Inverse Hyperbolic functions

Formulae:

1. sinh^-1 x = log_e (x + √(x² + 1))
2. cosh^-1 x = log_e (x + √(x² – 1)) for x ≥ 1
3. tanh-1 x = ½ log [(1 +x)/ (1 – x)] for x ϵ (-1, 1)
4. coth^-1 x = ½ log [(1 +x)/ (1 – x)] for |x| > 1
5. sech^-1 x = log_e (1 + √ (1 – x²)/ x) for x ϵ (0, 1]
6. cosech^-1 x = log_e (1 + √ (1 + x²)/ x) for x > 0
   = log_e (1 – √ (1 + x²)/ x) for x < 0
7. sinh^-1 x = cosh^-1 √(x² + 1) = cosech^-1 (1/ x) = tanh-1 (x / √ (1 + x²))
8. cosh^-1 x = sinh^-1 √(x² – 1) = sech^-1 (1/ x) = tanh-1 (√ (1 + x²) / x)

Function	domain	Range
sinh-1 x	R	R
cosh-1 x	[1, ∞)	[0, ∞)
tanh-1	(-1, 1)	R
coth-1 x	R – [-1, 1]	R – {0}
sech-1 x	(0,1]	[0, ∞)
cosech-1 x	R – {0}	R – {0}

Graphs of inverse hyperbolic functions:

i) y = sinh^-1 xii) y = cosh^-1 x
iii) y = tanh^-1 x
iv) y = sech^-1 x
v) y = cosech^-1 xvi) y = coth^-1 x