The Complex Exponential

Complex Analysis · Axiom Academy

LESSON The Complex Exponential What e^z means when z is complex — a scaling and a rotation, read straight off z = x + iy . 1. e^z Is a Scaling and a Rotation Split the exponent and apply the rule e^ a+b =e^a e^b : . The first factor e^x is a positive real number — it stretches . The second factor e^ iy is Euler's formula , a point on the unit circle — it rotates . So e^z takes 1 , scales it by e^x , and turns it through angle y . e^x sets the distance from the origin Hold x fixed and push y upward. Since y is the angle of the output, adding to y winds the image exactly once around the circle of radius e^x and lands it right back where it began. So — the complex exponential is periodic with period , something the real exponential could never do. The input slides straight up a vertical line; only the angle y changes, by a full turn. x is unchanged, so |e^z| = e^x stays fixed — the image rides one circle. Angle y sweeps , so the output winds around once and returns to its start. End point equals start point: for every z . The magnitude is |e^z| = e^x , and e^x > 0 for every real x . A modulus that is always positive can never be 0 , so for all z — the exponential misses the origin entirely. The two factors split the work cleanly. A vertical line ( x constant) fixes the radius e^x while y runs — its image is a circle . A horizontal line ( y constant) fixes the angle y while x runs the radius from 0 to — its image is a ray from the origin.

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