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$f$ and $g$ are entire function such that $|f^2+g^2|=1$ Then which of the following are correct?

  1. $f$ and $g$ are constant.

  2. $f$ and $g$ are bounded.

  3. $f$ and $g$ have no zeroes on unit circle.

  4. $ff'+gg'=0$

Well What I do is let $h(z)= f^2(z) + g^2(z)$ then clearly $h(z)$ is bounded entire hence constant by Liouville theorem, hence (4) is correct right?

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    ah I see, then $sin(z)$ and $cos(z)$2012-06-05

1 Answers 1

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The first option is wrong since for instance, $f(z) = \cos(z)$ and $g(z) = \sin(z)$.

The second option is wrong since for instance, $f(z) = \cos(z)$ and $g(z) = \sin(z)$.

The third option is wrong since for instance, $f(z) = \cos(\pi z)$ and $g(z) = \sin(\pi z)$. Note that $g(1) = 0$.

The fourth option is correct since $f^2 + g^2$ is again entire and by Liouville's theorem, we have that $f^2 + g^2 = c$, where $c$ is a constant such that $\lvert c \rvert = 1$. Hence, taking the derivative gives us $ff' + gg' = 0$.

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    Extremely Thank you for nice answer!2013-06-02