How Does the Function f(z) = (z + 1) / (z - 1) Map Complex Regions?

In summary, the function f(z) maps the set {z ϵ C : Re(z) < 0} into the set {w ϵ C : |w| < 1} by first finding the values of z+1 and z-1 and then multiplying them by -(x-1)-iy. This shows that the function is not necessarily 1-1, but rather onto, as it maps the entire set of negative real values onto the set of complex numbers with absolute value less than 1. The process is reversed for mapping the set {w ϵ C : |w| < 1} into the set {z ϵ C : Re(z) < 0}.
  • #1
jaejoon89
195
0
For z not equal to 1
f(z) = (z + 1) / (z - 1)

How do you show the function maps {z ϵ C : Re(z) < 0} into {w ϵ C : |w| < 1}
and
{w ϵ C : |w| < 1} into {z ϵ C : Re(z) < 0}?

----

I don't even know how to start this one besides that "into" means 1-1.

How do you show the mappings?
 
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  • #2
Well, the one thing you say you know is wrong! "Onto" means 1-1. "Into" does not have to be. If the real part of z is negative, we can write z as -x+ iy where x> 0. z+1= 1- x+ iy= (1-x)+ iy and z-1= -x+ iy- 1= (x-1)+ iy.

Now,
[tex]\frac{z+ 1}{z- 1}= \frac{(1-x)+iy}{-(x+1)+ iy}[/tex]
Multiply both numerator and denominator by -(x-1)- iy. What does that give you?
 

FAQ: How Does the Function f(z) = (z + 1) / (z - 1) Map Complex Regions?

1. What are complex variables?

Complex variables, also known as complex numbers, are numbers that have both a real and imaginary component. They are written in the form a + bi, where a is the real part and bi is the imaginary part, and i is the imaginary unit equal to the square root of -1.

2. Why are complex variables important in science?

Complex variables are important in science because they allow us to solve problems that cannot be solved with real numbers alone. They are used in fields such as physics, engineering, and mathematics to model and solve complex systems.

3. What are some real-world applications of complex variables?

Complex variables have numerous real-world applications, such as in electrical engineering for analyzing circuits, in physics for studying wave phenomena, in economics for analyzing stock market fluctuations, and in fluid dynamics for modeling fluid flow.

4. How do you perform operations with complex variables?

To perform operations with complex variables, you can use the rules of arithmetic for real numbers, combined with some additional rules for dealing with the imaginary unit i. For example, to add or subtract complex numbers, simply add or subtract the real and imaginary parts separately.

5. Can complex variables be graphed?

Yes, complex variables can be graphed on a complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This allows for visualizing and analyzing complex numbers and their relationships to each other.

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