Complex Trigonometric Functions

Complex Analysis · Axiom Academy

LESSON Complex Trigonometric Functions Sine and cosine, rebuilt from the exponential, so they accept any complex number — and behave in one surprising new way. 1. Built From Two Exponentials Euler's formula and its mirror, , are two equations in the two unknowns and . Add them and the terms cancel; subtract them and the terms cancel. Solving gives definitions that never mention a triangle — so they work for any complex z . cosine = the averaged sum of the two phasors sine = the averaged difference, divided by 2i 2. It Extends the Real Cosine — Identities Intact Plug a real number into the new and you recover the ordinary cosine exactly: the definition extends , it does not replace. And because everything is built from e^ iz , the algebra that proved the old identities still goes through unchanged — the Pythagorean identity, the period, even/odd symmetry, and the derivatives all survive. holds for every complex z (verified by substituting the definitions). and — the period carries over untouched. and , exactly as on the real line. Solving over the complex numbers gives only — the same real zeros as before. Extending to adds no new roots to or ; every zero still sits on the real axis. 3. Unbounded Off the Real Axis Here is what is genuinely new. Walk straight up the imaginary axis, z = iy , and the exponentials e^ iz =e^ -y and e^ -iz =e^ y stop oscillating and start growing . The result: and , both of which run off to infinity. The bound is purely a real-axis phenomenon.

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