Taylor Series for sin(z), cos(z)

Complex Analysis · Axiom Academy

EXAMPLE Taylor Series for sin(z) and cos(z) Deriving the entire-function expansions of sine and cosine from their derivative cycle Both and are entire (analytic on all of ), so each equals its Taylor series about z = 0 . Use the Maclaurin formula to find both expansions and their radius of convergence. Nice work — you derived the Taylor series for both sine and cosine straight from their derivative cycle. Cyclic derivatives: differentiating four times returns to , so f^ (n) (0) repeats with period 4 — the source of the alternating signs. Odd vs. even: keeps only the ODD powers while keeps only the EVEN powers — exactly the parity of each function. Entire functions: the factorial denominators crush every term, so the ratio test gives — both series converge for ALL complex z . Euler's bridge: adding the two series with an i recovers — the exponential and the trig functions are one object in . The same Maclaurin machine that handles e^z produces and — and ties all three together.

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