Zeros of functions on the complex plane

In summary, the statement discusses the relationship between a function having all its zeros on a specific line in the complex plane and having all its zeros on the unit circle. The speaker suggests that there may be a transformation of coordinates involved, specifically using the function f(z)= (z-A+r)/(z-A-r) with real numbers A and r, to map the line onto the unit circle. This is possible because the line and the unit circle have similar properties, with inverse points lying on either side for the line and at the centre and infinity for the circle. By mapping the line onto the circle, it becomes a generalized circle.
  • #1
zetafunction
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what is the relationship (if any) of the following statement

- A function has ALL the zeros on the line (complex plane) [tex] Re (z) = A [/tex] for some Real A

- A function has ALL the zeros on the unit circle defined by [tex] |z| \le 1 [/tex]

i think there is a transformation of coordinates so the line Re (z) = A would become the unit circle but i do not know what is.
 
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  • #2
Think it's f(z)= (z-A+r)/(z-A-r) where both A and r are real numbers. The line has inverse points lying either side of it and mirrored by it. The unit circle has its centre and inf as inverse points. So you can map the A-r to the centre of the circle and A+r to infinity, then scale the circle to the right size by mapping A to 1. It works because the line is a generalized circle.
:-)
 

1. What are zeros of a function on the complex plane?

Zeros of a function on the complex plane are points where the function is equal to zero. These points can be plotted on the complex plane, which is a two-dimensional coordinate system where the horizontal axis represents the real numbers and the vertical axis represents the imaginary numbers.

2. How are zeros of a function on the complex plane related to the roots of the function?

Zeros of a function on the complex plane are also known as roots or solutions of the function. This means that when the function is evaluated at these points, it will result in an output of zero. In other words, the zeros of a function are the values of the independent variable that make the function equal to zero.

3. Can a function have more than one zero on the complex plane?

Yes, a function can have multiple zeros on the complex plane. This is because a function can have multiple values of the independent variable that make the function equal to zero. The number of zeros a function has on the complex plane is equal to its degree, which is the highest power of the variable in the function.

4. How can the zeros of a function on the complex plane be found?

The zeros of a function on the complex plane can be found by solving the function for the values of the independent variable that make the function equal to zero. This can be done algebraically by factoring the function or using other methods such as the quadratic formula or the rational root theorem.

5. What is the significance of zeros of a function on the complex plane?

The zeros of a function on the complex plane are important because they can provide valuable information about the behavior of the function. For example, the number of zeros a function has can determine the number of intersection points it has with the x-axis. Zeros can also be used to find the minimum and maximum values of a function, as well as its intervals of increasing and decreasing.

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