Definition. A point $z_0$ is called a singular point or a singularity of a function $f$ if $f$ fails to be analytic at $z_0$ but is analytic at some point in every neighbourhood of $z_0$. A singularity is said to be isolated if in addition there is a deleted neighbourhood $0<|z-z_0|<\epsilon$ of $z_0$ throughout which $f$ is analytic.

Example. $f(z)=\frac{z+1}{z^3(z^2+1)}$ has three isolated singularities $z=0$ and $z=\pm i$.

Example. $\mathrm{Log}z=\ln r+i\Theta$ ($r>0$, $-\pi<\Theta<\pi$), the principal branch of the logarithmic function $\log z$ has a singularity at the origin but it is not isolated.

Example. $f(z)=\frac{1}{\sin\left(\frac{\pi}{z}\right)}$ has singularities $z=0$ and $z=\frac{1}{n}$ ($n=\pm 1, \pm 2, \cdots$). Each singularity except $z=0$ is isolated.

If $z_0$ is an isolated singularity of a function $f$, then there exists $R>0$ such that $f$ is analytic throughout which $0<|z-z_0|<R$, so $f(z)$ is represented by a Laurent series
\begin{align*}
f(z)&=\sum_{n=0}^\infty a_n(z-z_0)^n+\frac{b_1}{z-z_0}+\frac{b_2}{(z-z_0)^2}+\cdots+\frac{b_n}{(z-z_0)^n}+\cdots\\
&(0<|z-z_0|<R)\\
b_n&=\frac{1}{2\pi i}\oint_C\frac{f(z)}{(z-z_0)^{-n+1}}dz\ (n=1,2,\cdots)
\end{align*}
where $C$ is any positively oriented simple closed contour around $z_0$ and lying in the punctured disk $0<|z-z_0|<R$. The coefficient $b_1$ of $\frac{1}{z-z_0}$ in the above Laurent series expansion is called the residue of $f$ at the isolated singularity $z_0$. The residue is important since from the integral formula for $b_n$ above, we find
$$\oint_C f(z)dz=2\pi i b_1.$$
That is, the contour integral $\oint_C f(z)dz$ can be evaluated using the residue $b_1$. The residue $b_1$ of $f(z)$ at $z=z_0$ is also denoted by $\mathrm{Res}_{z=z_0}f(z)$.

Example. Consider $\oint_C\frac{dz}{z(z-2)^4}$ where $C$ is the positively oriented circle $|z-2|=1$. The integrand $\frac{1}{z(z-2)^4}$ is analytic everywhere except at the isolated singularities $z=0$ and $z=2$, so it has a Laurent series representation in the punctured disk $0<|z-2|<2$.
\begin{align*}
\frac{1}{z(z-2)^4}&=\frac{1}{(z-2)^4}\cdot\frac{1}{2+(z-2)}\\
&=\frac{1}{2(z-2)^4}\cdot\frac{1}{1-\left(-\frac{z-2}{2}\right)}\\
&=\frac{1}{2(z-2)^4}\sum_{n=0}^\infty\frac{(-1)^n}{2^n}(z-2)^n\\
&=\sum_{n=0}^\infty\frac{(-1)^n}{2^{n+1}}(z-2)^{n-4}\ (0<|z-2|<2)
\end{align*}
The residue of $\frac{1}{z(z-2)^4}$ at $z=2$ is $-\frac{1}{16}$, hence
$$\oint_C\frac{dz}{z(z-2)^4}=2\pi i\left(-\frac{1}{16}\right)=-\frac{\pi i}{8}.$$

Example. Evaluate $\oint_C e^{\frac{1}{z^2}}dz$ where $C$ is the unit circle $|z|=1$.

Solution. $e^{\frac{1}{z^2}}$ is analytic everywhere except at $z=0$. $e^{\frac{1}{z^2}}$ has an Laurent series expansion
$$e^{\frac{1}{z^2}}=1+\frac{1}{1!z^2}+\frac{1}{2!z^4}+\cdots\ (0<|z|<\infty).$$
Since $b_1=0$, $\oint_C e^{\frac{1}{z^2}}dz=0$.

If a function fails to be analytic at a point $z_0$, we cannot apply Taylor’s theorem at that point. However, it may be possible to find a series representation for $f(z)$ involving both positive and negative powers of $z-z_0$.

