TheoremStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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Logarithm formulas for inverse sinh, inverse cosh, and inverse tanh on their natural domains

Statement

The strictly increasing bijections $\sinh:\mathbb R\to\mathbb R$ , $\cosh:[0,\infty)\to[1,\infty)$ , and $\tanh:\mathbb R\to(-1,1)$ have inverse functions satisfying $\operatorname{arsinh}u=\log(u+\sqrt{u^2+1}),$ $\operatorname{arcosh}u=\log(u+\sqrt{u^2-1})\quad(u\ge1),$ $\operatorname{artanh}u=\frac12\log\frac{1+u}{1-u}\quad(|u|<1).$

Facts & Assumptions

Given: A real $u$ in the stated domain.

[L1]

The hyperbolic identities hold, and $\sinh:\mathbb R\to\mathbb R$ , $\cosh:[0,\infty)\to[1,\infty)$ , and $\tanh:\mathbb R\to(-1,1)$ are strictly increasing bijections (Addition formulas, identities, parity, and derivatives of the hyperbolic functions, The six hyperbolic functions and their natural domains).

[L2]

$\log$ is the inverse of $\exp$ and satisfies its product and reciprocal laws (Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm).

[L3]

Positive-base real powers are continuous, and every nonnegative real has its nonnegative square root (Continuity and derivatives of positive-base real powers, Square roots exist: a unique $\sqrt{a} \ge 0$ with $(\sqrt{a})^2 = a$ ; the positives are $\{x^2 : x \neq 0\}$ ).

Proof

technique · direct

1.1

Solving $\sinh z=u$ after putting $v=\exp z>0$ gives $v^2-2uv-1=0$ , hence $v=u+\sqrt{u^2+1}>0$ and $z=\log(u+\sqrt{u^2+1})$ .

L1L2L3

1.2

Solving $\cosh z=u$ with $z\ge0$ gives $v^2-2uv+1=0$ and the allowed root $v=u+\sqrt{u^2-1}$ , hence the displayed arcosh formula.

L1L2L3

1.3

Solving $\tanh z=u$ gives $v^2=(1+u)/(1-u)>0$ , so $z=\frac12\log((1+u)/(1-u))$ .

L1L2

2.1

The strict monotonicity and stated ranges in [L1] make each algebraic solution the unique inverse value on its declared domain.

step 1.1step 1.2step 1.3L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 78 results over 18 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)