Checking Holomorphicity of functions?

Question

I was checking the Holomorphicity of $f(z) = (z-3)^i$ which I wrote it as $e^{i log(z-3)}$, so as $Log(z-3) $ is holomorphic over $\Bbb{C}\setminus(-\infty,3)$ so is $(z-3)^i$. But I am thinking about $f(z) = \frac{\sin(z)}{z^3 + 1}$ and $f(z) = \operatorname{Log}(z-2i+1)$?, I thought of using Cauchy Riemann equations to rule out Holomorphic, like if it doesnot satisfy CR eqaution then it is not holomorphic, but it seems it may run into complicated calculations?

EDIT -

Just out of curiosity, I thought of this - As $(z-3)^i$ is holomorphic over $\Bbb{C}\setminus(-\infty,3)$, then what can we say about the Holomorphic nature of $i^{z-3}$, just the positions interchanged!!!

Answer 1

In both cases, the functions are holomorphic because they can be obtained from holomorphic functions using sums, subtractions, multiplications, divisions and composition. Each of these operations preserve being holomorphic.

Answer 2

$i^{z-3} = \exp((z-3) \log(i)) $ is holomorphic on all of $\mathbb C$.