Properties of exp(z)

Complex Analysis · Axiom Academy

The complex exponential keeps the algebra you know — and gains a brand-new one: it spirals, repeats every , and never hits zero. The single most important rule survives the jump to complex exponents: adding the exponents multiplies the values . Writing each factor in polar form makes the geometry pop — the modulus of the product is the product of the moduli, and the argument of the product is the sum of the arguments. Split each exponent into real and imaginary parts and regroup: . The real parts add in the size e^ x_1+x_2 ; the imaginary parts add in the angle y_1+y_2 . 2. Modulus, Argument — and Never Zero Reading directly: the part out front, e^ x , is the length , and the part inside, y , is the angle . So the size of e^z depends only on the real part of z , and the direction depends only on the imaginary part. Because |e^ z |=e^ x and the real exponential is strictly positive, |e^ z |>0 for every z . A modulus that is never zero means everywhere — the image always sits on a circle around the origin, never on it. 3. Periodicity: Step Up by 2πi, Land in the Same Place This is what the real exponential never does. Adding to the exponent leaves the value completely unchanged , because the extra angle is one full turn. Slide z straight up the imaginary axis by and its image e^ z comes right back to where it started. The reason is the addition law plus one full rotation: , since .

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