Coth

Generated from the engine's own definition: compute-engine's symbol, which we neither extend nor document by hand.

Hyperbolic cotangent, the reciprocal of hyperbolic tangent.

✓ checked against that system · ! checked, with disagreements (hover a chip)DLMF4.28.E7↗FungrimCoth✓↗Fungrim entry03e2a6✓↗22a9cd✓↗3ac0ce✓↗Wolfram LanguageCoth✓↗

Coth(complex | signed_infinity) -> numberHyperbolic cotangent, the reciprocal of hyperbolic tangent.

SageMathcoth(complex | signed_infinity)SymPycoth(complex | signed_infinity)mpmathcoth(complex | signed_infinity)Juliaenumeratio_ball(coth, complex | signed_infinity)

Typed signatures

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

Domain: Compute engine

Examples

Basic #

live ↗

In$\operatorname{Series}(\coth(x), x, 0, 10)$Out$\frac{1}{x}+\frac{x}{3}-\frac{x^3}{45}+\frac{2x^5}{945}-\frac{x^7}{4\,725}+\frac{2x^9}{93\,555}+O\left(x^{11}\right)$

live ↗

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

Scope #

Fifty digits of $\coth(6/5)$

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In$N(\coth(\frac{6}{5}), 50)$Out$1.199\,537\,544\,192\,350\,766\,749\,919\,931\,568\,098\,165\,628\,059\,955\,188\,6$

Threads over a matrix; the zero entry is a pole, complexInfinity

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

Tends to 1 at $+\infty$

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In$\coth(∞)$Out$1$

A pole at the origin

live ↗

In$\coth(0)$Out$\tilde∞$

Derivative is $-\operatorname{csch}^2 x$

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In$\frac{\mathrm{d}\coth(x)}{\mathrm{d}x}$Out$-\operatorname{csch}(x)^2$

As a matrix function the off-diagonal entry is a divided difference

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

live ↗

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

Properties #

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In$N(\coth(1)\coth(\frac{1}{2}))$Out$2.841\,347\,188\,415\,584\,637\,899\,128\,662\,683\,723\,870\,522\,8$

Known value · Wolfram MathematicalFunctionData, Coth, FunctionalEquations

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In$N(\coth(\frac{1}{2}+\frac{1}{2}i π))$Out$0.462\,117\,157\,260\,009\,74$

Known value · Wolfram MathematicalFunctionData, Coth, FunctionalEquations

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In$N(\coth(2)\coth(4))$Out$1.038\,010\,914\,919\,035\,549\,626\,687\,667\,882\,357\,812\,570\,8$

Known value · Wolfram MathematicalFunctionData, Coth, FunctionalEquations

live ↗

In$N(\coth(2+\frac{1}{2}i π))$Out$0.964\,027\,580\,075\,816\,9$

Known value · Wolfram MathematicalFunctionData, Coth, FunctionalEquations

About

Examples	13
Checked against	Wolfram Language 13 agreeSageMath 11 agreeSymPy 12 agreempmath 6 agree2 errorJulia 5 agree3 errorRust 3 agree1 error
Crosswalk	DLMFFungrimFungrim entry ×3Wolfram LanguageSageMathSymPympmathJulia

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

The reciprocal of tanh.