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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.
show/hide this revision's text 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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