Inverse hyperbolic sine: Series representations (formula 01.25.06.0036)

ArcSinh

$Mathematica Notation$

Elementary Functions

ArcSinh[z]

Series representations

Generalized power series

Expansions on branch cuts

For the function itself

In the upper half-plane

http://functions.wolfram.com/01.25.06.0036.01

Input Form

ArcSinh[z] == -((Pi I)/2) + (2 Pi I^Floor[Arg[I (z - x)]/(2 Pi)] Floor[Arg[I (z - x)]/(2 Pi)] + (1/2) Exp[Pi I Floor[Arg[I (z - x)]/(2 Pi)]] (Pi I + Sqrt[Pi] Sum[((2^k x^(1 - k))/k!) HypergeometricPFQRegularized[ {1/2, 1/2, 1}, {1 - k/2, (3 - k)/2}, -x^2] (z - x)^k, {k, 0, Infinity}])) /; Element[I x, Reals] && I x < -1

Standard Form

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MathML Form

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Rule Form

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

2007-05-02