Residue Theorem and Homologous to zero

Question

This is a very basic question and I couldn't find it posted yet but here it goes; The Residue Theorem states that if $f:G\to \mathbb{C}$ is analytic on $G$- a region and $f$ has isolated singularities $b_1,...,b_k$ and $\gamma$ is homologous to 0, then $$\displaystyle\int_{\gamma}f=2\pi i\sum_1^kn(\gamma;b_s)\mathrm{Res}(f;b_s).$$ To my understanding, if $\gamma$ "wraps" around an isolated singularity, $b_i$, (say 1 time), then $\gamma$ would not be homologous to 0 since $n(\gamma;b_i)=1$ in this case ($b_i\in \mathbb{C}-G$). I think I need a better picture of the situation in this case. Thank you for your help!

Answer 1

To be homologous to zero, means we can deform the closed curve $\gamma$ to a point while staying in the region.

Pretend $H$ is a singularity. Then $C_2$ can't be shrunk to a point instead it can be shrunk only to the boundary of $H$. If we shrink $C_2$ to a point (homologous to zero), we leave the region $K$. However, $C_1$ encircles no deleted neighborhoods or singularities so $C_1$ is homologous to zero.

Now the winding number is simply the number of times you circle the singularity.

---

Cauchy theorem states:

If $f(z)$ is analytic in $\Omega$, then $$ \int_{\gamma}f(z)dz = 0\tag{1} $$ for every cycle $\gamma$ which is homologous to zero in $\Omega$.

Residue theorem:

Let $f(z)$ be analytic except for isolated singularities $z_i$ in a region $\Omega$. Then $$ \frac{1}{2i\pi}\int_{\gamma}f(z)dz = \sum_in(\gamma,i)\operatorname{Res}\{f(z);z_i\}\tag{2} $$ for any cycle $\gamma$ which is homologous to zero in $\Omega$ and does not pass through and of the points $z_i$.

If every path in the residue theorem was homologous to zero, then equation $(1)$ would be equation $(2)$ and every integral you face would be zero. The winding number around the singularities will be $\pm 1$ depending on the direction of travel. If the curve is similar to $C_1$ in the image above, it is homologous to zero and we have equation $(1)$.

Answer 2

Your idea is mistaken. The fact that a curve has nonzero winding number around a single point gives no information about it being nullhomolgous. Rather, you want the following

A curve $\gamma$ in an open set $G$ of the plane is nullhomologous in $G$ if and only if $W(\gamma,z)=0$ for every $z\notin G$.

You can state the residue theorem as follows

Let $G$ be a region and $f:G\to \Bbb C$ be analytic except possibly at a finite set $A\subset G$. Let $\gamma:[0,1]\to G$ be a curve nullhomologous in $G$. Then $$\frac{1}{2\pi i}\int_\gamma f=\sum_{z\in A}W(\gamma,z)R(f,z)$$