Why Does Kirchoff's Loop Rule Apply to AC Circuits Using Complex Numbers?

[ougoah]

I was wondering why Kirchoff's loop rule holds for the complex representations of voltage drops in AC circuits. That is, sum of complex voltage drops = 0.

I tried working it out from Kirchoff's rule in DC, but am not sure how to proceed properly.

 

[robb_]

The loop rule is a statement of conservation of energy.