Section 3.3 Exponential Functions
The discussion of the previous section applies to general power series. It is more interesting to discuss some special power series that arise from important applications or have special properties. The exponential and logarithmic functions are two families of such functions.
In most elementary calculus treatment, the definition of the exponential functions \(a^{x}\) and proof of their properties use some hand waving at some points; similarly, the definition and properties of the trigonometric functions rely on some geometric arguments, instead of purely analytical ones. It should be a rewarding experience to review such a treatment and pinpoint such places.
The exponential functions arise most naturally as solutions of the ODE \(y'(x)=ry(x)\text{.}\) In looking for a solution of the form \(y=\sum_{n=0}^{\infty}c_{n} x^{n}\text{,}\) one finds
\begin{equation*}
y'(x)=\sum_{n=0}^{\infty}c_{n} n x^{n-1} =ry(x)= r \sum_{n=0}^{\infty}c_{n} x^{n},
\end{equation*}
from which one concludes that \(c_{n} n =rc_{n-1}\text{.}\) Then by induction one gets
\begin{equation*}
c_{n}=c_{0}\frac{r^{n}}{n!}.
\end{equation*}
Thus \(y=c_{0}\sum_{n=0}^{\infty}\frac{r^{n}x^{n}}{n!}\) should be a solution with \(y(0)=c_{0}\text{.}\) To justify the argument, one checks that this power series has its radius of convergence equal to \(\infty\text{,}\) so all the derivations are justified.
How was the solution \(c_{0}e^{rx}\) introduced in calculus? And how does this power series solution relate to \(c_{0}e^{rx}\text{?}\) Is this the only solution with \(y(0)=c_{0}\text{?}\)
The answer lies in developing properties of this solution. Denoting \(E(x)=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}\text{.}\) The key properties are
\begin{align*}
E'(x) \amp = E(x), \; \forall x,\\
E(x+y) \amp=E(x) E(y),\; \forall x, y,\\
E(0)\amp=1.
\end{align*}
It then follows that \(E(-x)E(x)=E(0)=1\) for any \(x\text{,}\) so \(E(x)\ne 0\) for any \(x\) (even complex valued). When \(x\) takes real values, \(E(x)\) also takes real values by construction. For \(x\gt 0\text{,}\) the power series for \(E(x)\) shows that \(E(x)\gt 0\text{.}\) One then shows using \(E(-x)E(x)=E(0)=1\) that \(E(x)>0\) for all real \(x \lt 0\text{.}\) Then the property \(E'(x)=E(x)>0\) shows that \(E(x)\) is monotone increasing for real valued \(x\text{.}\)
At this point, there is a well defined inverse function of \(E(x)\) for \(x\in \mathbb R\text{.}\) Call it \(\ln y\) for \(y > 0\text{.}\) Then \(E(\ln y)=y\) and \(\ln E(x) =x\text{.}\) It remains to establish that \(\ln y\) is defined for all \(y>0\text{.}\) As a consequence of \(E(x+y) =E(x) E(y)\text{,}\) we will have \(\ln (u v)=\ln u + \ln v\) for \(u, v>0\text{.}\)
Based on the properties of \(E(x)\text{,}\) one establishes that for any rational \(x=\frac pq\text{,}\)
\begin{equation*}
E(x)=[E(\frac 1q)]^{p}=[E(1)]^{\frac pq}=[E(1)]^{x}.
\end{equation*}
If one can establish that the function \(a^{x}\) is well defined for any real \(a > 0\) and any real \(x\) and that it is a continuous function for \(x\in \bbR\text{,}\) then one can use the continuity to show that the above equality holds for all \(x\text{.}\) However it is not a trivial task to define \(a^{x}\) for any real \(a > 0\) and any real \(x\) and prove that it is a continuous function of \(x\text{.}\)
Recall that there is an arithmetic procedure for computing \(a b\) and \(a^{m}\) only when \(a, b\) are rational numbers and \(m\) is an integer, that the definition of \(a^{1/n}\) for any positive real (even a rational number) \(a\) and positive integer \(n\) requires a limiting process and completeness of \(\bbR\text{.}\) Once this is defined, one can use the continuity and monotonicity of the power function \(x\mapsto x^{k}\) for any positive integer \(k\) and positive real \(x\) to define \(a^{m/n}\) for any positive real \(a\) and positive integers \(m, n\text{,}\) namely, \(a^{r}\) for any positive real \(a\) and positive rational \(r\text{.}\) Additional work is needed to define \(a^{x}\) for any positive real \(a\) and any real \(x\text{,}\) as was done in Exercise 6 of Chapter 1 of Rudin’s text for the case of \(a > 1\text{.}\)
Using properties of power series, the definition and properties of \(E(x)\) can be developed in a routine way, which is how Rudin develops this material. Rudin also sketches an argument to show that the treatment following Exercise 6 of Chapter 1 produces the same function as that using \(E(x)\text{.}\)
