The Complex Logarithm Challenge

Complex Analysis · Axiom Academy

The Complex Logarithm Challenge Ask your calculator for log(−1) and it says "Error." Complex analysis has an answer — in fact, it has infinitely many. One input, a whole ladder of answers For real numbers, the logarithm undoes the exponential and gives back exactly one number: . Push that idea into the complex plane and something strange happens. The exponential e^z is periodic — going up by lands you right back where you started, — so asking "which z has e^z = w ?" no longer has a single answer. Watch the complex logarithm of one fixed point produce its answers. Each value is across (the real part, the same for all of them) and up the imaginary axis. They appear one rung at a time, stacked exactly apart — an endless ladder where the real had just one step. The real logarithm gives one answer; the complex logarithm gives infinitely many, each higher than the last. Pin down one answer: the principal value To get a single, well-defined answer we make a choice: take the angle in the range . That gives the principal value . Drag the point z around the plane (or use the sliders) and watch its size set the real part and its angle set the imaginary part. Cross the negative real axis and the angle — and the imaginary part of Log z — jumps from near to near . That seam is the principal branch's edge.

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