28 Jul 2025

Singularities, Branch Points, and Branch Cuts

Removable singularities, poles, essential singularities, and the cuts that make multivalued functions single-valued.

bsc semester-ii mathematical-physics singularities branch-cuts

A point $z_0$ is an isolated singularity of $f$ if $f$ is analytic in a punctured disk

\[0<\lvert z-z_0\rvert<R\]

but is not analytic at $z_0$ itself.

Removable singularity

If $\lim_{z\to z_0}f(z)=L$ is finite, define $f(z_0)=L$. The singularity is then removed.

For

\[f(z)=\frac{z^2-1}{z-1},\]

factorization gives, for $z\ne1$,

\[f(z)=\frac{(z-1)(z+1)}{z-1}=z+1.\]

Therefore

\[\lim_{z\to1}f(z)=2.\]

Setting $f(1)=2$ makes the function analytic at $z=1$.

Pole

The point $z_0$ is a pole of order $m$ if

\[\boxed{ \lim_{z\to z_0}(z-z_0)^mf(z)=A, \qquad 0<\lvert A\rvert<\infty},\]

and no smaller positive integer has this property.

For

\[f(z)=\frac{1}{(z-z_0)^m},\]

$z_0$ is a pole of order $m$. The magnitude diverges as $\lvert z-z_0\rvert^{-m}$.

Essential singularity

An isolated singularity that is neither removable nor a pole is essential. For

\[f(z)=e^{1/z},\]

take $z=x>0$ and let $x\to0$: then $e^{1/x}\to\infty$. Along $z=x<0$, however, $e^{1/x}\to0$. The limit is not finite, so the singularity is not removable. For every fixed $m$,

\[z^me^{1/z}\to\infty\]

along $z=x\to0^+$, so no finite pole order exists. Therefore $z=0$ is essential.

Branch points and branch cuts

Write

\[z=re^{i(\theta+2\pi k)}.\]

Then

\[\sqrt z=\sqrt r\,e^{i(\theta+2\pi k)/2} =(-1)^k\sqrt r\,e^{i\theta/2}.\]

After one circuit around the origin, $\theta\mapsto\theta+2\pi$ and $\sqrt z$ changes sign. The value does not return to itself, so $z=0$ is a branch point.

Similarly,

\[\log z=\ln r+i(\theta+2\pi k)\]

changes by $2\pi i$ after one circuit. A branch cut removes a curve joining branch points or extending from a branch point to infinity, preventing such a circuit inside the chosen domain. For the principal branches, one common choice is the negative real axis, with

\[-\pi<\arg z<\pi.\]

An isolated singularity has a complete punctured disk around it. A branch point does not: a cut is introduced so a circuit cannot change the chosen value of \(\sqrt z\) or \(\log z\).

The cut is a convention; the multivalued behavior at the branch point is intrinsic.