Cosh

Hyperbolic cosine: $\cosh(x) = \frac{e^x + e^{-x}}{2}$.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaHyperbolic functions↗MathWorldHyperbolicCosine↗HyperbolicFunctions↗WikidataQ204034↗Britannicatopic/hyperbolic-functions↗DLMF4.28.E2↗FungrimCosh✓↗Fungrim entryWolfram LanguageCosh!↗

Cosh(x)the hyperbolic cosine of x.

SageMathcosh(x)SymPycosh(x)mpmathcosh(x)Juliaenumeratio_ball(cosh, x)Rustcosh(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: $\cosh(x) = \frac{e^x + e^{-x}}{2}$.
• Even function: $\cosh(-x) = \cosh(x)$.
• Exact at $x = 0$ as Cos is: $\cosh(0) = 1$; a floating-point argument, or N(...), gives a numeric result.

Examples

Basic #

N(...) gives the numeric value

live ↗

In$N(\cosh(0))$Out$1.0$

live ↗

In$N(\cosh(1))$Out$1.543\,080\,634\,815\,243\,778\,48$

Exact at 0, as Cos is

live ↗

In$\cosh(0)$Out$1$

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

live ↗

In$\cosh(1.2)$Out$1.810\,655\,567\,324\,374\,793\,09$

live ↗

In$\operatorname{Series}(\cosh(x), x, 0, 10)$Out$1+\frac{x^2}{2}+\frac{x^4}{24}+\frac{x^6}{720}+\frac{x^8}{40\,320}+\frac{x^{10}}{3\,628\,800}+O\left(x^{12}\right)$

Scope #

Arbitrary precision via N(...)

live ↗

In$N(\cosh(\frac{6}{5}))$Out$1.810\,655\,567\,324\,374\,793\,09$

Evaluates at a complex argument

live ↗

In$\cosh(2.5+i)$Out$3.313\,290\,146\,113\,224\,5+5.091\,071\,522\,949\,698i$

$\cosh(i\pi) = \cos\pi = -1$

live ↗

In$\cosh(i π)$Out$-1$

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

live ↗

In$\cosh(\bigl\lbrack\bigl\lbrack\frac{i×1π}{3}, -(i×1π)/3\bigr\rbrack, \bigl\lbrack0, -(i×1π)/6\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\frac{1}{2}, \frac{1}{2}\bigr\rbrack, \bigl\lbrack1, \frac{\sqrt{3}}{2}\bigr\rbrack\bigr\rbrack$

Interval arithmetic: $\cosh[0, \ln 3] = [1, \tfrac53]$, the minimum at 0 included

live ↗

In$\cosh(\mathrm{Interval}(0, \ln(3)))$Out$\mathrm{Interval}(1, \frac{5}{3})$

Uncertainty propagation, to first order: $\cosh(2/\sqrt3 \pm 0.1) \approx 1.744 \pm 0.143$

live ↗

In$\cosh(\mathrm{Around}(\frac{2}{\sqrt{3}}, 0.1))$Out$\mathrm{Around}(1.744\,112\,480\,153\,778, 0.142\,896\,058\,148\,157\,56)$

live ↗

In$\cosh(∞)$Out$∞$

Evenness drops the sign of a symbolic argument: $\cosh(-x) = \cosh x$

live ↗

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

$\cosh(\operatorname{arcosh} x) = x$ for every $x$

live ↗

In$\cosh(\mathrm{arcosh}(x))$Out$x$

$\cosh(\operatorname{arcosh}(2)) = 2$

live ↗

In$\cosh(\mathrm{arcosh}(2))$Out$2$

$\cosh(\operatorname{arcosh}(3i)) = 3i$

live ↗

In$\cosh(\mathrm{arcosh}(3i))$Out$3i$

Derivative: $\sinh x$

live ↗

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

Antiderivative

live ↗

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

To 30 significant digits, every one correctly rounded

live ↗

In$N(\cosh(1), 30)$Out$1.543\,080\,634\,815\,243\,778\,477\,905\,620\,76$

Fifty digits of a rational argument

live ↗

In$N(\cosh(\frac{6}{5}), 50)$Out$1.810\,655\,567\,324\,374\,793\,087\,872\,518\,342\,433\,390\,022\,428\,027\,85$

