Math Problem Statement
Solution
To solve this problem, we need to find the analytic function , where the real part is given, and . This problem can be approached using the Cauchy-Riemann equations to find the imaginary part and then determine the full analytic function.
Given:
- implies that at , .
Step 1: Use the Cauchy-Riemann equations.
For to be analytic, the Cauchy-Riemann equations must hold:
First, we compute the partial derivatives of .
Step 2: Use Cauchy-Riemann equations to find .
From :
Now integrate with respect to :
Next, use :
Equating this to , we have:
Simplifying:
Integrating with respect to :
Thus, .
Step 3: Determine the constant .
We know that . At (i.e., and ):
Since and , we have:
Thus, .
Step 4: Write the analytic function.
The analytic function is:
Or in terms of , this is the final analytic function.
Do you have any questions or need further details?
Here are 5 related questions:
- What are the Cauchy-Riemann equations and how are they used to determine analyticity?
- How do you integrate functions involving multiple variables like in this problem?
- What is the significance of the condition ?
- How can the real and imaginary parts of an analytic function be interpreted geometrically?
- Can the form of an analytic function always be uniquely determined from its real part?
Tip: When solving for analytic functions, always start by verifying the Cauchy-Riemann equations before attempting to integrate for the imaginary part.
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Math Problem Analysis
Mathematical Concepts
Complex Analysis
Cauchy-Riemann Equations
Analytic Functions
Partial Derivatives
Formulas
Cauchy-Riemann equations: ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x
v(x, y) = ∫ (partial derivative of u w.r.t x or y)
w(z) = u(x, y) + iv(x, y)
Theorems
Cauchy-Riemann Theorem
Analytic Function Theorem
Suitable Grade Level
Undergraduate
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