Pi as twice the smallest positive zero of cosine

Depends on

• Cosine has a smallest positive zero, lying strictly between zero and two

Used by

• A uniform limit of smooth functions need not be differentiable anywhere Corollary
• exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0 Corollary
• Pi is the first positive zero of sine Corollary
• The central binomial coefficient is asymptotic to 4ⁿ divided by the square root of pi n Corollary
• The normalized integral around a positively oriented circle centred at a is 1 Corollary
• x sin(1/x) extended by zero is continuous but not differentiable at zero Corollary
• The circular curve defeats the equality form of the vector-valued mean value theorem Counterexample
• The continuous path γ(x)=(x,x sin(1/x)) on [0,1], with γ(0)=(0,0), is not rectifiable Counterexample
• The map (z₁,z₂)↦(e^z₁,z₂) has invertible complex Jacobian everywhere and is not injective Counterexample
• The topologist's sine curve is connected but not path connected Counterexample
• The vortex field is closed but not exact on the punctured plane Counterexample
• Volterra's function is differentiable everywhere with bounded derivative, but its derivative is not Riemann integrable Counterexample
• Circular arcs, circumference as arc length, and diameter Definition
• The classical Weierstrass function Definition
• G(x)=x² sin(1/x) has a bounded derivative discontinuous at 0 that is nevertheless Riemann integrable, and Newton–Leibniz evaluates its integral Example
• r² sin(1/r) is differentiable at the origin with a discontinuous gradient Example
• Tangent identifies a bounded incomplete interval with the unbounded complete real line Example
• The arc length of one sine period is 4√2 E(1/√2) Example
• The Bartle-Sherbert bounds 2.828 < pi < 3.185 Example
• The family sin(nx)sin(ny) is uniformly bounded but not equicontinuous Example
• The four Dini derivatives of x sin(1/x) at 0 take two distinct values Example
• The sine harmonics are pointwise bounded but have no uniformly convergent subsequence Example
• The Weierstrass function with a=1/2 and b=15 Example
• x² sin(1/x²) has an unbounded, non-Riemann-integrable derivative Example
• xy sin(1/(x²+y²)) is differentiable at the origin with unbounded partial derivatives nearby Example
• A sector-area squeeze proves lim sin(x)/x=1 without first calibrating angle measure False statement
• False: any positive zero of sine characterizes pi False statement
• False: circumference divided by radius equals pi False statement
• False: every closed C1 field on a connected open set is exact False statement
• FALSE: every continuous function on a compact interval has a rectifiable graph False statement
• FALSE: every pointwise bounded sequence of continuous functions has a uniformly convergent subsequence False statement
• FALSE: spherical coordinates are globally injective False statement
• Low-frequency bound for the Weierstrass difference quotient Lemma
• The topologist's sine curve is connected Lemma
• The Weierstrass tail has one sign and dominates at the probe points Lemma
• Wallis integrals satisfy the two-step recurrence, closed forms, and the adjacent-integral squeeze Lemma
• [t]↦(cos 2π t,sin 2π t) is a homeomorphism from ℝ/ℤ to the unit circle Theorem
• Analytic sine and cosine agree with right-triangle ratios Theorem
• Inscribed regular-polygon perimeters increase to 2 pi, while circumscribed perimeters decrease to 2 pi Theorem
• Pi is equivalently the first sine zero, twice the first cosine zero, and half the least common period Theorem

…and 5 more results.

Dependency tree · two levels

4 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, Chapter 4 (standard reference, not scraped)
• C. Schmeiser, Introduction to Analysis (standard reference, not scraped)