Alphabeta MathHow statement and proof provenance work
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The principal inverse tangent
Definition
By Tangent is a continuous strictly increasing bijection from onto , tangent restricts to a continuous strictly increasing bijection
Its inverse is the principal inverse tangent
Thus for every real , while precisely for in the displayed principal interval. The inverse is continuous and strictly increasing by Continuous inverse theorem: a continuous injective on an interval is a bijection onto the order-convex set , and the inverse is continuous and strictly monotone in the same sense as .
Depends on
- Tangent is a continuous strictly increasing bijection from $(-\pi/2,\pi/2)$ onto $\mathbb R$
- Continuous inverse theorem: a continuous injective $f$ on an interval $I$ is a bijection onto the order-convex set $f[I]$, and the inverse $g : f[I] \to I$ is continuous and strictly monotone in the same sense as $f$
Used by
- Iid strong law fails at infinite absolute mean Counterexample
- Infinite variance can defeat square-root-n CLT scaling Counterexample
- Cauchy law and its characteristic function Example
- Machin's formula π/4=4 arctan(1/5)-arctan(1/239) Example
- Tangent identifies a bounded incomplete interval with the unbounded complete real line Example
- Uniform sine integral bound and dirichlet value Lemma
- Brownian positive occupation time has the arcsine law Theorem
- Principal arctangent: derivative, integral, power series, and the Gregory–Leibniz series Theorem
Dependency tree · two levels
21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- NIST Digital Library of Mathematical Functions, §4.23 Inverse Trigonometric Functions (standard reference, not scraped)