Computing log(-1)

Complex Analysis · Axiom Academy

Take the logarithm of a negative number with the formula log z = ln|z| + i(arg z + 2πk). Compute : find all of its values, and the principal value . In the complex plane -1 sits on the unit circle (so |-1| = 1 ), one step left of the origin along the negative real axis. The angle from the positive real axis around to it is — that angle is its argument. Nice work — you took the logarithm of a negative number, something the real-valued can never do. The formula: splits into a real part and an imaginary part built from the angle. It is multi-valued: because k ranges over all integers, gives — infinitely many values. Principal value: taking k = 0 (so in ) gives . And is exactly Euler's identity read backwards: , i.e. .

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