Why the $e$ in $e^z$?

Question

I am studying complex analysis, and I am learning about the complex-valued function $e^z$. But why not use another value other than $e^z$? In other words, why not use $2^z$, or $10^z$, for example? Why is the value of $e$ so special in this case?

Answer 1

The fact that $\frac d {dz} e^{z}=e^{z}$ makes it very useful. This property is not shared by $a^{z}$ for $a \neq e$.

Answer 2

First of all, there is $\frac{d}{dz}e^z = e^z$ while $\frac{d}{dz}10^z = (\log_e10) 10^z$.

There is also $e^{i\theta}=\cos \theta + i \sin \theta$ while $10^{i\theta}=\cos((\log_e 10)\theta) + i \sin((\log_e 10)\theta)$.

Also $e^z = 1 + z + \frac 12 z^2 + \frac 16 z^3 + \frac{1}{24} z^4 + \cdots$

while $10^z = 1 + z \log_e(10) + \frac 12 z^2 \log_e^2(10) + \frac 16 z^3 \log_e^3(10) + \frac{1}{24} z^4 log_e^4(10) + \cdots$

Answer 3

The exponential function is defined for complex values as $$\exp(z):=\sum_{n=0}^\infty \frac{z^n}{n!}, $$ and it enjoyes many properties, such as $\exp'=\exp$, and $\exp(z_1+z_2)=\exp(z_1) \exp(z_2).$

Using the exponential function we can define complex powers as $$\alpha^\beta := \exp(\beta \log \alpha). $$ Notice however, that this definition of powers is possibly multivalued --- it might depend on which branch of the logarithm is used. For example $$ \mathrm{e}^{1/2}=\exp \left( \frac{1}{2} \log(\mathrm{e}) \right)= \exp \left( \frac{1}{2} \left(\ln \mathrm{e}+2 \pi i n \right) \right)=\pm \exp\left( \frac{1}{2} \right),$$ has two possible values.

Typically, the principal branch of the logarithm is what is meant in the power $\mathrm{e}^z$. But, in any case, the way I organize it in my head is that the function $\exp$ comes first, and it is unambiguous. Powers can be defined based on this function, but they could take a variety of values.