Describing the Image of a Complex Function on the Unit Circle

Click For Summary
SUMMARY

The discussion focuses on describing the image of the complex function f(z) = (z-i)/(z+i) where z lies on the unit circle. Participants suggest simplifying the expression by multiplying the numerator and denominator by the conjugate of the denominator, (z + i)bar. This technique effectively eliminates the complex component from the denominator, facilitating a clearer analysis of the function's behavior on the unit circle.

PREREQUISITES
  • Understanding of complex functions and their properties
  • Familiarity with the unit circle in the complex plane
  • Knowledge of complex conjugates and their application
  • Basic algebraic manipulation of complex expressions
NEXT STEPS
  • Study the properties of complex functions on the unit circle
  • Learn about the geometric interpretation of complex mappings
  • Explore the concept of complex conjugates in detail
  • Investigate other methods for simplifying complex functions
USEFUL FOR

Mathematicians, physics students, and anyone studying complex analysis or working with complex functions on the unit circle.

naggy
Messages
58
Reaction score
0
How do I describe the image of a function like.
f(z) = (z-i)/(z+i) given that the length of z is equal to one. The domain is the unit circle.

Is the best way just let z=x+yi and then see what comes out? Or is there a simpler way of doing it.
 
Physics news on Phys.org
naggy said:
How do I describe the image of a function like.
f(z) = (z-i)/(z+i) given that the length of z is equal to one. The domain is the unit circle.

Is the best way just let z=x+yi and then see what comes out? Or is there a simpler way of doing it.

Hi naggy! :smile:

Hint: always get rid of the complex bit from the bottom if you can …

so multiply top and bottom by (z + i)bar :wink:
 

Similar threads

Graduate Existence of a limit implies that a function can be harmonic extended
  • · Replies 1 ·
Replies
1
Views
3K
Graduate How to take the spatial derivative of quaternions
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Find f(z) given f(x, y) = u(x, y) + iv(x,y)
  • · Replies 8 ·
Replies
8
Views
1K
Graduate Question about proof of Great Picard Theorem
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Expansion of a complex function around branch point
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Partial derivatives of the function f(x,y)
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad How to know if a complex root is inside the unit circle
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad About convergence of complex power series on the circle ##|z - z_0|=r##
  • · Replies 22 ·
Replies
22
Views
5K
High School Question about units for "area under curve"
  • · Replies 2 ·
Replies
2
Views
3K
Graduate What is the Image of the Complex Function f(z)=z+1/z on |z|>1?
  • · Replies 1 ·
Replies
1
Views
2K