What should e^z even mean?
On the real line e^x has a clear meaning, but "e raised to a complex power" is not something you can build by repeated multiplication — you cannot multiply e by itself "i times." So we do not define the complex exponential by analogy with arithmetic; we define it by the one property of e^x that survives the trip to the plane: its power series. We set e^z = sum z^n / n! over n from 0, the same series you met for e^x, now fed a complex z. This series converges for every z in the plane, so e^z makes sense everywhere.
Because the series converges everywhere, e^z is differentiable everywhere — it is an entire function, holomorphic on the whole plane with no bad points at all. And it keeps the headline property we want: term-by-term differentiation shifts the series back onto itself, so (e^z)' = e^z. The exponential is still its own derivative, just now over the plane. It also keeps the addition law e^(z + w) = e^z e^w, which falls straight out of multiplying the two series; this single identity is the engine behind almost every computation in this rung.
Splitting e^z into modulus and angle
The series is the honest definition, but it is not how you compute. Write z = x + i y and use the addition law: e^z = e^(x + i y) = e^x e^(i y). The first factor e^x is the ordinary real exponential, a positive number. The second factor e^(i y) is handled by Euler's formula, e^(i y) = cos y + i sin y, which you met on the previous rung. So the complex exponential factors completely into a real growth term and a pure rotation.
z = x + i y e^z = e^x ( cos y + i sin y ) |e^z| = e^x (the modulus, always positive) arg e^z = y (the argument, an angle)
Read off the geometry directly: the modulus |e^z| equals e^x, and the argument arg e^z equals y. In other words, e^z is exactly the point sitting at distance e^x from the origin, at angle y. This is precisely the polar form of a complex number, handed to you for free — which is why the exponential is the natural language for writing any complex number as r e^(i theta). A tiny example: e^(1 + i pi/2) = e^1 (cos(pi/2) + i sin(pi/2)) = e times i = e i, sitting straight up the imaginary axis at height e.
The surprise: e^z repeats
Now the fact that has no real-variable shadow. Add 2 pi i to the input: e^(z + 2 pi i) = e^z e^(2 pi i). But e^(2 pi i) = cos(2 pi) + i sin(2 pi) = 1 + 0 i = 1. So e^(z + 2 pi i) = e^z. The complex exponential is periodic, with period 2 pi i. Step straight up (or down) the imaginary direction by 2 pi, and you land on exactly the same output. This is the periodicity of the exponential, and it is genuinely strange the first time you see it: the function that on the real line never once repeats becomes, in the plane, a function that repeats endlessly.
The reason is no mystery once you have the polar picture. Changing y by 2 pi changes the argument by a full turn — and a full turn brings every point back to where it started. The growth factor e^x is untouched because x did not change. So adding 2 pi i spins the output once all the way around and sets it down on itself. The real exponential never noticed periodicity only because the real line gives you no room to rotate; the imaginary direction is exactly that missing room.
Two more facts you must not forget
First: e^z is never zero. Its modulus is |e^z| = e^x, and the real exponential e^x is strictly positive for every real x — it can shrink toward 0 as x runs to negative infinity, but it never reaches 0. Since the size of the output is never 0, the output itself is never the point 0. This the exponential is never zero looks like a small remark, but it is load-bearing: it is exactly why the logarithm can be defined on the whole punctured plane, and why log z must skip the single point 0.
Second, a warning about reach: e^z does not cover the plane the way you might expect. Because |e^z| = e^x can be any positive number and arg e^z = y can be any angle, the outputs sweep out every non-zero complex number — but only the non-zero ones, and each of them is hit infinitely often (once per strip). So the exponential maps the plane onto the punctured plane, infinitely-to-one. Contrast this with the real case, where e^x is one-to-one and misses every non-positive number. The same symbol, a completely different reach.
These two facts together set up the very next chapters. Infinitely-to-one means the exponential has no single, honest inverse — its undoing, the complex logarithm, is forced to be multivalued, one value per strip. And never-zero means that inverse is defined for every target except 0. You can already feel the shape of guide 3 forming: log z will be "which strip did this output come from?", a question with infinitely many right answers.
Where periodicity ripples outward
Almost every elementary function in this rung is built from e^z, so the 2 pi i periodicity is contagious. The complex trigonometric functions are defined through the exponential — cos z = (e^(i z) + e^(-i z))/2 and sin z = (e^(i z) - e^(-i z))/(2 i) — so they inherit a periodicity of 2 pi along the real direction, just as you expect. But because they are exponentials in disguise, they also inherit unbounded growth in the imaginary direction, the surprise of the very next guide.
Here is a quick, reliable routine for using the exponential in practice — turning any z into the size-and-angle form, and back.
- Given z = x + i y, split it: the real part x and the imaginary part y.
- Compute the modulus |e^z| = e^x (a positive real number) and the argument arg e^z = y (an angle).
- Assemble the value: e^z = e^x (cos y + i sin y). Reduce y modulo 2 pi if you only need where the output sits.
- Going backward, to write a given non-zero w as e^z, take e^x = |w| (so x = ln|w|) and y = arg w — but remember y is only determined up to adding 2 pi, the multivaluedness that becomes the logarithm.