2.1.3—Hyperbolic functions

The inverse functions are defined on restricted domains: asinh x=ln(x+√(x²+1)), acosh x=ln(x+√(x²−1)) for x≥1, and atanh x=½ln((1+x)/(1−x)) for |x|<1.

The domain restrictions make the square roots real and select a one-to-one branch. Differentiate or substitute back to verify a result, and preserve absolute-value conditions when integrating logarithms.

asinh 0=0. For x=1, acosh1=0 because ln(1+0)=0; atanh x cannot accept x=1 because its denominator in the logarithmic form vanishes.

Inverse hyperbolic notation means inverse function, not reciprocal, and the domains are not optional.