On the imaginary axis it becomes ordinary cosine, $\cos(\pi/5) = \frac{1+\sqrt5}{4}$

live ↗

In$\cosh(\frac{1}{5}i π)$Out$\frac{1}{4}+\frac{\sqrt{5}}{4}$

Rewritten as a cosine; no radical form is found

live ↗

In$\cosh(\frac{1}{24}i π)$Out$\cos(\frac{π}{24})$

A matrix function with an imaginary-argument entry

live ↗

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

Properties #

Even function: $\cosh(-x) = \cosh(x)$

live ↗

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

Hyperbolic Pythagorean identity: $\cosh^2 x - \sinh^2 x = 1$

live ↗

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

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

live ↗

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

$\cosh^2 x - \sinh^2 x$ simplifies to 1, as the circular identity does

live ↗

In$\mathrm{Simplify}(\cosh(x)^2-\sinh(x)^2)$Out$1$

$\cos(\pi/12)$ in radicals

live ↗

In$\cosh(\frac{1}{12}i π)$Out$\frac{\sqrt{2}}{4}+\frac{\sqrt{6}}{4}$

Known value · Wolfram MathematicalFunctionData, Cosh

$\cos(\pi/10)$ in nested radicals

live ↗

In$\cosh(\frac{1}{10}i π)$Out$\frac{1}{4}\sqrt{2}\sqrt{5+\sqrt{5}}$

Known value · Wolfram MathematicalFunctionData, Cosh

Becomes plain cosine; no closed radical form

live ↗

In$\cosh(\frac{1}{9}i π)$Out$\cos(\frac{π}{9})$

Known value · Wolfram MathematicalFunctionData, Cosh

$\cos(\pi/8) = \frac12\sqrt{2+\sqrt2}$

live ↗

In$\cosh(\frac{1}{8}i π)$Out$\frac{\sqrt{2+\sqrt{2}}}{2}$

Known value · Wolfram MathematicalFunctionData, Cosh

Becomes plain cosine, as $\cos(\pi/7)$ has no radical form

live ↗

In$\cosh(\frac{1}{7}i π)$Out$\cos(\frac{π}{7})$

Known value · Wolfram MathematicalFunctionData, Cosh

$\cos(\pi/6) = \frac{\sqrt3}{2}$

live ↗

In$\cosh(\frac{1}{6}i π)$Out$\frac{\sqrt{3}}{2}$

Known value · Wolfram MathematicalFunctionData, Cosh

Becomes plain cosine for $2\pi/9$

live ↗

In$\cosh(\frac{2}{9}i π)$Out$\cos(\frac{2π}{9})$

Known value · Wolfram MathematicalFunctionData, Cosh

live ↗

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

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

live ↗

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

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

live ↗

In$N(-2\cosh(1)\cosh(1/2)^2+2\cosh(1/2)^2+\cosh(1)^2)$Out$0.999\,999\,999\,999\,999\,999\,991\,568\,056\,313\,347\,813\,32$

Known value · Wolfram MathematicalFunctionData, Cosh, FunctionalEquations

live ↗

In$N(\cosh(1/2+(1/2)i π)^2)$Out$-0.271\,540\,317\,407\,621\,929\,236$

Known value · Wolfram MathematicalFunctionData, Cosh, FunctionalEquations

About

Examples	38
Checked against	Wolfram Language 37 agree1 differSageMath 35 agreeSymPy 35 agree1 differmpmath 28 agreeJulia 27 agree1 errorRust 13 agree
Crosswalk	WikipediaMathWorld ×2WikidataBritannicaDLMFFungrimFungrim entry ×10Wolfram 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, Tanh, Exp