List of basic formulas and concepts related to the display of lines and areas using the function ( w = f(z) ), where ( z = x + iy ) (a complex number):
- Complex function:
[
w = f(z) = u(x, y) + iv(x, y)
]
where ( u(x, y) ) and ( v(x, y) ) are the real and imaginary parts of the function, respectively. - Display of level lines:
- The level lines for ( u(x, y) = c_1 ) and ( v(x, y) = c_2 ) are displayed in the region ( w ) as lines corresponding to fixed values ( c_1 ) and ( c_2 ).
- Display image:
- If ( z ) varies along some line (such as a straight line or a circle), then ( w ) will be displayed along another line in the region ( w ).
- Cauchy-Riemann formula:
- For a function ( f(z) ) to be analytic, the following conditions must be satisfied:
[
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
]
- Jacobian:
- The Jacobian of the map ( J ) can be used to study the change in areas:
[
J = \frac{\partial(u, v)}{\partial(x, y)} = \frac{\partial u}{\partial x} \cdot \frac{\partial v}{\partial y} – \frac{\partial u}{\partial y} \cdot \frac{\partial v}{\partial x}
]
- Displaying circles and lines:
- Lines and circles in the domain ( z ) can be mapped to other curves in the domain ( w ) depending on the shape of the function ( f(z) ).
- Examples of displays:
- Linear mapping: ( w = az + b ) (where ( a ) and ( b ) are complex numbers).
- Formation of circles: (w = z^2) (maps circles to other circles or lines).