Characterizations of the exponential function

In mathematics, the exponential function can be characterized in many ways. This article presents some common characterizations, discusses why each makes sense, and proves that they are all equivalent.

The exponential function occurs naturally in many branches of mathematics. Walter Rudin called it "the most important function in mathematics".[1] It is therefore useful to have multiple ways to define (or characterize) it. Each of the characterizations below may be more or less useful depending on context. The "product limit" characterization of the exponential function was discovered by Leonhard Euler.[2]

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