Let $f(z)$ be an even entire function, assume that $f'(z)$ is also entire, show that $g(z)=f(\sqrt z) $ obey the Cauchy-Riemann equations in the complex plane, and $g_x(z)=-ig_y(z)=\dfrac{f^\prime(\sqrt z) }{2\sqrt z}$ if $z\ne0$, and $g_x(0)=-ig_y(0)=\dfrac{f''(0)}{2}$
here, $\sqrt z$ use nonpositive real axis $(-\infty, 0]$ as a branch cut; $g_x(z)=u_x(z)+iv_x(z),g_y(z)=u_y(z)+iv_y(z)$
Only the limits and continuity of functions, the Cauchy-Riemann equations can be allowed, the complex integration and Taylor series are forbidden
I think, as the problem Complex valued function $\cos\sqrt z$, if we permit the power series, the work will be easy
$$f(z)=\sum_{n=0}^\infty \frac{f^{(2n)}(0) z^{2n}}{(2n)!}, $$
then, $f(\sqrt z) $ is entire.
But, with the definition of analytic , the Cauchy-Riemann equations and $\sqrt z$ is continuous in $\Bbb C\setminus(-\infty,0]$, I cannot solve the problem.
Thank you very much for your help
