Maclaurin Series: Taylor series with $z_0=0$.

Laurent Series: If $f(z)$ is analytic in an annulus $R_1 < |z-z_0| < R_2$, then $f(z) = \sum_{n=-\infty}^{\infty} a_n (z-z_0)^n$, where $a_n = \frac{1}{2\pi i} \oint_C \frac{f(\zeta)}{(\zeta-z_0)^{n+1}} d\zeta$.

6. Singularities and Residues

Isolated Singularity: A point $z_0$ where $f(z)$ is not analytic but is analytic in a punctured disk $0 < |z-z_0| < R$.
Types of Isolated Singularities:
- Removable: Laurent series has no negative powers. $\lim_{z \to z_0} f(z)$ exists.
- Pole of Order $m$: Laurent series has a finite number of negative powers, with $a_{-m} \neq 0$. $\lim_{z \to z_0} (z-z_0)^m f(z)$ exists and is non-zero.
- Essential: Laurent series has infinitely many negative powers. $\lim_{z \to z_0} f(z)$ does not exist.
Residue: For an isolated singularity $z_0$, the residue is the coefficient $a_{-1}$ of the Laurent series. $\text{Res}(f, z_0) = a_{-1}$.
Formulas for Residues:
- Simple Pole ($m=1$): $\text{Res}(f, z_0) = \lim_{z \to z_0} (z-z_0)f(z)$.
- Pole of Order $m$: $\text{Res}(f, z_0) = \frac{1}{(m-1)!} \lim_{z \to z_0} \frac{d^{m-1}}{dz^{m-1}} [(z-z_0)^m f(z)]$.
Cauchy's Residue Theorem: If $f(z)$ is analytic inside and on a simple closed contour $C$, except for a finite number of isolated singularities $z_k$ inside $C$, then $\oint_C f(z) dz = 2\pi i \sum_{k=1}^n \text{Res}(f, z_k)$.

7. Laplace Transform

Definition: $\mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} e^{-st} f(t) dt$
Inverse Laplace Transform: $\mathcal{L}^{-1}\{F(s)\} = f(t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} e^{st} F(s) ds$ (Bromwich Integral)

Properties of Laplace Transform

Property	Time Domain $f(t)$	Frequency Domain $F(s)$
Linearity	$af(t) + bg(t)$	$aF(s) + bG(s)$
Time Scaling	$f(at)$	$\frac{1}{a}F(\frac{s}{a})$
Frequency Shifting	$e^{at}f(t)$	$F(s-a)$
Time Shifting	$f(t-a)u(t-a)$	$e^{-as}F(s)$
Differentiation (time)	$f'(t)$	$sF(s) - f(0)$
Integration (time)	$\int_0^t f(\tau) d\tau$	$\frac{1}{s}F(s)$
Multiplication by $t^n$	$t^n f(t)$	$(-1)^n \frac{d^n}{ds^n} F(s)$
Division by $t$	$\frac{f(t)}{t}$	$\int_s^{\infty} F(\sigma) d\sigma$
Convolution	$(f*g)(t) = \int_0^t f(\tau)g(t-\tau)d\tau$	$F(s)G(s)$

Common Laplace Transform Pairs

$f(t)$	$F(s) = \mathcal{L}\{f(t)\}$
$\delta(t)$ (Dirac Delta)	$1$
$u(t)$ (Unit Step)	$\frac{1}{s}$
$t^n$, $n \in \mathbb{Z}_{\ge 0}$	$\frac{n!}{s^{n+1}}$