Inverse hyperbolic cosine: Representations through equivalent functions (formula 01.26.27.2486)

ArcCosh

$Mathematica Notation$

Elementary Functions

ArcCosh[z]

Representations through equivalent functions

With inverse function

Involving cosh^-1(cosh(z))

http://functions.wolfram.com/01.26.27.2486.01

Input Form

ArcCosh[Cosh[z]] == Piecewise[ {{(-(-1)^Floor[(Pi - I z)/Pi]) (z - Pi I Floor[(Pi - I z)/Pi] + (Pi I)/2) + (Pi I)/2, Re[z] == 0}, {Sqrt[z^2] (1 - ((2 Pi I)/z) Floor[(Im[z] - Pi)/(2 Pi)]), Element[(Im[z] + Pi)/(2 Pi), Integers] && Re[z] > 0}, {Sqrt[z^2] (1 - ((2 Pi I)/z) Floor[(Im[z] + Pi)/(2 Pi)]), True}}]

Standard Form

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MathML Form

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</mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mi> z </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - 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Rule Form

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Date Added to functions.wolfram.com (modification date)

2007-05-02