The exponential addition formula $\exp (x+y)=\exp (x)\exp (y)$

Depends on

• The exponential series converges absolutely for every real argument
• Inside the common radius the product of two power-series sums is represented by the Cauchy product of their coefficients
• The binomial theorem in $\mathbb{R}$: $(x+y)^{n} = \sum_{k<n+1} \iota\!\binom{n}{k}\, x^{k} y^{\,n-k}$
• $\binom{n}{k}\,k!\,(n-k)! = n!$ for $k \le n$; hence $\binom{n}{k}\,k! = n^{\underline{k}}$, the quotient $n!/(k!(n-k)!)$ is a natural number, and $\binom{n}{k} = \binom{n}{n-k}$
• The set $[A]^{k}$ of $k$-element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$
• 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
• Finite sums and finite products, by recursion
• Laws of finite sums and finite products

Used by

• The exponential is positive and satisfies exp(-x)=1/exp(x) Corollary
• The symmetric Lovász Local Lemma under ep(d+1)≤1 Corollary
• A discontinuous positive solution of F(x+y)=F(x)F(y) Counterexample
• Assuming choice, a Hamel-basis additive map transported through exp gives a discontinuous logarithmic function that is not c log Counterexample
• The convergence (1+x/n)ⁿ→exp x is not uniform on ℝ Counterexample
• A positive non-log-convex solution of the Gamma functional equation Example
• Geodesics in the Poincare upper half-plane Example
• The additive and multiplicative real Lie groups Example
• The log-free product limit (1-2/n)ⁿ→exp(-2) Example
• 1+x≤exp(x) for every real x, hence (1-p)ᵐ≤exp(-mp) Lemma
• Chernoff bound for independent bernoulli trials Lemma
• Exponential contraction of projection away from a quasiconvex set Lemma
• Local convergence of the Baker–Campbell–Hausdorff series Lemma
• The moment generating function of a finite sum of independent variables is the product of their moment generating functions Lemma
• The square of the one-dimensional Gaussian integral is the plane Gaussian integral Lemma
• Addition formulas, identities, parity, and derivatives of the hyperbolic functions Theorem
• If k≥1 and n≥3k² 2ᵏ, an n-vertex tournament with property Sₖ exists Theorem
• Morse stability with explicit parameter dependence Theorem
• Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm Theorem
• Regular normalized multiplicative Cauchy equations characterize the exponential Theorem
• The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents Theorem
• Under square summability, the signed product of (1+pₙ) converges iff the series of pₙ converges Theorem

Dependency tree · two levels

55 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

• MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)
• J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
• J. K. Hunter, An Introduction to Real Analysis, Chapter 10 (standard reference, not scraped)