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I have been looking at the proof of the existence of $e^x$ and its properties, and I understand the induction argument which yields the Taylor series expansion around $x=0$. For example,

$E_1(x)=1 + x$, $E_{n+1}(x)=1 + \int_0^x E_n(t)$, etc.

However, I wonder how this argument was developed informally before the proof. For example, how was $E_1(x)$, etc. chosen?

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    Should this be tagged [math-history]?2011-08-31

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One way of solving a differential equation of the form $ \begin{align} &\frac{dy}{dx}=F(y(x)),\\ &y(0)=a, \end{align} $ is to rewrite it in integral form $ y(x)=a+\int_0^xF(y(u))\,du. $ Here, we are solving for functions $y\colon\mathbb{R}^+\to\mathbb{R}$ and $F\colon\mathbb{R}\to\mathbb{R}$ is given. The integral form can be solved iteratively. First choose any (continuous) initial guess $y_0\colon\mathbb{R}^+\to\mathbb{R}$ then iteratively define $y_{n+1}(x)$ in terms of $y_n$ $ y_{n+1}(x)=a+\int_0^xF(y_n(u))\,du. $ This is method is known as Picard iteration, and is guaranteed to converge to the unique solution to the differential equation for a large class of functions $F$. For example, it always converges if $F$ is Lipschitz continuous.

The exponential function $y(x)=\exp(x)$ satisfies $\frac{dy}{dx}=y$ and $y(0)=1$. This differential equation can be solved by Picard iteration by taking $F(y)=y$ and using $y_0=0$ or $y_0=1$ gives the iteration described in the question.

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    But, as mentioned in the Wikipedia link, e was first used explicitly by Leibniz. As he developed differential/integral calculus, I suppose that he could have used an argument like this (non-rigorously).2011-08-31
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I like starting with the functional equation $f(x+y)=f(x)f(y)$. From this, assuming differentiability, we can show that f'(x) = f'(0)f(x). ($f(x+h)-f(x) = f(x)f(h)-f(x) = f(x)(f(h)-1)$ so $(f(x+h)-f(x))/h = f(x)(f(h)-1)/h$, and let $h \rightarrow 0$)

This also works for the log, inverse tan, and other functions.

Once you have the differential equation, proceed as usual.

Of course, I claim no originality for this - I just like it.

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For example, how was $E_1(x)$, etc. chosen?

As long as $E_1(x)$ has the correct value at $x=0$, the iteration will converge to the same target.