Math Problem Statement
show that the function f(z)=(z bar)/|z| satisfies the C-R equation at the origin.but f'(0) does not exist
Solution
To analyze the given function with respect to the Cauchy-Riemann (C-R) equations and the existence of , we proceed as follows:
Step 1: Rewrite in terms of
Let , where , so:
- ,
- .
Thus:
This can be separated into real and imaginary parts: where:
Step 2: C-R equations at the origin
The Cauchy-Riemann equations are:
At the origin :
- When , . However, at , is undefined because in the denominator. We analyze the C-R equations separately under the assumption and are set to zero at .
The partial derivatives at the origin:
Thus, the C-R equations: are satisfied trivially at .
Step 3: Check if exists
The derivative of is given by:
At , is undefined because leads to a division by zero in . Even if we define artificially, the value of depends on the direction in which .
To see this, consider in polar coordinates. Then: As along different angles , the value of does not converge to a unique value. Therefore, does not exist.
Conclusion
- The Cauchy-Riemann equations are satisfied at the origin , but this is trivial since and are zero at that point.
- The derivative does not exist because is undefined at , and the directional limit depends on .
Would you like me to expand on any part of the proof?
Here are related questions to explore:
- What is the significance of the Cauchy-Riemann equations in complex analysis?
- How is the derivative of a complex function defined differently from real functions?
- Can a function be differentiable at a point without satisfying the Cauchy-Riemann equations?
- Why is the directionality of the limit crucial for determining ?
- How can the concept of holomorphic functions be applied here?
Tip: When analyzing functions involving division by , always check the behavior at carefully for continuity and differentiability.
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Math Problem Analysis
Mathematical Concepts
Complex Analysis
Cauchy-Riemann Equations
Complex Differentiation
Formulas
f(z) = (\overline{z}) / |z|
Cauchy-Riemann equations: \partial u/\partial x = \partial v/\partial y and \partial u/\partial y = -\partial v/\partial x
Complex derivative: f'(z) = lim_{h -> 0} (f(z+h) - f(z))/h
Theorems
Cauchy-Riemann Equations
Conditions for Differentiability of Complex Functions
Suitable Grade Level
Undergraduate Level