Check if a function is entire

Question

I wonder why $f(z)=2^{z^2}$ is entire and $g(z)=z^{2z}\sin z$ is not analytic. For these functions, I cannot get the explicit real and imaginary parts. I wonder how in general to check functions like these are entire. Thanks.

Answer 1

Let $z = x + \mathrm{i}y$. \begin{align*} 2^{z^2} &= \mathrm{e}^{z^2 \ln 2} \qquad \text{visibly entire, but ...} \\ &= \mathrm{e}^{(x + \mathrm{i}y)^2 \ln 2} \\ &= \mathrm{e}^{(x^2 - y^2 + 2\mathrm{i}xy) \ln 2} \\ &= \mathrm{e}^{(x^2 - y^2)\ln 2} \mathrm{e}^{\mathrm{i} \cdot 2xy \ln 2} \\ &= \mathrm{e}^{(x^2 - y^2)\ln 2} \cos(2 xy \ln 2) + \mathrm{i} \mathrm{e}^{(x^2 - y^2)\ln 2} \sin(2 xy \ln 2) \end{align*}

This should be sufficient example to handle finding the real and imaginary parts of the other expression you gave.