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The power-series, product-limit, IVP, functional-equation, and Picard definitions agree
Statement
The following descriptions give the same function : the power series ; the product limit ; the normalized solution of ; the normalized continuous multiplicative function; and the compact-uniform limit of the Picard iterates.
Facts & Assumptions
Given: The five displayed constructions.
The series-defined exponential satisfies and , and the product, ODE-uniqueness, functional-equation, and Picard characterizations are The exponential function is smooth and , For every real , , The exponential is the unique solution of with , Regular normalized multiplicative Cauchy equations characterize the exponential, and Picard iteration from produces the exponential partial sums.
Proof
Each theorem in [L2] identifies its construction with the series-defined function in [L1], with exactly the normalization stated here.
Equality with a common function is transitive, so all five descriptions are equivalent.
Depends on
- The real exponential function and the number $e$ by a power series
- The exponential function is smooth and $(\exp)'=\exp$
- For every real $x$, $(1+x/n)^n\to\exp x$
- The exponential is the unique solution of $y'=y$ with $y(0)=1$
- Regular normalized multiplicative Cauchy equations characterize the exponential
- Picard iteration from $1$ produces the exponential partial sums
Used by
- Differentiation under an improper integral can fail without uniform domination Counterexample
- Gaussian even moments for Brownian increments Lemma
- The plane Gaussian integral equals π by polar coordinates Lemma
- The standard normal density has total mass one Lemma
- Roadmap and non-circularity of the logarithm characterisations Remark
- The exponential roadmap and its circularity hazards Remark
Dependency tree · two levels
43 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, Analytic Functions (standard reference, not scraped)
- University of Pennsylvania MATH 3600, Section 34 (standard reference, not scraped)
- J. Lebl, Basic Analysis, Picard's Theorem (standard reference, not scraped)
- J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)