Describing the Image of a Complex Function on the Unit Circle

In summary, The image of the function f(z) = (z-i)/(z+i) on the unit circle can be described by multiplying the top and bottom by (z + i)bar. This simplifies the expression and removes the complex part from the denominator.
  • #1
naggy
60
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.
 
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  • #2
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:
 

FAQ: Describing the Image of a Complex Function on the Unit Circle

What are complex functions and how are they represented?

Complex functions are mathematical functions that involve both real and imaginary numbers. They are represented using the complex plane, which consists of a horizontal axis representing the real component and a vertical axis representing the imaginary component.

What is the significance of images of complex functions?

The images of complex functions provide visual representations of the behavior of these functions. They help us understand the relationship between the input and output values and the overall behavior of the function.

How are images of complex functions generated?

Images of complex functions are generated by plotting points on the complex plane and connecting them to form a continuous curve. This curve is known as the graph of the complex function.

What do the colors in the images of complex functions represent?

The colors in the images of complex functions represent the values of the function at different points on the complex plane. Typically, warmer colors (such as red) indicate higher values, while cooler colors (such as blue) indicate lower values.

How are images of complex functions useful in scientific research?

Images of complex functions are useful in scientific research as they provide a visual representation of complex mathematical relationships. They can help researchers identify patterns and make predictions about the behavior of these functions. They are also used in fields such as physics, engineering, and economics to model real-world phenomena.

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