The complex exponential by its power series

Depends on

• Complex series, absolute convergence, complex power series, and radius of convergence
• Integer powers in the complex field
• The complex numbers as $\mathbb R[x]/(x^2+1)$, with the real embedding and imaginary unit $i$
• $\mathbb C=\mathbb R[x]/(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)/(a^2+b^2)$
• The factorial $n!$ and the falling factorial $n^{\underline{k}}$, defined by recursion in $\mathbb{N}$
• The canonical natural $\iota(n) = n \cdot 1_F$ of a field
• Canonical naturals are positive and strictly increasing
• The complex exponential series converges absolutely for every complex argument

Used by

• Finite-order Padé scaling-and-squaring exponential Algorithm
• Quantum phase-estimation spectral sampler Algorithm
• Trigonometric polynomials are uniformly dense in continuous functions on the torus Corollary
• Distinct pure frequencies differing by a reciprocal-lattice shift have identical samples Counterexample
• Sharp frequency cutoffs have kernels that are not in L1 Counterexample
• Complex logarithms, the principal logarithm, and principal and multivalued complex powers Definition
• Complex sine, cosine, hyperbolic sine, and hyperbolic cosine from the complex exponential Definition
• Continuous logarithms and continuous arguments along a contour Definition
• Diagonal Padé approximant to the exponential Definition
• Dirichlet series Definition
• Eigenphase estimation access model Definition
• Fourier coefficients and trigonometric polynomials on the torus Definition
• Fourier transform on complex L1 classes Definition
• Full-rank lattices, covolume, and the dual lattice Definition
• Hamiltonian-simulation polynomial target Definition
• Inner, singular inner and outer functions Definition
• Spherical Bessel and outgoing Hankel function conventions Definition
• The Bernoulli numbers are defined by the generating series t/(eᵗ-1) Definition
• The combined-field operator for smooth-boundary Helmholtz scattering Definition
• The complex-time heat kernel on a proper sector Definition
• The Jacobi theta function θ(t)=∑_n∈ℤ e^-π n² t for t>0 Definition
• The normalised sinc function Definition
• The phi-function family by everywhere-convergent series Definition
• The recursive radix-two fast Fourier transform Definition
• The Riemann zeta function on the half-plane Res>1 Definition
• The three-dimensional volume integral scattering model Definition
• The unitary discrete Fourier transform on ℤ/Nℤ Definition
• The unnormalised engineering DFT and its conversion to the unitary transform Definition
• Type-1, type-2, and type-3 nonuniform Fourier transforms Definition
• A continuous argument computed along a spiralling contour Example
• A simple zero of the Weierstrass sigma function on the square lattice Example
• A stable-spectrum nonnormal exponential has a large transient norm Example
• An outer function with a prescribed power of a vanishing modulus Example
• Arcsine equilibrium measure and capacity of a segment Example
• Boundary vanishing of a nonzero Hardy function is confined to a null set Example
• Capacity of a disc and its circular equilibrium measure Example
• Characteristics of a monomial, an exponential and a tangent Example
• Chebyshev extremals and the exact disk Fekete polynomial Example
• Componentwise holomorphy checked for an explicit map ℂ²→ℂ³ Example
• Cyclic convolution on ℤ/4ℤ via the DFT Example

…and 59 more results.

Dependency tree · two levels

44 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)