Complex Powers

Complex Analysis · Axiom Academy

What does it even mean to raise one complex number to another? One formula — and the famous result that i^i is real. For real numbers, — exponent and logarithm undo each other. We promote that to a definition for complex and any complex w . The animation runs the machine left to right: take , multiply by w , then feed the result to . The one definition — everything else follows from it The catch lives entirely in . Its imaginary part is the angle of z , and an angle is only fixed up to full turns of . So is really a whole ladder of values, one per integer k — and each one, run through the machine, gives a different value of z^w . Each integer k is a different branch → a different z^w How many values you actually get depends on the exponent w : The shifts wrap around and cancel: z^n is single-valued (the ordinary power). Exactly n distinct values: the n roots . Finitely many values — exactly q of them. Infinitely many values — the branches never repeat. 3. The Famous Result: i^i Is Real Apply the machine to z=w=i . Since , its log is . Multiplying by i turns that imaginary number real (because ), and the of a real number is real. The animation drops the whole family onto the real axis. Every one is a positive real number. The principal value ( k=0 ) is . Now a base that stays positive and real. Here (no angle to worry about), so — a pure rotation . By Euler's formula it lands on the unit circle at angle radians. Watch the point swing out from 1 to that angle.

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