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

Used by

• Trigonometric polynomials are uniformly dense in continuous functions on the torus Corollary
• 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
• Dirichlet series Definition
• Fourier coefficients and trigonometric polynomials on the torus Definition
• Fourier transform on complex L1 classes Definition
• The Bernoulli numbers are defined by the generating series t/(eᵗ-1) Definition
• The Jacobi theta function θ(t)=∑_n∈ℤ e^-π n² t for t>0 Definition
• The Riemann zeta function on the half-plane Res>1 Definition
• A continuous argument computed along a spiralling contour Example
• Componentwise holomorphy checked for an explicit map ℂ²→ℂ³ Example
• Normalized Hermite Fourier eigenfunctions 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
• The unit-circle integral of exp(z)/z is 2 pi i by uniform termwise integration Example
• Finite tori are compact Hausdorff spaces separated by characters Lemma
• Products of near-one characteristic factors Lemma
• The complex exponential series converges absolutely for every complex argument Lemma
• The Dirichlet series for zeta converges absolutely and locally uniformly on the half-plane Res>1 Lemma
• Branch-defined complex powers agree with integer powers Theorem
• Euler's formula: exp(iθ)=cosθ+i sinθ for every real θ Theorem
• exp(z+w)=exp z exp w, and the complex exponential extends the real exponential Theorem
• Feller converse to Lindeberg-Feller Theorem
• Gleason Kahane Zelazko Theorem
• The Dirichlet eta series is holomorphic on Res>0 and equals the prefactor times zeta there Theorem
• The exponential definitions of complex sine, cosine, hyperbolic sine, and hyperbolic cosine equal their entire power series Theorem

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38 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)