The Complex Exponential Function

Core Idea

The complex exponential is defined by e^z = e^x (cos y + i sin y) for z = x + iy. It is entire (holomorphic everywhere), satisfies (e^z)' = e^z, and is periodic with period 2πi: e^z+2πi = e^z. The exponential is surjective but not injective; its image avoids 0.

How It's Best Learned

Verify that e^iy lies on the unit circle. Compute e^1+iπ/4 and e²⁺ⁱ⁰ to see how the real and imaginary parts of the exponent affect magnitude and direction.

Common Misconceptions

Assuming e^z behaves like the real exponential; it is periodic with period 2πi, not monotonic. Forgetting that |e^x+iy| = e^x independent of y, so e^z is not bounded as y varies.

Explainer

You already know Euler's formula from complex exponential form: e^iθ = cos θ + i sin θ, which places e^iθ on the unit circle at angle θ. The complex exponential generalizes this to all complex inputs. For z = x + iy, define e^z = e^x (cos y + i sin y). The real part x controls the *magnitude* (e^x), and the imaginary part y controls the *angle* (y radians from the positive real axis). So e^z is a point at distance e^x from the origin, rotated y radians counterclockwise.

This decomposition has a striking consequence: the magnitude |e^z| = e^x depends only on the real part of z, never on the imaginary part. The imaginary part shifts the angle but leaves the radius unchanged. As a result, e^z is never zero — e^x > 0 for all real x, so no choice of y can make the magnitude vanish. This is why the image of the complex exponential is ℂ \ {0}: it covers every nonzero complex number, but zero is permanently excluded.

The most important property distinguishing the complex exponential from its real counterpart is periodicity: e^{z + 2πi} = e^z for all z. Because e^2πi = cos(2π) + i sin(2π) = 1, adding 2πi to z completes a full 360° rotation — returning to the same image point. The real exponential is strictly increasing and injective; the complex exponential is surjective but not injective. The vertical strip 0 ≤ Im(z) < 2π is a fundamental domain: every nonzero complex number has exactly one preimage there. Shift vertically by any multiple of 2π and you land on the same output.

Since e^z satisfies (e^z)' = e^z and is holomorphic everywhere — meaning differentiable at every point in ℂ — it is an entire function with no singularities, no branch cuts, and no restricted domain. This makes it the simplest and best-behaved transcendental function in complex analysis. All the other elementary transcendental functions are built from it: the complex sine and cosine are cos z = (e^iz + e^−iz)/2 and sin z = (e^iz − e^−iz)/(2i), and the complex logarithm is the inverse of e^z. Understanding e^z is the foundation for everything that follows.