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.
Complex analysis is a branch of mathematics that deals with the study of functions of complex numbers. It involves the analysis of how these functions behave and interact with each other.
Analytic functions are complex functions that can be expressed as a power series. They are functions that can be differentiated infinitely many times and are defined on an open subset of the complex plane.
This equation represents a geometric shape known as an ellipse in the complex plane. The points on the ellipse satisfy the condition that the sum of the distances from the points to the points 1 and -1 is equal to 4.
Some examples of analytic functions that satisfy this equation are ƒ(z) = 2z, ƒ(z) = z^2, and ƒ(z) = 2z + 2z^2. These functions all have different shapes for their corresponding ellipses, but they all satisfy the given equation.
To solve this equation, we can use complex analysis techniques such as Cauchy-Riemann equations and the Cauchy integral formula. These techniques allow us to find analytic functions that satisfy the given equation and determine their specific forms.