$\exp (z+w)=\exp z\exp w$, and the complex exponential extends the real exponential

Depends on

• The complex exponential by its power series
• The complex exponential series converges absolutely for every complex argument
• The Cauchy product of two absolutely convergent complex series converges absolutely to the product of their sums
• The real exponential function and the number $e$ by a power series
• The binomial theorem over the complex field
• $\binom{n}{k}\,k!\,(n-k)! = n!$ for $k \le n$; hence $\binom{n}{k}\,k! = n^{\underline{k}}$, the quotient $n!/(k!(n-k)!)$ is a natural number, and $\binom{n}{k} = \binom{n}{n-k}$

Used by

• Buffered certified rational phase approximation Algorithm
• A nonvanishing holomorphic function on such a domain has holomorphic roots of every positive order Corollary
• Complex de Moivre formula for every integer exponent Corollary
• exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0 Corollary
• Holomorphic roots of a nonvanishing function on a disc Corollary
• The addition formulas for complex trigonometric and hyperbolic functions Corollary
• The spherical complementary series converge to the trivial representation Corollary
• Distinct pure frequencies differing by a reciprocal-lattice shift have identical samples Counterexample
• e^1/z has an essential singularity at 0 and omits the value 0 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
• Generator convention matters Counterexample
• Infinite variance can defeat square-root-n CLT scaling Counterexample
• Irrational rotation is ergodic but not weakly mixing Counterexample
• log|z| has no global harmonic conjugate on C{0 Counterexample
• Sharp frequency cutoffs have kernels that are not in L1 Counterexample
• The disc algebra is unital and separating but not self-adjoint or dense Counterexample
• The map (z₁,z₂)↦(e^z₁,z₂) has invertible complex Jacobian everywhere and is not injective Counterexample
• The regular representation of R is not a Hilbert direct sum of irreducibles Counterexample
• Complex powers defined from a holomorphic logarithm branch Definition
• Controlled Phase Rotation Definition
• Finite Hermitian matrix dynamics Definition
• Hamiltonian-simulation polynomial target Definition
• QSP signal matrix and ordered phase sequence Definition
• Roots of a compact connected Lie group Definition
• The Jacobi theta function θ(t)=∑_n∈ℤ e^-π n² t for t>0 Definition
• The recursive radix-two fast Fourier transform Definition
• The unitary discrete Fourier transform on ℤ/Nℤ Definition
• The unnormalised engineering DFT and its conversion to the unitary transform Definition
• A continuous argument computed along a spiralling contour Example
• A Fourier integral missed by one sample but captured by Filon moments Example
• A half-circle configuration loop traces an elementary half twist Example
• A nonnormal action grows transiently while its Arnoldi residual certificate remains valid Example
• A one-dimensional Arnoldi projection misses a slow mode of a stiff diagonal matrix Example
• A pure two-strand full twist as an ordered loop Example
• A simple zero of the Weierstrass sigma function on the square lattice Example
• Characteristic functions of bernoulli binomial and poisson laws Example
• Characteristics of a monomial, an exponential and a tangent Example
• Clustered sources and absolute l1 error Example
• Cyclic convolution on ℤ/4ℤ via the DFT Example
• Independent sums via characteristic functions Example

…and 127 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)