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*}

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

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

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

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

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

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

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