# The Exponential Function The exponential function is a function $\exp(x)$ defined as $\exp (x) = \sum_{n=0}^\infty \frac {x^n} {n!}$ ^fo2wdr (where $!$ denotes is the [[Factorial]]) Because $\exp(0) = 1$, and because $\exp(x)\exp (y) = \exp(x + y)$ for all complex numbers $x$ and $y$, **the exponential function shares all properties of exponentiation** (see [[Exponent Rules]]). **That is why $e^x$ is used as a shorthand for $\exp$** (one of the definitions of $e$ is $\exp(1)$, see [[The Natural Constant]]). **This also makes perfect sense for real inputs ($\exp (x) = e^x$ for all real $x$), but for complex numbers it is nonsensical** (how do you multiply $e$ by itself $i$ times?). The shorthand is, however, still used even for complex $\exp$ inputs. **As such, whenever presented with $e^i$, one should think of the exponential function, and not of exponentiation.** ^vm4wyz If you differentiate individual elements of $\exp(x)$ using the basic rules of [[Derivative|derivatives]], you'll see that the first derivative of $\exp(x)$ is itself $\exp(x)$. In fact **$\exp(x)$ is the only function that is its own derivative.** Whereas exponentiation only takes real inputs, the domain of the exponential function are all complex numbers. ## Showing that $\exp(0) = 1$ $ \begin{align} \exp(0) &= \sum_{n=0}^\infty \frac {0^n} {n!} \\ &= \frac {0^0} {0!} + \sum_{n=1}^\infty \frac {0^n} {n!} && \text{$0^0 = 1$ per exponent rules and $0! = 1$ per factorial rules} \\ &= 1 + \sum_{n=1}^\infty \frac {0^n} {n!} && \text{$0^x = 0$ when $x\ge1$ } \\ &= 1 \end{align} $ ## Proof of $\exp(x)\exp (y) = \exp(x + y)$ The proof uses [[Cauchy Product]], the [[Binomial Theorem]] and [[Combinations|the binomial coefficient]]. $ \begin {align} \exp (x) \exp(y) &= \left(\sum_{n=0}^\infty \frac {x^n} {n!} \right)\left(\sum_{m=0}^\infty \frac {y^m} {m!} \right) \\ &= \sum_{n=0}^\infty \sum_{m=0}^n {\frac {x^m} {m!} \frac {y^{n-m}} {(n-m)!}} && \text {(Cauchy product)} \\ &= \sum_{n=0}^\infty \sum_{m=0}^n {\frac {x^m} {m!} \frac {y^{n-m}} {(n-m)!}} \frac {n!}{n!} \\ &= \sum_{n=0}^\infty \sum_{m=0}^n \binom n m {{x^m y^{n-m}} \frac 1{n!}} \\ &= \sum_{n=0}^\infty (x+y)^n {\frac 1{n!}} && \text{(binomial theorem)} \\ &= \exp(x+y) \end {align} $