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# 8.5: Singularities

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## Definition: Singular Function

A function $$f(z)$$ is singular at a point $$z_0$$ if it is not analytic at $$z_0$$

## Definition: Isolated Singularity

For a function $$f(z)$$, the singularity $$z_0$$ is an isolated singularity if $$f$$ is analytic on the deleted disk $$0 < |z - z_0| < r$$ for some $$r > 0$$.

## Example $$\PageIndex{1}$$

$$f(z) = \dfrac{}{}$$ has isolated singularities at $$z = 0$$, $$\pm i$$.

## Example $$\PageIndex{2}$$

$$f(z) = e^{1/z}$$ has an isolated singularity at $$z = 0$$.

## Example $$\PageIndex{3}$$

$$f(z) = \log (z)$$ has a singularity at $$z = 0$$, but it is not isolated because a branch cut, starting at $$z = 0$$, is needed to have a region where $$f$$ is analytic.

## Example $$\PageIndex{4}$$

$$f(z) = \dfrac{1}{\sin (\pi /z)}$$ has singularities at $$z = 0$$ and $$z = 1/n$$ for $$n = \pm, \pm 2, ...$$ The singularities at $$\pm 1 /n$$ are isolated, but the one at $$z = 0$$ is not isolated. Figure $$\PageIndex{1}$$: Every neighborhood of 0 contains zeros at $$1/n$$ for large $$n$$. (CC BY-NC; Ümit Kaya)

8.5: Singularities is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff (MIT OpenCourseWare) via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.