Inverse Hyperbolic Functions

The inverse hyperbolic functions arise when solving equations like sinh x = k. Unlike inverse trig functions, they can be expressed as natural logarithms, making them very useful in integration.

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1 Definitions and Logarithmic Forms

arsinh x = ln(x + sqrt(x^2 + 1)) -- Domain: all reals, Range: all reals

arcosh x = ln(x + sqrt(x^2 - 1)) -- Domain: [1, infinity), Range: [0, infinity)

artanh x = (1/2) ln((1 + x)/(1 - x)) -- Domain: (-1, 1), Range: all reals

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2 Deriving arsinh x

Let y = arsinh x, so sinh y = x.

(e^y - e^(-y))/2 = x

Multiply by e^y: e^(2y) - 2x e^y - 1 = 0

Quadratic in e^y: e^y = x + sqrt(x^2 + 1) (taking the positive root since e^y > 0)

arsinh x = ln(x + sqrt(x^2 + 1))

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3 Deriving arcosh x

Let y = arcosh x, so cosh y = x (with y >= 0).

e^(2y) - 2x e^y + 1 = 0

e^y = x + sqrt(x^2 - 1) (taking the larger root for y >= 0)

arcosh x = ln(x + sqrt(x^2 - 1))

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