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.
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@conditionalMethod Thank you! – Analyst_311419 Oct 25 '19 at 03:11
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See https://math.stackexchange.com/questions/1942685/proving-gz-z2z-sinz-is-not-an-entire-function?rq=1 – Brian Moehring Oct 25 '19 at 03:14
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$z^{2z}$ is not a well defined function from $\mathbb C$ to $\mathbb C$, so the question of it being an entire function does not arise. – Kavi Rama Murthy Oct 25 '19 at 05:45
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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.
Eric Towers
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