The Complex Logarithm

Complex Analysis · Axiom Academy

Inverting e^z : why has infinitely many values, and how we pick one. Write z in polar form and look for w = u + iv with e^w = z . Then , so matching size and angle gives e^u = |z| and . The modulus becomes the real part ; the argument becomes the imaginary part. — the real part comes from the length — the imaginary part comes from the angle The angle of z is only defined up to full turns: and point the same way. So for a fixed z there is not one logarithm but a whole ladder of them, each differing by — because e^ w and are equal. Every value shares the same — the ladder is a single vertical line in the w -plane. Climbing one rung adds a full turn: , i.e. . Each integer k names one rung; k = 0 is the rung closest to the real axis. e^z is -periodic, so its inverse must be many-valued — there is no way around it. Since |1| = 1 and , we get . Not just 0 — every multiple of , because for all integers k . To get a single-valued function we keep exactly one rung of the ladder — the one whose angle lands in the strip . That choice is the principal value , written with a capital L : , where is the principal argument. The strip: the horizontal band contains exactly one value of for every — that is the value returns. Lowercase vs. capital: is the full multi-valued set; is the single principal value living inside the strip. You've built the complex logarithm from e^w = z , seen why it carries infinitely many values, and singled out the principal value inside the strip .

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