What does $\exp(f)$ mean?

Question

In several posts around this site, I have encountered the expression $\exp(x)$ where $x$ is an arbitrary expression. What does this notation mean?

Answer 1

$\exp(x) = e^x{}{}{}{}{}{}{}{}{}{}{}{}{}$

It's often used when we have $\exp(f(x)) = e^{f(x)}$ and $f(x)$ is complicated function. The focus, then, becomes the function in the exponent of $e$ (not to mention that it helps readers to not have to squint to read, say when $f$ is a rational function, $e^{f(x)}$).

Answer 2

It denotes the exponential function with base $e$ defined as:

$$\exp(z):=e^z.$$

Answer 3

This is the exponential function. In other words, $$\mathrm{exp}\,(x)$$ is another way of writing $$e^x,$$ where $$e=2.7182818284590452353...$$

Answer 4

Usually $1+x+\frac{x^2}{2!} + \frac{x^3}{3!} + \ldots$, if $x$ is something that can be multiplies by itself, divided by an integer, finite sums are defined, and there is some notion of the sums convergence. It makes sense for all finite-dimensional matrices, some operators in infinite dimensional spaces and also for other things.