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This is in reference to this $f:\mathbb{C}\rightarrow\mathbb{C}$ entire function and $f(z)=u(x)+iv(y)$ then $f$ is a polynomial

Since $f$ is entire it satisfies Cauchy Riemann so $u_x=v_y \& 0=u_y=v_x$

But how does that show $f$ is a polynomial?

Any help please

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$u_x$ and $v_y$ are functions only of $x$ and of $y$ respectively, so the only way they’re equal is if they are constants.

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  • $\begingroup$ And a constant function as long as it is not zero is a polynomial. $\endgroup$ – Erock Brox Jan 20 at 6:26
  • $\begingroup$ @ErockBrox ??? $0$ is certainly a polynomial. $\endgroup$ – David C. Ullrich Jan 20 at 18:42

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