinh(\frac{1}{2})^2+\sinh(\frac{1}{2})^4)\sinh(1)^2$

Known value · Wolfram MathematicalFunctionData, Sinh, FunctionalEquations

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In$N((\sinh(1/2)+\cosh(1/2))^2)$Out$2.718\,281\,828\,459\,045\,235\,37$

Known value · Wolfram MathematicalFunctionData, Sinh, de Moivre's formula

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In$N((\cosh(1/2)-\sinh(1/2))^2)$Out$0.367\,879\,441\,171\,442\,321\,601$

Known value · Wolfram MathematicalFunctionData, Sinh, de Moivre's formula

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In$N(\sinh(1)^2)$Out$1.381\,097\,845\,541\,815\,729\,78$

Known value · Wolfram MathematicalFunctionData, Sinh, FunctionalEquations

About

Examples	40
Checked against	Wolfram Language 39 agree1 differSageMath 37 agreeSymPy 37 agree1 differmpmath 29 agreeJulia 28 agree1 errorRust 14 agree1 differ
Crosswalk	WikipediaMathWorld ×2WikidataBritannicaDLMFFungrimFungrim entry ×16Wolfram 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: Cosh, Tanh, Exp