Csch

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

Hyperbolic cosecant, the reciprocal of hyperbolic sine.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageCsch✓↗

Csch(complex | signed_infinity) -> numberHyperbolic cosecant, the reciprocal of hyperbolic sine.

SageMathcsch(complex | signed_infinity)SymPycsch(complex | signed_infinity)mpmathcsch(complex | signed_infinity)Juliaenumeratio_ball(_csch, 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

1 more kept as data, checked against the oracles

Basic #

live ↗

In$\operatorname{Series}(\operatorname{csch}(x), x, 0, 10)$Out$\frac{1}{x}-\frac{x}{6}+\frac{7x^3}{360}-\frac{31x^5}{15\,120}+\frac{127x^7}{604\,800}-\frac{73x^9}{3\,421\,440}+O\left(x^{11}\right)$

live ↗

In$\operatorname{Normal}(\operatorname{Series}(\operatorname{csch}(x), x, 2i π, 3))$Out$\frac{1}{360}×7(x-2i π)^3-\frac{1}{6}(x-2i π)+\frac{1}{x-2i π}$

Scope #

Fifty digits of $\operatorname{csch}(9/5)$

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In$N(\operatorname{csch}(\frac{9}{5}), 50)$Out$0.339\,884\,691\,415\,493\,707\,140\,758\,149\,466\,532\,117\,984\,776\,132\,436\,71$

Threads over a matrix; the zero entry is a pole

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

Decays to 0 at $+\infty$

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In$\operatorname{csch}(∞)$Out$0$

A pole at the origin

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In$\operatorname{csch}(0)$Out$\tilde∞$

Derivative is $-\operatorname{csch}x\coth x$

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

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

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In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\operatorname{csch}(\operatorname{\_1}), \bigl\lbrack\bigl\lbrack(1/3)i π, (-(1/3)i)π\bigr\rbrack, \bigl\lbrack0, (-(1/6)i)π\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack-\frac{2×3^{1/2}}{3}i, \frac{4×3^{1/2}}{9}i+\frac{4}{3}i\bigr\rbrack, \bigl\lbrack0, 2i\bigr\rbrack\bigr\rbrack$

live ↗

In$\int\!\operatorname{csch}(x)\, \mathrm{d}x$Out$-\ln(\coth(\frac{\vert x\vert}{2}))$

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

Properties #

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In$\operatorname{csch}(\frac{1}{2})^8-2(2+2\operatorname{csch}(\frac{1}{2})^2)\operatorname{csch}(1)^2\operatorname{csch}(\frac{1}{2})^4$Out$\operatorname{csch}(\frac{1}{2})^8-4(1+\operatorname{csch}(\frac{1}{2})^2)\operatorname{csch}(1)^2\operatorname{csch}(\frac{1}{2})^4$

Known value · Wolfram MathematicalFunctionData, Csch, FunctionalEquations

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In$N(4(1+\operatorname{csch}(1/2)^2)\operatorname{csch}(1)^2)$Out$13.562\,237\,873\,143\,914\,211\,582\,403\,390\,594\,531\,801\,748\,96$

Known value · Wolfram MathematicalFunctionData, Csch, FunctionalEquations

live ↗

In$N(\operatorname{csch}(1/2+(1/2)i π)^2)$Out$-0.786\,447\,732\,965\,927\,572\,156$

Known value · Wolfram MathematicalFunctionData, Csch, FunctionalEquations

live ↗

In$N(4(1+\operatorname{csch}(2)^2)\operatorname{csch}(4)^2)$Out$0.005\,779\,318\,611\,928\,637\,296\,920\,375\,930\,644\,908\,284\,074\,784$

Known value · Wolfram MathematicalFunctionData, Csch, FunctionalEquations

About

Examples	15
Checked against	Wolfram Language 13 agree1 unevaluatedSageMath 11 agree1 unevaluatedSymPy 12 agree1 differ1 unevaluatedmpmath 6 agree2 errorJulia 5 agree3 errorRust 3 agree1 differ1 error
Crosswalk	Wolfram 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 sinh.