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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 by a power series
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
Picard iteration from produces the exponential partial sums