Range of z+1/z (complex function)

[gop]

Homework Statement

Give the range of the function
$$f(z)=z+1/z$$
$$z \in \mathbf{C} , Im(z)>0$$

Homework Equations

The Attempt at a Solution

tried to use polar form to either separate real and imaginary part or get r*exp(I*phi) both failed to yield suitable closed form expressions. Also tried to just solve w=z+1/z which gives
$$z=1/2(w+-sqrt(w^2-4))$$ where especially the w^2-4 term makes it difficult to say what really happens.

thx

 

[mighty2000]

for any help in advance

Thank you for your interesting question. The range of the function f(z)=z+1/z where z is a complex number with Im(z)>0 can be found by considering the behavior of the function in the complex plane.

First, let's rewrite the function in terms of polar coordinates:

f(z) = r*exp(i*phi) + 1/(r*exp(i*phi)) = r*exp(i*phi) + 1/r*exp(-i*phi)

= (r+1/r)*exp(i*phi)

We can see that the magnitude of the function is given by (r+1/r), which means that the range of the function will be all non-negative real numbers. However, to determine the exact range, we need to consider the behavior of the function for different values of the argument phi.

For phi=0, the function becomes f(z) = (r+1/r), which has a minimum value of 2 when r=1. This means that the range of the function includes all values greater than or equal to 2.

For phi=pi/2, the function becomes f(z) = i*(r-1/r), which has a minimum value of 0 when r=1. This means that the range of the function also includes all non-negative imaginary numbers.

For phi=pi, the function becomes f(z) = -(r+1/r), which has a maximum value of -2 when r=1. This means that the range of the function also includes all values less than or equal to -2.

Therefore, the range of the function f(z) = z+1/z where z is a complex number with Im(z)>0 is all non-negative real numbers and all non-negative imaginary numbers, as well as all values less than or equal to -2.

I hope this helps you with your question. Let me know if you have any further questions.