Deriving Euler's Formula

Complex Analysis · Axiom Academy

EXAMPLE Deriving Euler's Formula Substitute an imaginary input into the exponential series and watch it split into cosine and sine. Starting only from the Taylor series of the exponential function, show that . We will plug into the series for e^x , use the cycle of powers of i , and watch the terms sort themselves into the series for and . Expand term by term. The even powers of carry no leftover i — they stack up into the series for . The odd powers each keep one factor of i — together they form . Nice work. You derived one of the most famous identities in mathematics straight from a power series — no new axioms, just the exponential series and the powers of i . The series do the work: e^x , , and each have a Taylor series, and those series stay valid for an imaginary input. Powers of i cycle: i^0, i^1, i^2, i^3 = 1, i, -1, -i and then it repeats every four — that pattern is what sorts the terms. Parity splits the parts: even powers of have no leftover i and rebuild ; odd powers each keep one i and rebuild . The result: , and setting gives Euler's identity . This is why the complex exponential is the natural language for rotation: is exactly the point on the unit circle.

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