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

• A sample-size consequence for compression schemes Corollary
• 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
• Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
• The convergence (1+x/n)ⁿ→exp x is not uniform on ℝ Counterexample
• The Jacobi theta function θ(t)=∑_n∈ℤ e^-π n² t for t>0 Definition
• A biased one-pixel CNN approximates but is not the exact MMSE denoiser Example
• A finite Euler adjoint gradient can differ from the exact flow gradient Example
• A finite identifiable model with posterior consistency Example
• A finite-mean polynomial-tail reward is not sub-Gaussian Example
• A Gaussian maximum without independence Example
• A mean-matched count law can be overdispersed relative to Poisson Example
• A mixture of straight Gaussian paths has a nonlinear marginal velocity Example
• A one-observation Poisson-log Fisher-scoring step Example
• A positive non-log-convex solution of the Gamma functional equation Example
• A proper Gaussian likelihood gives a proper posterior Example
• A stable coupled model has an unstable explicit fast microstep Example
• A wide tanh network stays lazy on a fixed time horizon Example
• Boundary-incompatible forcing reduces Strang order to five quarters in a Dirichlet modal model Example
• Calibration examples under validation and distribution shift Example
• Euler has state error but exact log-density quadrature for a linear flow Example
• Exact and pseudo likelihoods for finite-window Poisson and Strauss models Example
• Exact scalar MMSE denoiser for a two-point prior Example
• Exponential singular decay, logarithmic balls, and spectral gaps Example
• Exponential singular-value decay and an exponentially weighted truth class Example
• Fixed-time error slopes for Euler, midpoint, and classical RK4 Example
• Geodesics in the Poincare upper half-plane Example
• Heat and Poisson semigroups as Fourier multipliers Example
• The additive and multiplicative real Lie groups Example
• The Gaussian integral on the line with a certified sinh-sinh truncation tail Example
• The log-free product limit (1-2/n)ⁿ→exp(-2) Example
• Too few sweeps lower the order from Gauss collocation Example
• Cohomologous symplectic forms on a noncompact manifold are always isotopic False statement
• 1+x≤exp(x) for every real x, hence (1-p)ᵐ≤exp(-mp) Lemma
• A binomial support-count bound Lemma
• A binomial-sum bound for compression sample sizes Lemma
• A one-coordinate projection risk bound Lemma
• Bernstein tail bound for independent bounded sums Lemma

…and 74 more results.

Dependency tree · two levels

56 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)