Quarter-turn values and shifts by pi/2 and pi

Depends on

• Pi as twice the smallest positive zero of cosine
• Cosine has a smallest positive zero, lying strictly between zero and two
• The addition formulas for sine and cosine
• Parity and the Pythagorean identity for sine and cosine
• Sine is positive and cosine is strictly decreasing on (0,2), with cos 2 at most -1/3

Used by

• 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 normal component of the curl is the limiting circulation per unit area of shrinking discs Corollary
• x sin(1/x) extended by zero is continuous but not differentiable at zero Corollary
• A map with two preimages but degree zero Counterexample
• A surjective map need not be a fibration Counterexample
• Agreement accumulating only at the boundary does not force a holomorphic identity Counterexample
• An invertible derivative at one point does not give a local inverse without C¹ regularity Counterexample
• Antipodal points on a round sphere have many minimizing geodesics Counterexample
• f(x+iy)=eˣ(cos 2y+i sin 2y) is continuous, satisfies f(z+w)=f(z)f(w) and f(1)=e, but is not the standard complex exponential Counterexample
• Principal arcsine has no finite derivative at -1 or 1 Counterexample
• sin(1/x) has no limit as x tends to zero Counterexample
• sin(3π/2)=-1 shows that the positive square root is not an unconditional half-angle formula Counterexample
• The continuous path γ(x)=(x,x sin(1/x)) on [0,1], with γ(0)=(0,0), is not rectifiable Counterexample
• The topologist's sine curve is connected but not path connected Counterexample
• u=v=π/4 shows that the tangent addition formula cannot omit its domain restrictions Counterexample
• Volterra's function is differentiable everywhere with bounded derivative, but its derivative is not Riemann integrable Counterexample
• Principal inverse sine and inverse cosine Definition
• A closed cylinder as a finitely patched oriented surface Example
• A continuous argument computed along a spiralling contour Example
• A positive non-log-convex solution of the Gamma functional equation Example
• A right circular cylinder is an elementary solid region, presented by two caps and four side quarters Example
• Addition and half-angle identities compute the sine, cosine, and tangent of π/12 Example
• F(x)=x² sin(1/x²) has an unbounded derivative whose Henstock–Kurzweil integral is sin 1 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
• Hopf circle fibration Example
• Mobius band as an interval bundle with monodromy Example
• Normal coordinates on the round sphere Example
• r² sin(1/r) is differentiable at the origin with a discontinuous gradient Example
• Stokes' theorem on a flat disc and on a hemisphere with the same induced boundary circle Example
• Surface area and flux on a sphere, with scalar integrals on a hemisphere Example
• The closed ball is an elementary solid region, presented by the eight spherical octants Example
• The extension of x² sin(1/x) by zero is differentiable but its derivative is discontinuous at zero 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 Fourier series of a sawtooth and the Basel sum Example
• The Fourier series of a square wave and the odd reciprocal-square sum Example
• The Mobius band presented by two regular patches, with normal comparison on the interiors of the overlap components Example
• The outward flux of the inverse-square field through a sphere centred at the origin is 4π Example
• The scalar line integral of x over the right unit semicircle equals two Example

…and 19 more results.

Dependency tree · two levels

16 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)