oddiseas said:Homework Statement
Show that |e^iz|<=1 for Im(z)>=0
b)show that if |z|>=3 then the following inequality holds
Homework Equations
The Attempt at a Solution
|e^iz|=e^-b(cosa+isina)
and since the inmaginery part is always one, the magnitude is always less or equal to one' provided b is <=0. Now the second part i am having trouble with.
What following inequality? It doesn't appear in either your original post or the edited post.oddiseas said:b)show that if |z|>=3 then the following inequality holds
Inequality in complex numbers refers to the comparison of two complex numbers using the symbols <, >, ≤, or ≥. This comparison is based on the magnitude of the complex numbers, which is determined by their distance from the origin on the complex plane.
To graph inequalities in complex numbers, you first plot the complex numbers on the complex plane. Then, you shade the region that satisfies the inequality. For example, if the inequality is z < 3, you would shade the region inside the circle with radius 3 centered at the origin.
The absolute value of a complex number is its distance from the origin on the complex plane. Inequalities in complex numbers are often expressed in terms of absolute values, such as |z| < 3. This means that the distance of z from the origin is less than 3.
Complex numbers can be used to represent inequalities in real-life situations, such as in economics, engineering, and physics. For example, in economics, complex numbers can be used to model supply and demand inequalities. In engineering, they can be used to represent constraints in optimization problems. In physics, they can be used to represent inequalities in electromagnetic fields.
One common misconception is that inequalities in complex numbers only involve the imaginary part of the complex numbers. In reality, inequalities also consider the real part of the complex numbers. Another misconception is that all complex numbers can be compared using inequalities. This is not true as some complex numbers are not comparable due to their different magnitudes and directions on the complex plane.