Extending the Exponential Function

Complex Analysis · Axiom Academy

Extending the Exponential Function We know e^x for real x . Here is what happens when the exponent goes complex — and why the answer is a scale and a spin. From a line to the whole plane The real exponential e^x only ever lands on the positive part of one line: , , always a positive real number. So what could e^ i possibly mean? The trick is to feed in a complex exponent z = x + iy and let the algebra of exponents carry us off the line. Split the exponent and the exponent rule does the rest: . Read that right-to-left as two moves. The real part x sets a magnitude e^ x ; the imaginary part y is an angle you rotate through. Watch the point start at 1 , grow out to e^ x , then swing through the angle y — landing on . A scale, then a spin — that is all the complex exponential ever does. Drag the two sliders. The x slider sets the distance from the origin, e^ x ; the y slider sets the angle . The dot is always , and the readout is computed live from e^ x , , — nothing is pre-baked. Notice and , and that setting gives e^ 0 =1 . Pure rotation when x=0 (the dot rides the unit circle), pure growth when y=0 (it stays on the real axis). You just saw and drove the definition: turns the real part into a magnitude and the imaginary part into an angle. That single move — scale then spin — is the bridge from the line to the plane, and it is where Euler's formula comes from as the special case x=0 .

This is the written version of the interactive lesson above. See the full Complex Analysis course.