Complex Analysis
Complex Numbers
Definition - Rectangular Form
\(z=a+bi\) where \(a,b\in\mathbb{R}\) and \(i\) satisfies \(i^2=-1\).
Complex Conjugate
Definition
\(\overline{z} = a-bi\).
Basic Algebra
- Addition. \(z_1+z_2 = (a_1+a_2) + (b_1+b_2)i\)
- Multiplication. \(z_1z_2 = (a_1a_2 - b_1b_2) + (a_1b_2 + a_2b_1)i\)
- Division. \(\frac{z_1}{z_2} = \frac{z_1\overline{z}_2}{z_2\overline{z}_2} = \frac{z_1\overline{z}_2}{a_2^2+b_2^2} = \frac{(a_1a_2+b_1b_2) + (a_2b_1-a_1b_2)i}{a_2^2+b_2^2}\)
Exponential Representation - Polar Form
Definition
\(z=|z|\exp(i\theta)\), \(\theta = \arctan(b/a)\), \(|z| = \sqrt{z\overline{z}}\)
Conjugate
\(\overline{z} = |z|\exp(-i\theta)\)
Argand Diagram
`Complex Plane’
\textbf{Re}al and Complex (\textbf{Im}) axes.
Thus, complex numbers can be represented as a vector in the complex plane.
Geometry
Addition - Tail-Tip Addition
\(+(z_1,z_2) = (a_1+b_1) + (a_2+b_2)i\) - vector tail-tip addition
Addition Length Relation
\(|z_1 + z_2| \leq |z_1| + |z_2|\)
Subtraction - Tail-Tip addition using the opposite direction of the second operand
\(-(z_1,z_2) = +(z_1,-z_2) = (a_1-b_1) + (a_2-b_2)i\)
Subtraction Length Relation
\(|z_1 - z_2| \geq |z_1| - |z_2|\)
Multiplication
\(z_1z_2 = r_1r_2\exp(i(\theta_1+\theta_2))\)