Calculus Small Changes, Big Ideas

Section 20.5 Singularities and Laurent series

• A function is analytic on \(D\) if its power series converges to the function. Cauchy’s integral theorems imply that a function is holmorphic if and only if it is analytic and thus we can use the terms interchangeably.

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• If a function \(f\) is not holomorphic at \(z=z_{0}\text{,}\) we say \(f\) has a singularity at \(z_{0}\text{.}\)

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• If \(f\) has a singularity at \(z=z_{0}\text{,}\) we can represent it with a Laurent series, in which we allow negative powers of \((z-z_{0})\text{:}\)

  \begin{equation*} f(z)=\sum_{k=-\infty}^{\infty}a_{k}(z-z_{0})^{k}=\sum_{k=1}^{\infty}a_{-k}(z-z_{0})^{-k}+\sum_{k=0}^{\infty}a_{k}(z-z_{0})^{k} \end{equation*}

  The part with the negative powers is called the principal part and the part with the nonnegative powers is called the analytic part.

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• If \(f\) is holomorphic within the annular domain \(D\) defined by \(r<|z-z_{0}|<R\text{,}\) then \(f\) has a valid Laurent series representation, with the coefficeints given by

  \begin{equation*} a_{k}=\frac{1}{2\pii\ii}\oint_{C}\frac{f(z)}{(z-z_{0})^{k+1}}\dd{z}\text{,} \end{equation*}

  where \(C\) is a simple closed curve that lies entirely within \(D\) and has \(z_{0}\) in its interior.

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• Annular domains don’t have to be ring-shaped... they could be circles, complements of circles, or the entire complex plane.

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• Manipulations of power series by writing things in terms of \(z-z_{0}\) for some given \(z_{0}\)

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• Types of singularities:

  • If the principal part has no terms, the singularity is a removable singularity. In this case we can always redefine a single point and make it holomorphic.

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  • If the principal part has \(n>0\) terms, the singularity is a pole of order \(n\text{.}\) A pole of order \(1\) is a simple pole.

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  • If the principal part has infinitely many terms, the singularity is an essential singularity.

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• Zeros and poles:

  • A function \(f\) that is holomorphic in some disk \(|z-z_{0}|<R\) has a zero of order \(n\) if and only if \(f\) can be written

    \begin{equation*} f(z)=(z-z_{0})^{n}\phi(z) \end{equation*}

    where \(\phi\) is holomorphic at \(z=z_{0}\) and \(\phi(z_{0})\neq 0\text{.}\)

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  • A function \(f\) that is holomorphic in a punctured disk \(0<|z-z_{0}|<R\) has a pole of order \(n\) if and only if \(f\) can be written

    \begin{equation*} f(z)=\frac{\phi(z)}{(z-z_{0})^{n}} \end{equation*}

    where \(\phi\) is holomorphic at \(z=z_{0}\) and \(\phi(z_{0})\neq 0\text{.}\)

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  • If \(g\) and \(h\) are analytic at \(z=z_{0}\text{,}\) and \(h\) has a zero of order \(n\) at \(z=z_{0}\) and \(g(z_{0})\neq 0\text{,}\) then \(f(z)=g(z)/h(z)\) has a pole of order \(n\) at \(z=z_{0}\text{.}\)

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  • A function is meromorphic if it is analytic except at some set of isolated poles.

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• Would this be the right place to talk about limits at infinity?

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