Isolated Point

An isolated singular point is a point in some punctured neighborhood in which the function $f(z)$ ${\ displaystyle f (z)}$ $f (z)$ is unambiguous and analytic , but at the point itself it is either not defined or not differentiable .

Classification

If a $a$ ${\ displaystyle a}$ $a$ Is an isolated singular point for $f(z)$ ${\ displaystyle f (z)}$ $f(z)$ then $f(z)$ ${\ displaystyle f (z)}$ $f(z)$ Being analytic in some punctured neighborhood of this point, it decomposes into a Laurent series converging in this neighborhood.

$f(z)=\sum _{n\in \mathbb {Z} }a_{n}(z-a)^{n}=\sum _{n=0}^{\infty }a_{n}(z-a)^{n}+\sum _{n=1}^{\infty }{\frac {a_{-n}}{(z-a)^{n}}}$ ${\ displaystyle f (z) = \ sum _ {n \ in \ mathbb {Z}} a_ {n} (za) ^ {n} = \ sum _ {n = 0} ^ {\ infty} a_ {n} (za) ^ {n} + \ sum _ {n = 1} ^ {\ infty} {\ frac {a _ {- n}} {(za) ^ {n}}}}$ $f(z)=\sum _{{n\in \mathbb{Z } }}a_{n}(z-a)^{n}=\sum _{{n=0}}^{{\infty }}a_{n}(z-a)^{n}+\sum _{{n=1}}^{{\infty }}{\frac {a_{{-n}}}{(z-a)^{n}}}$ .

The first part of this expansion is called the regular part of the Laurent series, the second - the main part of the Laurent series.

The type of the singular point of the function is determined by the main part of this expansion.

Isolated Point

Classification

See also

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