Is a function holomorphic, if it satisfies the Cauchy-Riemann equations?

Question

I am trying to prove, that $f(z)=(z+n)^r$, where $n$ and $r$ are positive integers can be differentiated for any $n$ and $r$.

I have proven, that for any $n$ the function $f(z)=z+n$ will be differentiable by the use of mathematical induction and know, that holomorphic functions, when multiplied keep on being holomorphic, yet am not sure how to show that the function will be holomorphic.

How could I do that?

Answer 1

You know that $f(z) = z + n$ is holomorphic, and that the product of two holomorphic functions is holomorphic.

Can you prove that $g(z) = (z + n)^2$ is holomorphic? Hint - $(z + n)^2 = (z + n) \times (z + n)$.

What about $h(z) = (z + n)^3$?

Now can you see how to get a proof of the case for $(z + n)^r$?