Inverse hyperbolic functions

Builds on Hyperbolic sine Hyperbolic cosine Hyperbolic tangent Natural logarithm Square root function

Definition

The inverse hyperbolic sine $\operatorname{arsinh} \colon \mathbb{R} \to \mathbb{R}$ is the inverse of $\sinh$ . The inverse hyperbolic cosine $\operatorname{arcosh} \colon [1, \infty) \to [0, \infty)$ is the inverse of the restriction of $\cosh$ to $[0, \infty)$ . The inverse hyperbolic tangent $\operatorname{artanh} \colon (-1, 1) \to \mathbb{R}$ is the inverse of $\tanh$ .

Remarks

Solving a quadratic equation in $e^y$ gives the formulas $\operatorname{arsinh} x = \ln(x + \sqrt{x^2 + 1})$ and $\operatorname{arcosh} x = \ln(x + \sqrt{x^2 - 1})$ . Similarly, $\operatorname{artanh} x = \frac{1}{2} \ln \frac{1 + x}{1 - x}$ for $|x| < 1$ . Their derivatives are $\frac{1}{\sqrt{x^2 + 1}}$ , $\frac{1}{\sqrt{x^2 - 1}}$ for $x > 1$ , and $\frac{1}{1 - x^2}$ . Mathlib defines the first one as Real.arsinh.

Reading path

To understand this page from the ground up, read these 55 pages in order. The foundations come first.

Limit point of a set definition
Local maximum definition
Sequence definition
Strictly monotone function definition
Limit of a function definition
Local minimum definition
Real sequence definition
Subsequence definition
Continuity at a point definition
Derivative definition
Bounded sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Series definition
Continuous function definition
Fermat's theorem on stationary points statement
Sum of a finite geometric series statement
Monotone convergence theorem for sequences statement
Convergent sequence definition
Every sequence has a monotone subsequence statement
Squeeze theorem statement
Cauchy product definition
Partial sum definition
Power series definition
Intermediate value theorem statement
Bolzano–Weierstrass theorem statement
The sequence r^n tends to zero for |r| < 1 example
Radius of convergence definition
Convergent series definition
Boundedness theorem statement
Square root function definition
A series of nonnegative terms converges iff its partial sums are bounded statement
Absolute convergence definition
Sum of a series definition
Extreme value theorem statement
Comparison test statement
Geometric series example
Cauchy product of absolutely convergent series statement
Rolle's theorem statement
Ratio test statement
Mean value theorem statement
Ratio formula for the radius of convergence statement
A function with positive derivative is strictly increasing statement
The exponential series has infinite radius of convergence statement
Exponential function definition
Hyperbolic cosine definition
Hyperbolic sine definition
Exponential of a sum statement
Hyperbolic tangent definition
The exponential function is its own derivative statement
The exponential function is positive statement
The exponential function is strictly increasing statement
Natural logarithm definition
Inverse hyperbolic functions definition

Tag	`04D7CHT5V1`
URI	`m2://04D7CHT5V1`
Kind	definition
Writing	W1 stated
MSC	33B10
Defines	Inverse hyperbolic functions
Uses	Hyperbolic sine, Hyperbolic cosine, Hyperbolic tangent, Natural logarithm, Square root function

Inverse hyperbolic functions

Definition

Remarks

Reading path

Further reading

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