You should look at the Wikipedia article about characterizations of the exponential function; it has five. As for your question about whether a function can be expressed as a series or not, to …

2 Another Wikipedia article of relevance is Characterizations of the exponential function. Which characterization is most appropriate to take to be the definition depends on the context.

Aug 23, 2023 · For every real number $x$, we define the natural exponential function to be $e^ {x} = \exp (x)$. Where $\exp$ denotes the inverse function of $\ln$, and $e$ is a number such …

So here's my proof, using only the definition of the exponential function and elementary properties of limits. We use the following definition of the exponential function:

Aug 22, 2021 · Finding exponential function limit definition from definition of e Ask Question Asked 4 years, 2 months ago Modified 2 years, 8 months ago

Jan 3, 2023 · My question : (1) Do these definitions exhaust the ways the exponential function can be defined? (2) Are these definitions actually different at least conceptually ( though denoting …

Oct 16, 2020 · 1 There is a standard definition of exponential order used in the theory of Laplace transforms, which is a hypothesis in various theorems about them.

Nov 7, 2018 · The exponential function is the unique smooth group isomorphism from the additive group of reals to the multiplicative group of positive reals whose derivative at zero is one.

Feb 16, 2013 · Just a quick note on the comment, there is no need to assume that the exponential is differentiable. Simply define the exponential function $\exp (x) = \sum_ {i=0}^ {\infty}$ and all …

Aug 30, 2023 · The function f(x) = ax f (x) = a x is generally taught to only allow a> 0 a> 0. This is usually justified by giving a few examples of complex points in cases where a<0 a <0.