Zeros, Singularities, Residues: Singularites
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The following theorem is the strong form of the Casorati Weierstrass theorem, tells the function nature in the vicinity of essential singularity. But its proof demands the concept of normal family and Montel-Caratheodory thereom which is beyond the scope of this introductory course.
(Great) Picard Theorem: Suppose that $f(z)$ has an essential singularity at $z_{0}$. In any neighborhood $N(z_{0})$ of $z_{0}$, the function $f(z)$ assumes every complex number as a value, with possibly one exception, an infinite number of times.

Singularities at Infinity: Suppose that $f$ is analytic in $\vert z \vert > R$. Then,

1. $f$ has a removable singularity at $z = \infty$ if $f(1/z)$ has a removable singularity at $z = 0$.
2. $f$ has a pole of order $m$ at $z = \infty$ if $f(1/z)$ has a pole of order $m$ at $z = 0$.
3. $f$ has an essential singularity at $z = \infty$ if $f(1/z)$ has an essential singularity at $z = 0$.

Examples: The function MATH has a removable singularity at $z = \infty$. A polynomial MATH with $n \ge 1$ and $a_{n} \neq 0$ has a pole of order $n$ at $z = \infty$. Every entire transcendental function MATH for all $z \in \QTR{Bbb}{C}$ (for example, $e^{z}$, $\sin z$, $\cos z$) have an essential singularity at $z = \infty$.

   
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