In crystallography, you'll frequently encounter complex exponential functions expressed using the form $e^{i\phi}$. For instance, the structure factor $F_{hkl}$ is often given as \begin{equation} F_{hkl}=\sum_{j} f_{j}(s)\,e^{2\pi i\,(hx_{j} + ky_{j} + lz_{j})}=|F_{hkl}|e^{i\phi_{hkl}} \end{equation} Unfortunately, the literature is rife with this misleading notation. This sparks confusion mostly because we tend to think of $e$ as a real, tangible number (i.e., $e = 2.71828\ldots$). Well, what could it possibly mean to raise $e$ to an imaginary power? Whenever you see an expression like $e^{i\phi}$ in crystallography, it's often helpful to dispense with your base instinct of conceptualizing $i\phi$ as an exponent, because this notation has nothing to do with iterative multiplication of $2.71828$. So if you've been asking yourself—how do we iteratively multiply something $i$ times, anyway? Doesn't that make zero sense? In a way, you'd be right! Mechanically speaking, $e^{i\phi}$ genuinely doesn't make any sense. This notation is simply a relic of a particular type of shorthand: something very specific to real arguments of a special infinite polynomial called exp, which is defined as: \begin{equation} \boxed{\exp(x)=\sum_{k=0}^{\infty} \frac{x^{k}}{k !}=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}+\frac{x^{5}}{120}+\cdots} \end{equation} When we plug in 1 to this power series, we obtain a familiar result: \begin{equation} \exp(1)=\sum_{k=0}^{\infty} \frac{1^{k}}{k !}=1+1+\frac{1}{2}+\frac{1}{6}+\frac{1}{24}+\frac{1}{120}+\cdots \approx 2.71828\ldots = e. \end{equation} This is one of a myriad of ways in which we can formally define $e$: it's equivalent to $\exp(1)$. Now it turns out $\exp$ has a slew of fairly unique properties. For instance, if we plug in 2, we get \begin{equation} \exp(2)=\sum_{k=0}^{\infty} \frac{2^{k}}{k !}=1+2+\frac{4}{2}+\frac{8}{6}+\frac{16}{24}+\frac{32}{120}+\cdots \approx 7.38905\ldots = e^{2}. \end{equation} Similarly, plugging in 3 yields \begin{equation} \exp(3)=\sum_{k=0}^{\infty} \frac{3^{k}}{k !}=1+3+\frac{9}{2}+\frac{27}{6}+\frac{81}{24}+\frac{243}{120}+\cdots \approx 20.0855\ldots = e^{3}. \end{equation} This rather unexpected pattern conveys a powerful and frankly somewhat weird result. As long as we're supplying exp with some real input, it turns out $\mathrm{exp}(x)$ always converges to $e^{x}$, which holds true even for negative exponents: \begin{equation} \exp(-1)=\sum_{k=0}^{\infty} \frac{(-1)^{k}}{k !}=1-1+\frac{1}{2}-\frac{1}{6}+\frac{1}{24}-\frac{1}{120}+\cdots \approx 0.36787\ldots = e^{-1}. \end{equation} Therefore, it became convenient shorthand to abbreviate any argument given to exp as $e$ raised to the power of that value. In fact, all we've really done here is rediscover an essential property of exponents: $\exp(a+b) = \exp(a)\cdot\exp(b)$. Nevertheless, this $e^{x}$ representation is only productive, at least in a mechanical sense, if $x$ is real. Much of this is abruptly nullified if we feed exp a complex argument, such as $i\phi$. Why? If we're attempting to parse $\mathrm{exp}(i\phi)$, we can no longer conceptualize the output of this function as a power of $e$. Let's demonstrate this by evaluating $\mathrm{exp}(i\phi)$: \begin{equation} \exp(i \phi)=\sum_{k=0}^{\infty} \frac{(i \phi)^{k}}{k !}=1+i \phi+\frac{(i \phi)^{2}}{2}+\frac{(i \phi)^{3}}{6}+\frac{(i \phi)^{4}}{24}+\frac{(i \phi)^{5}}{120}+\cdots \end{equation} We can simplify this expression by evaluating the various powers of $i$: \begin{equation} \exp(i \phi)=\sum_{k=0}^{\infty} \frac{(i \phi)^{k}}{k !}=1+i \phi-\frac{\phi^2}{2}-\frac{i \phi^3}{6}+\frac{\phi^{4}}{24}+\frac{i \phi^{5}}{120}-\cdots \end{equation} This particular form makes it clear that the even powers of $\phi$ remain real, whereas the odd powers of $\phi$ retain coefficients of $i$ and thus become complex quantities. If we group the real and imaginary terms separately (by factoring out $i$), we arrive at \begin{equation} \boxed{\exp (i \phi)=\underbrace{\left(1-\frac{\phi^{2}}{2} +\frac{\phi^{4}}{24} -\frac{\phi^{6}}{720} +\cdots\right)}_\text{Maclaurin expansion of cos($\phi$)} + \,i\underbrace{\left(\phi-\frac{\phi^{3}}{6} +\frac{\phi^{5}}{120}- \frac{\phi^{7}}{5040} +\cdots\right)}_\text{Maclaurin expansion of sin($\phi$)}} \end{equation} Miraculously, we've somehow regenerated the Maclaurin expansions of the sine and cosine functions. This forms the crux of what we now know as Euler's formula: \begin{equation} \mathrm{exp}(i\phi) = \mathrm{cos}(\phi) + i\,\mathrm{sin}(\phi), \end{equation} which for our purposes just provides an expeditious way of encoding the amplitude and phase of a diffracted wavevector. Sadly, this is frequently occluded by the ubiquitous $e^{i\phi}$ notation, which in this specific context is not particularly helpful. Explicitly articulating $\mathrm{exp}(i\phi)$ lets us better appreciate the underlying beauty of the provenance of Euler's formula.