Applications of complex numbers to solve non-complex problems Recently I asked a question regarding the diophantine equation $x^2+y^2=z^n$ for $x, y, z, n \in \mathbb{N}$, which to my surprise was answered with the help complex numbers. I find it fascinating that for a question which only concerns integers, and whose answers can only be integers, such an elegant solution comes from the seemingly unrelated complex numbers - looking only at the question and solution one would never suspect that complex numbers were lurking behind the curtain!
Can anyone give some more examples where a problem which seems to deal entirely with real numbers can be solved using complex numbers behind the scenes? One other example which springs to mind for me is solving a homogeneous second order differential equation whose coefficients form a quadratic with complex roots, which in some cases gives real solutions for real coefficients but requires complex arithmetic to calculate.
(If anyone is interested, the original question I asked can be found here: $x^2+y^2=z^n$: Find solutions without Pythagoras!)
EDIT:
I just wanted to thank everyone for all the great answers! I'm working my way through all of them, although some are beyond me for now! 

$\newcommand{\SLp}[1]{\mathrm L^{#1}}\newcommand{\norm}[1]{\lVert#1\rVert}$
The famous Riesz-Thorin Interpolation Theorem:

Let $(X,\mathcal M,\mu),(Y,\mathcal N,\nu)$ be measure spaces, $p_0,p_1,q_0,q_1 \in [1,\infty]$. (If $q_0 = q_1 = \infty$, $\nu$ is also required to be semi-finite) Define, for $t \in \left]0,1\right[$,
  $$ \frac{1}{p_t} = \frac{1-t}{p_0} + \frac{t}{p_1}, \qquad \frac{1}{q_t} = \frac{1-t}{q_0} + \frac{t}{q_1}
$$
  If $\Phi\in\operatorname{Hom}(\SLp{p_0}(\mu) + \SLp{p_1}(\mu),\SLp{q_0}(\nu) + \SLp{q_1}(\nu))$ such that $\Phi\restriction_{\SLp{p_0}},\Phi\restriction_{\SLp{p_1}}$ are bounded. Then
  $$ \norm{\Phi}_{\SLp{p_t} \to \SLp{q_t}} \leq \norm{\Phi}_{\SLp{p_0} \to \SLp{q_0}}^{1-t}\norm{\Phi}_{\SLp{p_1} \to \SLp{q_1}}^t
$$

which does not seem to be related to complex analysis, is usually proved using the Three Lines Lemma (a.k.a. Hadamard Three Line Theorem):

Let $f$ be a bounded continuous complex function on the strip $E = \{z \in \mathbb{C}\mid a \leq \Re z \leq b\}$ that is holomorphic on $E^\circ$. Define $m(x) = \sup_{y \in \mathbb{R}} |f(x + iy)|$. Then $m(x)$ is logarithmically convex on $[a,b]$.

