Alphabeta MathHow 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.
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, , and
Statement
For real , and . In particular . The conventions and prerequisite facts used below are recorded in , and the complex exponential extends the real exponential, Euler's formula: for every real , Conjugation laws, , multiplicativity of modulus, and the triangle inequality, Pythagorean and parity identities for all six trigonometric functions on their natural domains, The exponential is positive and satisfies , Quarter-turn values and shifts by pi/2 and pi, Pi as twice the smallest positive zero of cosine.
Facts & Assumptions
Given: Reals .
Proof
Apply the addition law to and Euler's formula.
Multiplicativity of modulus, the Pythagorean identity, and positivity of give the modulus formula.
The defining quarter-turn value and give Euler's identity.
Depends on
- $\exp(z+w)=\exp z\,\exp w$, and the complex exponential extends the real exponential
- Euler's formula: $\exp(i\theta)=\cos\theta+i\sin\theta$ for every real $\theta$
- Conjugation laws, $z\overline z=|z|^2$, multiplicativity of modulus, and the triangle inequality
- Pythagorean and parity identities for all six trigonometric functions on their natural domains
- The exponential is positive and satisfies $\exp(-x)=1/\exp(x)$
- Quarter-turn values and shifts by pi/2 and pi
- Pi as twice the smallest positive zero of cosine
Used by
- 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
- 1=e^2π i does not imply 0=2π i: logarithms invert the exponential only modulo its kernel Example
- The logarithms of -1 are (2k+1)π i, k∈ℤ Example
- ker(exp)=2π iℤ, and exp z=exp w exactly when z-w∈2π iℤ Theorem
- The complex exponential maps ℂ onto ℂ∖{0} Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 127 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)