Theorem [Laurent’s Theorem]. Suppose that a function $f$ is analytic throughout an annular domain $R_1<|z-z_0|<R_2$, centered at $z_0$, and let $C$ denote any positively oriented simple closed contour around $z_0$ and lying in that domain. Then, at each point in the domain, $f(z)$ has the series representation
$$f(z)=\sum_{n=0}^\infty a_n(z-z_0)^n+\sum_{n=1}^\infty\frac{b_n}{(z-z_0)^n}\ (R_1<|z-z_0|<R_2),$$
where
\begin{align*}
a_n&=\frac{1}{2\pi i}\oint_C\frac{f(z)}{(z-z_0)^{n+1}}dz\ (n=0,1,2,\cdots),\\
b_n&=\frac{1}{2\pi i}\oint_C\frac{f(z)}{(z-z_0)^{-n+1}}dz\ (n=1,2,\cdots).
\end{align*}

The expansion can be also written as
$$f(z)=\sum_{n=-\infty}^\infty c_n(z-z_0)^n\ (R_1<|z-z_0|<R_2),$$
where
$$c_n=\frac{1}{2\pi i}\oint_C\frac{f(z)}{(z-z_0)^{n+1}}dz\ (n=0,\pm 1,\pm 2,\cdots).$$

Example. From the Maclaurin series expansion $e^z=\sum_{n=0}^\infty\frac{z^n}{n!}$, we obtain Laurent series expansion for $e^{\frac{1}{z}}$
$$e^{\frac{1}{z}}=\sum_{n=0}^\infty\frac{1}{n!z^n}=1+\frac{1}{1!z}+\frac{1}{2!z^2}+\cdots\ (0<|z|<\infty).$$
The coefficient $b_1$ is
$$b_1=\frac{1}{2\pi i}\oint_C e^{\frac{1}{z}}dz=1$$
for any positively oriented simple closed contour $C$ around $0$ and lying in the domain $0<|z|<\infty$. Hence, we obtain the integral
$$\oint_C e^{\frac{1}{z}}dz=2\pi i$$
for any positively oriented simple closed contour $C$ around $0$ and lying in the domain $0<|z|<\infty$.

Example. $f(z)=\frac{1}{(z-i)^2}$ is already in the form of a Laurent series, where $z_0=i$. From
$$f(z)=\sum_{n=-\infty}^\infty c_n(z-i)^n\ (0<|z-i|<\infty),$$
we find that $c_{-2}=1$ and all other coefficients are zero. Hence, we have
$$\oint_C\frac{dz}{(z-i)^{n+3}}=\left\{\begin{array}{ccc}
0 &\mbox{if}&n\ne -2,\\
2\pi i &\mbox{if}& n=-2
\end{array}
\right.$$
for any positively oriented simple closed contour $C$ around $i$ and lying in the domain $0<|z-i|<\infty$.

Example. Let $f(z)$ be the function
$$f(z)=\frac{-1}{(z-1)(z-2)}=\frac{1}{z-1}-\frac{1}{z-2}.$$
Since $f(z)$ has two singularities $z=1$ and $z=2$, we may consider the following three different domains to obtain Laurent series expansion in each domain
$$D_1: |z|<1,\ D_2: 1<|z|<2,\ D_3: |z|>2.$$

For $D_1: |z|<1$,
$$\frac{1}{z-1}=-\frac{1}{1-z}=-\sum_{n=0}^\infty z^n$$
and
$$-\frac{1}{z-2}=\frac{1}{2-z}=\frac{1}{2}\frac{1}{1-\frac{z}{2}}.$$
Since $|z|<1$, $|z|<2$ and so $\frac{1}{1-\frac{z}{2}}=\sum_{n=0}^\infty\left(\frac{z}{2}\right)^n$. Hence, we obtain the Taylor series expansion
\begin{align*}
f(z)&=-\sum_{n=0}^\infty z^n+\sum_{n=0}^\infty\frac{z^n}{2^{n+1}}\\
&=\sum_{n=0}^\infty(2^{-n-1}-1)z^n\ (|z|<1).
\end{align*}
For $D_2: 1<|z|<2$,
$$\frac{1}{z-1}=\frac{1}{z}\frac{1}{1-\frac{1}{z}}=\frac{1}{z}\sum_{n=0}^\infty\left(\frac{1}{z}\right)^n.$$
Hence, we obtain the Laurent series expansion
\begin{align*}
f(z)&=\frac{1}{z}\sum_{n=0}^\infty\left(\frac{1}{z}\right)^n+\frac{1}{2}\sum_{n=0}^\infty\left(\frac{z}{2}\right)^n\\
&=\sum_{n=0}^\infty\frac{z^n}{2^{n+1}}+\sum_{n=1}\frac{1}{z^n}\ (1<|z|<2).
\end{align*}
For $D_3: |z|>2$,
\begin{align*}
f(z)&=\frac{1}{z-1}-\frac{1}{z-2}\\
&=\frac{1}{z}\sum_{n=0}^\infty\left(\frac{1}{z}\right)^n-\frac{1}{z}\sum_{n=0}^\infty\left(\frac{2}{z}\right)^n\\
&=\sum_{n=1}^\infty\frac{1-2^{n-1}}{z^n}\ (2<|z|<\infty).
\end{align*}

