07 Complex Regions
Connected Set¶
A set where any 2 points can be joined without leaving the set
Refer to Types of Sets
- open' = closed
- closed' = open
Domain¶
a set that is both open and connected.
Limit/Accumulation Point¶
Deleted neighborhood of \(z_0\) contains atleast one point of \(S\)
closed set has all limit points
all interior points and boundary points are limit points
Properties of Functions¶
Differentiable¶
Consider derivative equation
\[ f'(z) = \lim_{\Delta z \to 0} \frac{ f(z + \Delta z) - f(z) }{ \Delta z } \]
A function is said to differentiable if \(f'(z)\) is unique
Analytic¶
Differentiable @ \(z_0\) and its neighborhood
Entire¶
Analytic Everywhere
Harmonic¶
\(u\) is harmonic if it satisfied Laplace equation, ie
\[ u_{xx} + u_{yy} = 0 \]
If \(f(z) = u+iv\), then
- \(f(z)\) is analytic
- Put \(y = 0, x = z \to f(z) = f(x)\) for shortcut
- real and imaginary parts are harmonic
- \(v\) is harmonic conjugate of \(u\)
Hyperbolic Function¶
\[ \begin{aligned} \cos(ix) &= \cosh(x) & \sin(ix) &= i \sinh(x) \\ \cosh(x) &= \frac{e^x + e^{-x}}{2} & \sinh(x) &= \frac{e^x - e^{-x}}{2} \\ \sinh'(x) &= \cosh(x) & \cosh'(x) &= \sinh(x) \end{aligned} \]
\[ \cosh^2(x) - \sinh^2(x) = 1 \]
CR Equation¶
Consider \(f(z) = u + iv\)
| Rectangular | Polar | |
|---|---|---|
| \(u_x = v_y\) | \(u_r = \frac{1}{r} v_\theta\) | |
| \(u_y = - v_x\) | \(u_\theta = -r v_r\) | |
| Continuous | \(u_x, u_y, v_x, v_y\) | \(u_r, u_\theta, v_r, v_\theta\) |
| \(f'(z)\) | \(u_x + i v_x\) | \((u_r + i v_r) e^{-i \theta}\) |