complex exponential
The exponential function e^x grows; the trigonometric functions sin and cos oscillate. The complex exponential is the single function that contains both, unified by stepping off the real line. Feed it an imaginary number and growth turns into rotation; feed it a complex number and you get a spiral. It is arguably the most useful function in applied mathematics, because it converts oscillation and decay — the language of waves, circuits, and vibrations — into simple algebra.
The definition extends e^z by its power series, and the result factors beautifully: for z = x + i y, e^z = e^x (cos y + i sin y). The real part e^x sets the magnitude; the imaginary part y sets the angle. The heart of it is Euler's formula, e^{i theta} = cos theta + i sin theta, which says the unit complex number at angle theta is reached by exponentiating i theta. Setting theta = pi gives the celebrated e^{i pi} + 1 = 0. The complex exponential is its own derivative, never zero, and periodic with period 2 pi i — a fact that quietly causes all the multivaluedness of the complex logarithm.
In engineering and physics the complex exponential is the workhorse for anything that oscillates. A signal A cos(omega t + phi) is written as the real part of a complex amplitude times e^{i omega t}; phasors in AC circuit analysis are complex exponentials; the Fourier transform decomposes a signal into e^{i omega t} building blocks; and a damped wave is e^{(-gamma + i omega) t}, decay and oscillation in one stroke. Differentiating an exponential just multiplies by a constant, so differential equations for oscillators collapse into algebra.
Compute e^{1 + i pi/2}. Using e^z = e^x (cos y + i sin y) with x = 1 and y = pi/2: e^1 (cos(pi/2) + i sin(pi/2)) = e (0 + i) = e i. So the point sits on the imaginary axis at height e, the magnitude e^1 scaled and the angle pi/2 rotated — growth times rotation, just as the formula promises.
The real part scales the magnitude, the imaginary part rotates the angle.
Unlike the real exponential, the complex exponential is periodic, with e^{z + 2 pi i} = e^z, so it is many-to-one: distinct inputs can give the same output. This periodicity is exactly why the inverse, the complex logarithm, is multivalued and needs a branch cut.