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I am trying to construct the exponential function on $\mathbb R$ by first finding all functions $f$ such that $f = f'$ (which should be all the constant multiples of $\exp$), then characterizing $\exp$ by the initial condition $f(0) = 1$.

I intend to use only the definitions and properties of derivatives and integrals, and the fundamental theorem(s) of calculus; in particular, I am trying to avoid Taylor series and/or the notion of uniform convergence. Of course, these restrictions are by no means precise, and I really don't know if it's possible to come up with a reasonably "simple" construction using only this set of tools, but I figured I'd give it a try.

So far I've only managed to prove uniqueness in the sense that if $f = f'$ and $c \in \mathbb R$, then $f$ is uniquely determined by $f(c)$. With this comes the corollary that if $f(c) = 0$ for any $c \in \mathbb R$, then $f = 0$. Existence seems pretty difficult to get to without any of the heavy machinery.

Here is the proof of uniqueness as described above:

Lemma. If $f = f'$, $g = g'$, and $c \in \mathbb R$, then $f + c \cdot g = f' + c \cdot g' = f' + (c \cdot g)' = (f + c \cdot g)'$.

Lemma. If $f = f'$ and $f(c) = 0$ for some $c \in \mathbb R$, then $f(x) = 0$ for all $x \leq c$. (I had trouble proving it for $x > c$.)

Proof. By the FTC, we have $\int_a^b f = f(b) - f(a)$. If $f(x) > 0$ for $c - \varepsilon \leq x \leq c$, then $\int_{c - \varepsilon}^c f > 0$, but $f(c) - f(c - \varepsilon) < 0$, a contradiction. Similar contradiction for $f(x) < 0$.

Theorem. If $f = f'$, $g = g'$, and $f(c) = g(c)$ for some $c \in \mathbb R$, then $f = g$.

Proof. Suppose $f(d) \neq g(d)$ for some $d \in \mathbb R$. WLOG, assume $g(d) \neq 0$. If $d < c$, then $(f - g)(d) \neq 0 = (f - g)(c)$; if $c < d$, then $$\left( f - \frac{f(d)}{g(d)} \cdot g \right)(c) \neq 0 = \left( f - \frac{f(d)}{g(d)} \cdot g \right)(d).$$ In any case, we contradict the second lemma.

Corollary. If $f = f'$ and $f(c) = 0$ for some $c \in \mathbb R$, then $f = 0$.

Proof. $0 = 0'$.

Now I want to prove the existence of $f$ for any initial condition $f(x) = y$, but I don't know how.

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Show that Euler's method converges. –  Qiaochu Yuan Jun 20 '12 at 12:06
It is a differential equations, and existence is something related to infinite-dimensional spaces. You need some tool from analysis, like Ascoli-Arzelà or some fixed-point theorem. –  Siminore Jun 20 '12 at 12:06
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1 Answer

Supoose $u$ and $v$ are functions so that $u'=u$, $v' = v$ and so $u(0)=v(0) = 1$. Put $\Phi(t) = u(t)v(-t)$ for $t\in\mathbb{R}$.

$$\Phi'(t) = u'(t)v(-t) - u(t)v'(-t) = u(t)v(-t) - u(t)v(-t) = 0.$$

The function $\Phi$ is constant. Since $\Phi(0) = 1$, $u(t)v(-t) = 1$ for all $t\in{\mathbb{R}}$.

We can apply this result to $v$ and $v$ to get $$v(-t) = {1\over v(t)},\qquad t\in\mathbb{R}.$$ We conclude that $u(t)/v(t) = 1$ for $t\in\mathbb{R}$, so $u = v$.

Here is one way to get existence.

Define $l(x) = \int_1^x{dt\over t}$, for $x > 0$. This function is continuous and differentiable on $(0,\infty)$.

Observe that $$l(xy) = \int_1^{xy} {dt\over t} = \int_1^x {dt\over t} + \int_x^{xy}{dt\over t} = l(x) + \int_x^{xy} {dt\over t}.$$

Using a change of variable, we get $$\int_x^{xy} {dt\over t} = \int_1^y {x\,dt\over xt} = l(y).$$ We have $l(xy) = l(x)+l(y)$ for $x, y > 0$. We clearly have $l(1) = 0$. And it's not hard to show that $l(x^\alpha) = \alpha l(x)$ for $\alpha\in\mathbb{R}$ and $x > 0$.

By the fundamental theorem of calculus we know that $$l'(x) = 1/x$$ for $x > 0$. This function is strictly increasing and therefore 1-1 on $(0,\infty)$.

The inverse function to $l$ will satisfy the the initial value problem $u'(t) = u(t)$ and $u(0) = 1$. This gives existence.

You can use the properties of $l$ to see this is an exponential function. The base for this exponential is $l^{-1}(1)$.

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This gives you uniqueness. –  ncmathsadist Jun 20 '12 at 13:03
This is explicitly what the OP doesn't need. –  mixedmath Jun 20 '12 at 13:06
Existence now added. This uses only the fundamental theorem of calculus, the chain rule and change of variable. –  ncmathsadist Jun 20 '12 at 13:20
I guess you forgot a '+' in $l(xy)=l(x)l(y)$ which should be $l(xy)=l(x)+l(y)$... =) –  bartgol Jun 20 '12 at 14:14
Fixed; thanks bartgol. –  ncmathsadist Jun 20 '12 at 14:31
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