[RCQ/CV-05001] Decoding analyticity of rational functions --- Short Questions

Decoding analyticity of rational functions .

Properties of derivative.

Complete the following sentences by indicating which property is used.

1. The function $z$ is analytic, therefore $z^2,z^3,\cdots, z^n$ are all analytic because____________________________
2. As $z,z^2,\cdots,z^n$ are analytic at all points in complex $z$ plane, all polynomials are analytic because_____________________
3. All rational functions of the form $P(z)/Q(z)$ are analytic at all points except where __________ because___________________
4. Composite function, (function of a function) $f(g(z))$ is analytic everywhere the following property holds.