Complex sine, cosine, hyperbolic sine, and hyperbolic cosine from the complex exponential

Depends on

• The complex exponential by its power series
• $\mathbb C=\mathbb R[x]/(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)/(a^2+b^2)$

Used by

• Exact success amplification by attenuating the good branch Algorithm
• Known-amplitude amplification with integer rounding Algorithm
• The addition formulas for complex trigonometric and hyperbolic functions Corollary
• 1/sin(1/z) has a nonisolated singularity at 0 Counterexample
• sin(1/z) has an essential singularity at 0 Counterexample
• 1/z is not uniformly approximable by polynomials on the unit circle Example
• Characteristics of a monomial, an exponential and a tangent Example
• Deficiencies of the exponential and sine Example
• The Gaussian integral on the line with a certified sinh-sinh truncation tail Example
• The rank-one cotangent and its conic Example
• Amplitude-amplification invariant plane Lemma
• Fresnel series geometric tail bounds Lemma
• The Lipschitz formula for the reciprocal-power sums Lemma
• Analytic ellipse data give exponential Chebyshev–Lobatto interpolation error Theorem
• Complex sine and cosine are unbounded on the complex plane Theorem
• Eisenstein series are modular forms; their Fourier coefficients Theorem
• Exact success for known positive amplitude with explicit dilution Theorem
• Exact success probability after amplitude amplification Theorem
• Expected cost of unknown-amplitude search Theorem
• Known-amplitude success bound Theorem
• Phase-matching condition for amplitude-amplification rotations Theorem
• Success bound under an explicit amplitude-amplification phase mismatch Theorem
• The exponential definitions of complex sine, cosine, hyperbolic sine, and hyperbolic cosine equal their entire power series Theorem
• The exponential formulas, real restrictions, and trigonometric-hyperbolic dictionary over ℂ Theorem
• The Mittag-Leffler expansion of pi cotangent Theorem
• The Riemann zeta function has the standard Bernoulli special values at the positive even and nonpositive integers Theorem
• The sine map biholomorphically sends an upper half-strip onto the upper half-plane Theorem
• The Weierstrass product for sine Theorem
• The zeros of complex sine are the integer multiples of pi, and the zeros of complex cosine are the odd half-integer multiples of pi Theorem
• Unknown-amplitude search has no false positives and halts almost surely Theorem

Dependency tree · two levels

13 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

• J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)