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

• Exact success amplification by attenuating the good branch Algorithm
• Finite truncated tanh–sinh, exp–sinh, and sinh–sinh trapezoid rules Algorithm
• Known-amplitude amplification with integer rounding Algorithm
• 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
• Two-dimensional simple symmetric walk is recurrent Corollary
• x sin(1/x) extended by zero is continuous but not differentiable at zero Corollary
• A real-valued optimum is not a fractional Grover iteration Counterexample
• Fidelity and trace-distance inequalities do not imply QRE data processing Counterexample
• I₂(5) admits no crystallographic scaling and no reduced crystallographic root system with that base pairing Counterexample
• 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
• Arcsine-weighted inner product on the Chebyshev interval Definition
• Circular arcs, circumference as arc length, and diameter Definition
• DCT-II and DST-I with fixed even and odd extension conventions Definition
• Standard Cauchy probability law Definition
• The classical Weierstrass function Definition
• The Jacobi theta function θ(t)=∑_n∈ℤ e^-π n² t for t>0 Definition
• The real Coxeter form, its radical, reflections, and form-preserving maps Definition
• Type-I cosine transform for Chebyshev–Lobatto data Definition
• A compactly supported contrast with volume-operator norm greater than two Example
• A cycle and an overlong arm: explicit non-positive witnesses Example
• A moved-space intersection in A₃ that is not the meet Example
• A unit contrast on a small cube lies in the Born regime Example
• A vector with mixed signs is not a root, while every root has a sign Example
• An indefinite Coxeter form: infinite, but not of affine type Example
• B-tilde versus C-tilde: the n=2 coincidence and the duality behind the difference Example
• Chamber faces and their stabilizers in A₂ Example
• Dihedral diagrams I₂(m): Gram determinants, the infinite case, and the low-rank coincidences Example
• Exact Grover rotation for four addresses and one marked item Example
• Extremal length of a rectangle and of a round annulus by hand Example
• 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
• G2 from I2(6): the scaled realization and its twelve roots Example

…and 66 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)