Sech

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

Hyperbolic secant, the reciprocal of hyperbolic cosine.

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

Sech(complex | signed_infinity) -> numberHyperbolic secant, the reciprocal of hyperbolic cosine.

SageMathsech(complex | signed_infinity)SymPysech(complex | signed_infinity)mpmathsech(complex | signed_infinity)Juliaenumeratio_ball(_sech, 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 #

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In$\operatorname{Series}(\mathrm{sech}(x), x, 0, 10)$Out$1-\frac{x^2}{2}+\frac{5x^4}{24}-\frac{61x^6}{720}+\frac{277x^8}{8\,064}-\frac{50\,521x^{10}}{3\,628\,800}+O\left(x^{12}\right)$

Scope #

A 50-digit value of $\operatorname{sech}(9/5)$

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In$N(\mathrm{sech}(\frac{9}{5}), 50)$Out$0.321\,804\,869\,506\,587\,868\,350\,220\,885\,541\,608\,016\,744\,096\,112\,598\,73$

Threads over a matrix, simplifying only the exact entries like $\operatorname{sech}(0)=1$

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In$\mathrm{sech}(\bigl\lbrack\bigl\lbrack(1/3)i π, (-(1/3)i)π\bigr\rbrack, \bigl\lbrack0, (-(1/6)i)π\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack2, 2\bigr\rbrack, \bigl\lbrack1, \frac{2\sqrt{3}}{3}\bigr\rbrack\bigr\rbrack$

Sech decays to 0 at $+\infty$

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In$\mathrm{sech}(∞)$Out$0$

Equals 1 at the origin

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In$\mathrm{sech}(0)$Out$1$

The derivative is $-\tanh x\,\operatorname{sech} x$

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

A matrix function that mixes the entries via the eigenstructure of an upper triangular matrix

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In$\mathrm{MatrixFunction}(\operatorname{\_1}↦\mathrm{sech}(\operatorname{\_1}), \bigl\lbrack\bigl\lbrack(1/3)i π, (-(1/3)i)π\bigr\rbrack, \bigl\lbrack0, (-(1/6)i)π\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack2, \frac{4×3^{1/2}}{9}-\frac{4}{3}\bigr\rbrack, \bigl\lbrack0, \frac{2\sqrt{3}}{3}\bigr\rbrack\bigr\rbrack$

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

Properties #

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In$N((2-\mathrm{sech}(1/2)^2)\mathrm{sech}(1))$Out$0.786\,447\,732\,965\,927\,410\,151\,558\,029\,603\,407\,693\,721\,77$

Known value · Wolfram MathematicalFunctionData, Sech, FunctionalEquations

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In$N(2\mathrm{sech}(1)\mathrm{sech}(1/2)^2+(\mathrm{sech}(\frac{1}{2})^4-2\mathrm{sech}(1/2)^2)\mathrm{sech}(1)^2-\mathrm{sech}(\frac{1}{2})^4)$Out$2.239\,910\,263\,484\,962\,572\,369\cdot10^{-21}$

Known value · Wolfram MathematicalFunctionData, Sech, FunctionalEquations

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In$N(\mathrm{sech}(1/2+(1/2)i π)^2)$Out$-3.682\,694\,376\,831\,169\,200\,97$

Known value · Wolfram MathematicalFunctionData, Sech, FunctionalEquations

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In$N((2-\mathrm{sech}(2)^2)\mathrm{sech}(4))$Out$0.070\,650\,824\,853\,164\,465\,686\,164\,927\,845\,968\,686\,899\,164\,62$

Known value · Wolfram MathematicalFunctionData, Sech, FunctionalEquations

About

Examples	12
Checked against	Wolfram Language 12 agreeSageMath 10 agreeSymPy 11 agreempmath 8 agreeJulia 7 agree1 errorRust 4 agree1 differ
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 cosh.