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Picard's Big Theorem says that if a function $f(z)$ has an isolated essential singularity at a point $w$, then in every neighborhood of $w$, $f(z)$ hits every complex number infinitely many times, with perhaps at most one exception.

Is there a version of Picard's theorem that goes something like this?

Let $V$ be an open disc (finite radius) such that $f(z)$ is holomorphic on $V - \lbrace w \rbrace$, and has an essential singularity at $w$. Let $0 \leq \theta < \phi < 2\pi$, and define $Cone(w,V,\theta,\phi)$ to be $V \cap \lbrace w + re^{i\varphi} \mid r > 0, \theta < \varphi < \phi \rbrace$. Think of this as a "pizza slice" of the disc $V$.

Is it true that there exists an $\alpha$ such that $f(z) = \alpha$ for infinitely many $z\in Cone(w,V,\theta,\phi)$?

I am certain this is true, though I cannot prove it.

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Picard's Big Theorem says that if a function $f(z)$ has an isolated essential singularity at a point $w$, then in every neighborhood of $w$, $f(z)$ hits every complex number infinitely many times, with perhaps at most one exception.

Is there a version of Picard's theorem that goes something like this?

Let $V$ be an open disc (finite radius) such that $f(z)$ is holomorphic on $V - \lbrace w \rbrace$. rbrace$, and has an essential singularity at$w$. Let$0 \leq \theta < \phi < 2\pi$, and define$Cone(w,V,\theta,\phi)$to be$V \cap \lbrace w + re^{i\varphi} \mid r > 0, \theta < \varphi < \phi \rbrace$. Think of this as a "pizza slice" of the disc$V$. Is it true that there exists an$\alpha$such that$f(z) = \alpha$for infinitely many$z\in Cone(w,V,\theta,\phi)$? I am certain this is true, though I cannot prove it. 2 Changed my question to be less trivial. Picard's Big Theorem says that if a function$f(z)$has an isolated singularity at a point$w$, then in every neighborhood of$w$,$f(z)$hits every complex number infinitely many times, with perhaps at most one exception. Can Is there a version of Picard's theorem be strengthened in the following fashionthat goes something like this? Let$V$be an open disc (finite radius) such that$f(z)$is holomorphic on$V - \lbrace w \rbrace$. Let$0 \leq \theta < \phi < 2\pi$, and define$Cone(w,V,\theta,\phi)$to be$V \cap \lbrace w + re^{i\varphi} \mid r > 0, \theta < \varphi < \phi \rbrace$. Think of this as a "pizza slice" of the disc$V$. Does Is it true that there exists an$f(z)$similarly hit every complex number (save perhaps one) \alpha$ such that $f(z) = \alpha$ for infinitely many times in $Cone(w,V,\theta,\phi)$?z\in Cone(w,V,\theta,\phi)\$?

I am certain this is true, though I cannot prove it.

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