$\exp (z+w)=\exp z\exp w$, 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

• 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
• 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
• 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
• 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
• Complex powers defined from a holomorphic logarithm branch Definition
• Roots of a compact connected Lie group Definition
• A continuous argument computed along a spiralling contour Example
• Characteristic functions of bernoulli binomial and poisson laws Example
• Independent sums via characteristic functions Example
• Morera proves holomorphy of z↦∫₀¹ tᶻ dt on Rez>1 Example
• The complex exponential satisfies the Cauchy–Riemann equations in Cartesian and polar form Example
• The power series of exp(z₀+z₁) on every bidisc Example
• Transform of an interval indicator Example
• FALSE: every entire function with an antiderivative is a polynomial False statement
• A nonvanishing holomorphic function on a disc has a holomorphic logarithm Lemma
• Basic properties of characteristic functions Lemma
• Characteristic function of a normal law Lemma
• Characteristic functions are positive definite Lemma
• Characteristic functions under affine maps and independent sums Lemma
• Finite simple analytic families and their exact endpoint norms Lemma
• Finite sums of the sine harmonics Lemma
• Fourier uniqueness for continuous functions on the Euclidean torus Lemma
• Moments give derivatives of the characteristic function Lemma
• Products of near-one characteristic factors Lemma
• Real L2 multipliers and unitary transport Lemma
• Zero free entire function of exponential type is an exponential Lemma
• A circle traversed k times has winding number k inside and 0 outside Theorem
• A nonvanishing holomorphic function on a homologically simply connected domain has a holomorphic logarithm Theorem
• A plane domain is homologically simply connected exactly when every harmonic function has a global conjugate Theorem
• Basic operations are continuous on Schwartz space Theorem
• Branch-defined complex powers agree with integer powers Theorem
• Complex sine and cosine are unbounded on the complex plane Theorem

…and 15 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)