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Could someone possible explain the differences between each of these;

Singularities, essential singularities, poles, simple poles.

I understand the concept and how to use them in order to work out the residue at each point, however, done fully understand what the difference is for each of these

As far as i understand a simple pole is a singularity of order $1$?

then we have poles of order $n$ which aren't simple?

not too sure about essential singularity

JJJ
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smith
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3 Answers3

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The point $z_{0}$ is an isolated singularity of $f(z)$ if $f(z)$ is analytic in $0 \lt |z-z_{0}| \lt r$ (a circle of radius r centered at $z_{0}$ with the point $z_{0}$ punched out). If one expands a function $f(z)$ in a Laurent series about the point $z_{0}$, $$f(z) = \sum\limits_{k=-\infty}^{\infty} a^{k} (z-z_{0})^{k}$$ we can classify isolated singularties into 3 cases:

  1. If there are no negative powers of $z-z_{0}$, then $z_{0}$ is a removable singularity and the Laurent series is a power series.

    • Example: $$\frac{\sin(z)}{z} = 1 - \frac{z^{2}}{3!} + \frac{z^{4}}{5!} - ...$$ has a removable singularity at 0.

  2. $f(z)$ has a pole of order m at $z_{0}$ if m is the largest positive integer such that $a_{-m} \ne 0$. A pole of order one is a simple pole. A pole of order two is a double pole, etc.

    • Example: $$f(z) = \frac{1}{(z-3i)^{7}}$$ has a pole of order 7 at $z=3i$

  3. If there are an infinite number of negative powers of $z-z_{0}$, then $z_{0}$ is an essential singularity.

    • Example: $$\mathrm{e}^{1/z} = 1 + \frac{1}{z} + \frac{1}{2!z^{2}} + ...$$ has an essential singularity at 0.
Saketh Malyala
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poweierstrass
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There are three kinds of singularities.

Removable singularity, which can be extended to a holomorphic function over that point.

Poles, which is removable after multiplying some $(z-a)^n$. The smallest $n$ is called the order of the pole, when $n=1$, it is called simple.

Essential singularity: neither of the above. For example $g(z)=e^{1/z}$ since $|g(z)z^l|$ is never bounded near $0$.

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    could you explain what a holomorphic function is? – smith May 16 '15 at 03:02
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    @smith See here: http://en.wikipedia.org/wiki/Holomorphic_function – bjd2385 May 16 '15 at 04:14
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    I like to call removable singularities "poles of order $0$" – reuns Jul 08 '16 at 03:48
  • @reuns high five for removable singularities as poles of order 0. are essential singularities poles of order negative infinity? and zeroes are poles of order positive infinity? – BCLC Oct 27 '21 at 20:51
  • @bjd2385 perhaps OP uses the definitions of analytic and holomorphic differently? – BCLC Oct 27 '21 at 20:52
  • @reuns and Qixiao, do you know the real analysis versions? – BCLC Oct 27 '21 at 20:53
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    @JohnSmithKyon Sure. Removable means that $f$ extends to an analytic function. Pole means that $1/f$ is analytic. Essential singularity means that... $f$ is given by a Laurent series around the point and (can you finish?) – reuns Oct 28 '21 at 01:16
  • @reuns oh thanks those are good ones! why don't you post as an answer? but wait...what's laurent series for $f: \mathbb R \to \mathbb R$? i don't think i learned this in real analysis but...i guess there are indeed laurent series in real analysis like $\frac1x$ itself is its own laurent series? – BCLC Oct 28 '21 at 11:28
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Singularity:

$\quad$ A point $a$ is said to be a singular point of a function $f$ if

i) f is not analytic at $a$ and

ii) if we can find a neighborhood of $f(a)$ such that there exists a point $b$ in which $f$ is analytic.

Essential Singularity:

$\quad$ A point $a$ is said to be a essential singular point of a function $f$ if

i) f is not analytic at $a$ and

ii) if every neighborhood of $f(a)$ contains infinte number of points in which $f$ is analytic.

Poles:

a point $a$ is said to be a pole if

i)it is a essential singularity and

ii)$\lim_{z \to a} f(z) = \infty$

A pole of order 1 is simple pole and double pole if it is order 2.

Sam Christopher
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  • well many people say a singularity for a point $a$ where $f(z)$ isn't analytic but it is analytic on an open $U$ with $a$ on the boundary (so $U$ doesn't have to contain a neighborhood of $a$, example a branch point) – reuns Oct 11 '16 at 01:13