Alphabeta MathThe Complex Exponential and Euler's Formula
About this page
How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
✓ 19 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 19 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Linear Algebra
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Integer powers in the complex field
Lemma
The binomial theorem over the complex field
Definition
The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane
Theorem
The complex plane is complete, and convergence is equivalent to convergence of real and imaginary parts
Definition
Complex series, absolute convergence, complex power series, and radius of convergence
Theorem
Every absolutely convergent complex series converges, and rearrangements preserve its sum
Theorem
Cauchy-Hadamard for complex power series, including zero and infinite radius
Lemma
The Cauchy product of two absolutely convergent complex series converges absolutely to the product of their sums
Lemma
The complex exponential series converges absolutely for every complex argument
Definition
The complex exponential by its power series
Theorem
exp(z+w)=exp z exp w, and the complex exponential extends the real exponential
Theorem
Euler's formula: exp(iθ)=cosθ+i sinθ for every real θ
Corollary
exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0
Theorem
ker(exp)=2π iℤ, and exp z=exp w exactly when z-w∈2π iℤ
Theorem
Every nonzero complex number has a unique polar form r(cosθ+i sinθ) with r>0 and -π<θ≤π
Theorem
The complex exponential maps ℂ onto ℂ∖{0}
Corollary
Complex de Moivre formula for every integer exponent
Theorem
The n-th roots of a complex number and the n distinct roots of unity for every n≥1
Corollary
For n≥2, the sum of all n-th roots of unity is zero
Definition
Complex logarithms, the principal logarithm, and principal and multivalued complex powers
Theorem
All logarithms of z≠0 are Logz+2π i k, k∈ℤ
Theorem
There is no continuous logarithm on all of ℂ∖{0}
Definition
Complex sine, cosine, hyperbolic sine, and hyperbolic cosine from the complex exponential
Theorem
The exponential formulas, real restrictions, and trigonometric-hyperbolic dictionary over ℂ
Definition
Formal complex polynomials, evaluation, degree, leading coefficient, and monic polynomials
Lemma
A nonconstant complex polynomial tends to infinite modulus and attains a global minimum modulus
Lemma
A nonzero value of a nonconstant complex polynomial cannot be a local minimum of its modulus
Theorem