$\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$

Depends on

• $\exp(z+w)=\exp z\,\exp w$, and the complex exponential extends the real exponential
• Euler's formula: $\exp(i\theta)=\cos\theta+i\sin\theta$ for every real $\theta$
• Conjugation is an involutive real-field automorphism, $z\overline z=|z|^2$, and modulus is definite, multiplicative, and subadditive
• Pythagorean and parity identities for all six trigonometric functions on their natural domains
• The exponential is positive and satisfies $\exp(-x)=1/\exp(x)$
• Quarter-turn values and shifts by pi/2 and pi
• Pi as twice the smallest positive zero of cosine

Used by

• Certified dyadic enclosures for unit-circle phases Algorithm
• Certified Gaussian gridding for type 1 and type 2 Algorithm
• QC24 State Preparation Algorithm
• Quantum phase-estimation spectral sampler Algorithm
• A holomorphic logarithm is a primitive of the logarithmic derivative Corollary
• A nonvanishing holomorphic function on such a domain has holomorphic roots of every positive order Corollary
• A plane harmonic function bounded above or below is constant Corollary
• Cartesian and polar forms of the Cauchy–Riemann equations agree away from the origin Corollary
• Contraction Hille-Yosida theorem Corollary
• The principal logarithm is the normalised holomorphic branch on the slit plane Corollary
• The winding number is the increment of a continuous argument divided by 2π Corollary
• Trigonometric polynomials are uniformly dense in continuous functions on the torus Corollary
• A holomorphic function on an annulus can have a nonzero closed-contour integral Counterexample
• A radial Poisson limit does not control a tangential path Counterexample
• Distinct pure frequencies differing by a reciprocal-lattice shift have identical samples 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
• Flat hyperplanes do not have spherical stationary-phase decay Counterexample
• Generator convention matters Counterexample
• Infinite variance can defeat square-root-n CLT scaling Counterexample
• Irrational rotation is ergodic but not weakly mixing Counterexample
• Sharp frequency cutoffs have kernels that are not in L1 Counterexample
• The exponential map is a holomorphic surjection C to C^× that is not an automorphism Counterexample
• The family eⁿᶻ converges chordally to infinity on the right half-plane without being holomorphically normal there Counterexample
• The map (z₁,z₂)↦(e^z₁,z₂) has invertible complex Jacobian everywhere and is not injective Counterexample
• The radix-two split fails for odd N Counterexample
• Characteristic function of a real random variable Definition
• Controlled Phase Rotation Definition
• Controlled-Z and SWAP Definition
• Exact one-qubit gates and rotations Definition
• Finite Hermitian matrix dynamics Definition
• Finite-interval oscillatory integral and phase convention Definition
• Fourier coefficients and trigonometric polynomials on the torus Definition
• Fourier transform on complex L1 classes Definition
• Hamiltonian-simulation polynomial target Definition
• Inner, singular inner and outer functions Definition
• Roots of a compact connected Lie group Definition
• Spectral measures and the known-mean periodogram Definition
• The complex-time heat kernel on a proper sector Definition
• The normalised sinc function Definition
• The one-dimensional torus and its normalized Haar integral Definition

…and 193 more results.

Dependency tree · two levels

32 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)