The discussion centers on finding all analytic functions ƒ: ℂ→ℂ that satisfy the equation |ƒ(z)-1| + |ƒ(z)+1| = 4 for all z∈ℂ, with the condition that ƒ(0) = √3 i. Participants note that the constant sum of distances indicates an elliptical shape in the complex plane. The challenge lies in deriving the explicit form of ƒ(z), with references to Liouville's theorem suggesting the potential for boundedness in the solution.
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MakVish said: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.