Theorem. Suppose that a function $f$ is analytic throughout a disk $|z-z_0|<R_0$ centered at $z_0$ and with radius $R_0$. Then $f(z)$ has the power series representation
$$f(z)=\sum_{n=0}^\infty a_n(z-z_0)^n\ (|z-z_0|<R_0),$$
where
$$a_n=\frac{f^{(n)}(z_0)}{n!}\ (n=0,1,2,\cdots).$$

Proof. First consider the case $z_0=0$ and show that
$$f(z)=\sum_{n=0}^\infty\frac{f^{(n)}(0)}{n!}z^n\ (|z|<R_0).$$
This particular Taylor series is called a Maclaurin series. Let us write $|z|=r$ and let $C_0$ denote any positively oriented circle centered at the origin and with radius $r_0$, where $r<r_0<R_0$.

Since $f$ is analytic inside and on the circle $C_0$ and since $z$ is interior to $C_0$, by the Cauchy Integral Formula, we have
$$f(z)=\frac{1}{2\pi i}\oint_{C_0}\frac{f(s)}{s-z}ds.$$
Note that $\frac{1}{s-z}$ may be written as
\begin{align*}
\frac{1}{s-z}&=\frac{1}{s}\frac{1}{1-\frac{z}{s}}\\
&=\sum_{n=0}^{N-1}\frac{z^n}{s^{n+1}}+z^N\frac{1}{(s-z)s^N}.
\end{align*}
Using this, $f(z)$ may be written as
\begin{align*}
f(z)&=\frac{1}{2\pi i}\oint_{C_0}\frac{f(s)}{s-z}ds\\
&=\sum_{n=0}^{N-1}\frac{1}{2\pi i}\oint_{C_0}\frac{f(s)}{s^{n+1}}ds z^n+\frac{z^N}{2\pi i}\oint_{C_0}\frac{f(s)}{(s-z)s^N}ds\\
&=\sum_{n=0}^{N-1}\frac{f^{(n)}(0)}{n!}z^n+\rho_N(z),
\end{align*}
where
$$\rho_N(z)=\frac{z^N}{2\pi i}\oint_{C_0}\frac{f(s)}{(s-z)s^N}ds.$$
We are done if we can show that $\displaystyle\lim_{N\to\infty}\rho_N=0$. By triangle inequality, $|s-z|\geq ||s|-|z||=r_0-r$. So, we obatin
\begin{align*}
|\rho_N(z)|&\leq \frac{r^N}{2\pi}\frac{M}{(r_0-r)r_0^N}2\pi r_0\\
&=\frac{Mr_0}{r_0-r}\left(\frac{r}{r_0}\right)^N.
\end{align*}
Since $r<r_0$, $\displaystyle\lim_{N\to\infty}|\rho_N(z)|=0$ i.e. $\displaystyle\lim_{N\to\infty}\rho_N=0$. Hence, we have proved that
$$f(z)=\sum_{n=0}^\infty\frac{f^{(n)}(0)}{n!}z^n.$$
Now, suppose that $f(z)$ is analytic when $|z-z_0|<R_0$. Then $g(z):=f(z+z_0)$ is analytic when $|z|=|(z+z_0)-z_0|<R_0$. So, $g(z)$ has the Maclaurin series representation
$$g(z)=\sum_{n=0}^\infty\frac{g^{(n)}(0)}{n!}z^n\ (|z|<R_0)$$
or
$$f(z+z_0)=\sum_{n=0}^\infty\frac{f^{(n)}(z_0)}{n!}z^n\ (|z|<R_0).$$
Replacing $z$ by $z-z_0$, we finally obtain the Taylor series representation
$$f(z)=\sum_{n=0}^\infty\frac{f^{(n)}(z_0)}{n!}(z-z_)^n\ (|z-z_0|<R_0).$$

Example. The function $f(z)=e^z$ is entire, so it has a Maclaurin series representation for all $z\in\mathbb{C}$.
$$e^z=\sum_{n=0}^\infty\frac{z^n}{n!}\ (|z|<\infty).$$

Example. $\sin z$ is defined as
$$\sin z=\frac{e^{iz}-e^{-iz}}{2i}.$$
So,
\begin{align*}
\sin z&=\frac{1}{2i}\left[\sum_{n=0}^\infty\frac{(iz)^n}{n!}-\sum_{n=0}^\infty\frac{(-iz)^n}{n!}\right]\\
&=\frac{1}{2i}\sum_{n=0}^\infty[1-(-1)^n]\frac{i^nz^n}{n!}\\
&=\sum_{n=0}^\infty(-1)^n\frac{z^{2n+1}}{(2n+1)!}\ (|z|<\infty).
\end{align*}
Using the definition $\cos z=\frac{e^{iz}+e^{-iz}}{2}$, one can similarly obtain the Maclaurin series representation for $\cos z$
$$\cos z=\sum_{n=0}^\infty(-1)^n\frac{z^{2n}}{(2n)!}.$$

