MATH2069 Complex Analysis

UNSW MATH2069 Complex Analysis and Vector Calculus Note

Complex Analysis

Basics

$$ e^{ix} = \cos x + i\sin x $$
$$ \sin x = {e^{ix} – e^{-ix} \over 2i}$$
$$ \cos x = {e^{ix} + e^{-ix} \over 2} $$
$$ \sinh x = \frac {e^x – e^{-x}} {2}$$
$$ \cosh x = \frac {e^x + e^{-x}} {2} $$
$$ \sinh x = -i \sin (i x) $$
$$ \cosh x = \cos (i x)$$

$$\cosh^2 x – \sinh^2 x = 1 $$

$$\sinh(x+iy) = \sinh(x) \cos(y) + i \cosh(x) \sin(y) $$
$$\cosh(x+iy) = \cosh(x) \cos(y) + i \sinh(x) \sin(y) $$

$$\sin(x+iy) = \sin(x) \cosh(y) + i \cos(x) \sinh(y) $$
$$\cos(x+iy) = \cos(x) \cosh(y) – i \sin(x) \sinh(y) $$

$$\frac{d}{dx}\sinh x = \cosh x$$
$$\frac{d}{dx}\cosh x = \sinh x $$


Complex Inequality

Triangle Inequality
$$ |z + w| \leq |z| + |w| $$

Circle Inequality
$$ | z + w | \geq |\,|z| – |w|\,| $$

Principal Argument

$$ -\pi \lt Arg(z) \leq \pi $$

Cauchy-Riemann

$$f\prime(z_0) = U_x(X_0,Y_0) + i U_x (X_0,Y_0) = V_x(X_0,Y_0) – i V_x (X_0,Y_0) $$

$$\begin{align*}
U_x = V_y & \qquad
U_y = – V_x
\end{align*}
$$

Analytic

Harmonic Function

$$\frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2} = 0 $$

Complex Lograrithm

$$\log z = \ln |z| + i Arg(z) $$

Taylor Series

$$ e^x =1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!}+\cdots\! = \sum^\infty_{n=0} {x^n\over n!}$$

$$\frac{1}{(1-x)}=1+x+x^2+x^3+x^4+ \cdots = \sum^\infty_{n=0} {x^n} $$

$$\sin x = x – \frac{x^3}{3!} + \frac{x^5}{5!} +\cdots = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} $$

$$\cos x = 1 – \frac{x^2}{2!} + \frac{x^4}{4!} +\cdots = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n}$$

$$\sinh x = x + \frac {x^3} {3!} + \frac {x^5} {5!} + \frac {x^7} {7!} +\cdots = \sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!}$$

$$\cosh x = 1 + \frac {x^2} {2!} + \frac {x^4} {4!} + \frac {x^6} {6!} + \cdots = \sum_{n=0}^\infty \frac{x^{2n}}{(2n)!}
$$