Prove Complex Inequality: Re z < 0

Thread starter smyroosh
Start date Nov 3, 2009
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Complex Inequality

In summary, a complex inequality involves both real and imaginary numbers and can be proved by showing it is true for all possible complex numbers. "Re z" in an inequality means the real part of a complex number and an example of a complex inequality is "|z + 2i| < |z + 3i|". Proving a complex inequality is important for determining the values of complex numbers that satisfy the inequality and has applications in areas such as complex analysis and differential equations.

Nov 3, 2009

smyroosh

How to prove the following inequality: for complex z such that Re z < 0 :

[tex]\left| e^z-1\right| < \left| z\right|[/tex] ?

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Nov 3, 2009

elibj123

use the identities

[tex]\left|z\right|^{2}=z\bar{z}[/tex]

and

[tex]\bar{e^{z}}=e^{\bar{z}}[/tex]

since both sides of the inequality are positive
you can square it up

Nov 4, 2009

smyroosh

What next?

1. What is a complex inequality?

A complex inequality is an inequality that involves complex numbers. It is similar to a regular inequality, but instead of just using real numbers, it also includes imaginary numbers.

2. How do you prove a complex inequality?

To prove a complex inequality, you need to show that it is true for all possible complex numbers. This can be done by using algebraic manipulations and properties of complex numbers, such as the triangle inequality and the modulus inequality.

3. What does "Re z" mean in the inequality "Re z < 0"?

"Re z" refers to the real part of the complex number z. In other words, it is the part of z that does not involve the imaginary unit, i. Therefore, the inequality "Re z < 0" means that the real part of z is less than 0.

4. Can you give an example of a complex inequality?

One example of a complex inequality is "|z + 2i| < |z + 3i|". This means that the distance between the complex number z and 2i is less than the distance between z and 3i.

5. Why is proving a complex inequality important?

Proving a complex inequality is important because it allows us to determine the values of complex numbers that satisfy the inequality. This can be useful in various areas of mathematics, such as complex analysis and differential equations.