Suppose that a function $f$ is analytic inside and on a positively oriented circle $C_R$, centered at $z_0$ and with radius $R$. Then by Cauchy Integral Formula
$$f^{(n)}(z_0)=\frac{n!}{2\pi i}\oint_{C_R}\frac{f(z)}{(z-z_0)^{n+1}}dz.$$
Since $C_R$ is compact (i.e. closed and bounded) and $f(z)$ is continuous, $|f(z)|$ assumes a maximum (and also a minimum) on $C_R$. Let us call the maximum value of $|f(z)|$ on $C_R$ $N_R$. On $C_R$, $|z-z_0|=R$. Hence, we have the upper bound for $|f^{(n)}(z_0)|$ as
\begin{align*}
|f^{(n)}(z_0)|&=\frac{n!}{2\pi}\left|\oint_{C_R}\frac{f(z)}{(z-z_0)^{n+1}}dz\right|\\
&\leq\frac{n!M_R}{R^n}
\end{align*}
for $n=1,2,\cdots$. This inequality is called Cauchy’s Inequality.

Using Cauchy’s Inequality, we can prove the following Liouville’s Theorem.

Theorem. If $f$ is entire and bounded in the complex plane $\mathbb{C}$, then $f(z)$ is constant.

Proof. Since $f$ is bounded, there exists $M>0$ such that $|f(z)|\leq M$ for all $z\in\mathbb{C}$. Since $f$ is entire, for any $R>0$,
$$|f'(z)|\leq\frac{M}{R}$$
by Cauchy’s Inequality. As $R\to\infty$, $\frac{M}{R}\to 0$, so we obtain $|f'(z)|=0$ for all $z\in\mathbb{C}$. This implies that $f'(z)=0$ for all $z\in\mathbb{C}$ and hence, $f(z)$ is constant.

Cauchy’s Integral Theorem says that if a function $f(z)$ is analytic throughout some simply connected domain $D$, then for any contour $C$ in $D$, $\oint_C f(z)dz=0$. It turns out that the converse of Cauchy’s Theorem is also true, namely

Theorem. If a function $f(z)$ is continuous in a simply connected domain $D$ and $\oint_C f(z)dz=0$ for every closed contour $C$ within $D$, then $f(z)$ is analytic throughout $D$.

This theorem is called Morera’s Theorem. Let us prove the theorem.

First we show that under the assumption the line integral $\int_\gamma f(z)dz$ is independent on the path $\gamma$. Let $\gamma$ be a path from $z_1$ to $z_2$ and choose $\gamma_1$ another path from $z_1$ to $z_2$ as shown in the figure.

Then $C: \gamma\cup (-\gamma_1)$ is a positively oriented contour. So, by assumption,
\begin{align*}
0&=\oint_C f(z)dz\\
&=\int_\gamma f(z)dz+\int_{-\gamma_1}f(z)dz\\
&=\int_\gamma f(z)dz-\int_{\gamma_1}f(z)dz.
\end{align*}
That is,
$$\int_\gamma f(z)dz=\int_{\gamma_1}f(z)dz.$$
Hence, the line integral $\int_\gamma f(z)dz$ does not depend on the path $\gamma$ but only on the endpoints $z_1$ and $z_2$.

Let us define $F(z)=\int_{z_0}^z f(w)dw$. Then
\begin{align*}
F(z+\Delta z)-F(z)&=\int_{z_0}^{z+\Delta z} f(w)dw-\int_{z_0}^z f(w)dw\\
&=\int_z^{z+\Delta z} f(w)dw.
\end{align*}
Since $\int_z^{z+\Delta z}dw=\Delta z$,
$$f(z)=\frac{1}{\Delta z}\int_z^{z+\Delta z}f(z) dw$$
and so, we have
$$\frac{F(z+\Delta z)-F(z)}{\Delta z}-f(z)=\frac{1}{\Delta z}\int_z^{z+\Delta z}[f(w)-f(z)]dw.$$
Since $f$ is continuous at $z$, given $\epsilon>0$ there exists $\delta>0$ such that $|f(w)-f(z)|<\epsilon$ whenever $|w-z|<\delta$. If $z+\Delta z$ is close enough to $z$ so that $|\Delta z|<\delta$, then
$$\left|\frac{F(z+\Delta z)-F(z)}{\Delta z}-f(z)\right|<\frac{1}{|\Delta z|}\epsilon|\Delta z|=\epsilon.$$
Therefore,
$$F'(z)=\lim_{\Delta z\to 0}\frac{F(z+\Delta z)-F(z)}{\Delta z}=f(z).$$
That is, $F(z)$ analytic in $D$. This means that $f(z)$ is analytic by Cauchy Integral Formula.