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Mathematics LibreTexts

8.5: Singularities

  • Page ID
    6520
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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.

    002 - (8.5.4).svg
    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.