If constant is analytic in a domain , then is also a domain. If only has poles in , this conclusion still holds.

holomorphic at , where it has a zero point of order , then there is a neighborhood of and a neighborhood of such that every point in except itself has precisely distinct inverse images in . If has pole of order at , conclusion still holds.

Suppose has pole , . Observe that carries into the disc . If , consider both transformations , .

If is univalent(injective) on a domain , then all the zeros and poles of in are simple.