Evaluating eᶻ at z = 2+iπ

Complex Analysis · Axiom Academy

Splitting the complex exponential into its real factor and Euler's formula. Evaluate the complex exponential , leaving the answer in the simplest exact form. In the complex plane the result has modulus and argument , so it points straight along the negative real axis — landing at . Nice work — you evaluated a complex exponential by splitting it into a real magnitude and a rotation. Split the exponent: . The real part becomes a magnitude, the imaginary part a rotation. Euler's formula does the rotation: , so . Geometry: the result has modulus and argument — it sits on the negative real axis. This is Euler's identity in disguise: because , the factor e^2 simply gets flipped to the negative real axis.

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