I.Complex Function

1. Inequalities for module of complex numbers

Basic:

Re⁡(z)≤∣Re⁡(z)∣≤∣z∣,Im⁡(z)≤∣Im⁡(z)∣≤∣z∣\operatorname{Re}(z) \leq|\operatorname{Re}(z)| \leq|z|, \quad \operatorname{Im}(z) \leq|\operatorname{Im}(z)| \leq|z|

Triangle inequality: z=0,z=z1z=0, z=z_{1} and z=z1+z2z = z_{1} + z_2 are three vertices of a triangle with sides ∣z1∣,∣z2∣\left|z_1\right|,\left|z_2\right| and ∣z1+z2∣\left|z_1+z_2\right|

∣z1+z2∣≤∣z1∣+∣z2∣\left|z_1+z_2\right| \leq\left|z_1\right|+\left|z_2\right|

Other variants of triangle inequality:

∣∣z1∣−∣z2∣∣≤∣z1+z2∣,∣∣z1∣−∣z2∣∣≤∣z1−z2∣≤∣z1∣+∣z2∣\left|\left|z_1\right|-\left|z_2\right|\right| \leq\left|z_1+z_2\right|, \quad| | z_1\left|-\left|z_2\right|\right| \leq\left|z_1-z_2\right| \leq\left|z_1\right|+\left|z_2\right|

EXERCISE 1.1: Find an upper bound for ∣−1z4−5z+1∣\left|\frac{-1}{z^4-5 z+1}\right| if ∣z∣=2|z|=2.

\begin{align} \left|z^4-(5 z-1)\right|&\geq\left|\left|z^4\right|-|5 z-1|\right| \\ &\geq\left||z|^4-(5|z|+1)\right|\\ &=\left|2^4-5(2)-1\right| \\ &=5 \end{align}

∣−1z4−5z+1∣≤15\left|\frac{-1}{z^4-5 z+1}\right| \leq \frac{1}{5}

2. Argument of complex numbers

Argument of a complex number: arg(z)arg(z) represents the inclination of the vector zz (measured in radians) from the positive real axis (positive in the counterclockwise sense)

cos⁡(arg⁡(z))=Re⁡(z)∣z∣=xx2+y2,sin⁡(arg⁡(z))=Im⁡(z)∣z∣=yx2+y2\cos (\arg (z))=\frac{\operatorname{Re}(z)}{|z|}=\frac{x}{\sqrt{x^2+y^2}}, \quad \sin (\arg (z))=\frac{\operatorname{Im}(z)}{|z|}=\frac{y}{\sqrt{x^2+y^2}}

Properties:

arg(z1z2)=arg(z1)+arg(z2)arg(z_1z_2) = arg(z_1) + arg(z_2)