Complex Analysis: Find Analytic Functions w/ |ƒ(z)-1| + |ƒ(z)+1| = 4

[MakVish]

Homework Statement

Find all analytic functions ƒ: ℂ→ℂ such that
|ƒ(z)-1| + |ƒ(z)+1| = 4 for all z∈ℂ and ƒ(0) = √3 i

The Attempt at a Solution

I see that the sum of the distance is constant hence it should represent an ellipse. However, I am not able to find the exact form for ƒ(z). Any help is appreciated. Thanks.

 

[Dick]

Homework Statement

Find all analytic functions ƒ: ℂ→ℂ such that
|ƒ(z)-1| + |ƒ(z)+1| = 4 for all z∈ℂ and ƒ(0) = √3 i

The Attempt at a Solution

I see that the sum of the distance is constant hence it should represent an ellipse. However, I am not able to find the exact form for ƒ(z). Any help is appreciated. Thanks.

Do you know Liouville's theorem?