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 f(z)=zzf(z) = \frac{\overline{z}}{|z|} with respect to the Cauchy-Riemann (C-R) equations and the existence of f(0)f'(0), we proceed as follows:


Step 1: Rewrite f(z)f(z) in terms of zz

Let z=x+iyz = x + iy, where x,yRx, y \in \mathbb{R}, so:

  • z=xiy\overline{z} = x - iy,
  • z=x2+y2|z| = \sqrt{x^2 + y^2}.

Thus: f(z)=zz=xiyx2+y2.f(z) = \frac{\overline{z}}{|z|} = \frac{x - iy}{\sqrt{x^2 + y^2}}.

This can be separated into real and imaginary parts: f(z)=u(x,y)+iv(x,y),f(z) = u(x, y) + iv(x, y), where: u(x,y)=xx2+y2,v(x,y)=yx2+y2.u(x, y) = \frac{x}{\sqrt{x^2 + y^2}}, \quad v(x, y) = -\frac{y}{\sqrt{x^2 + y^2}}.


Step 2: C-R equations at the origin

The Cauchy-Riemann equations are: ux=vy,uy=vx.\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.

At the origin (x,y)=(0,0)(x, y) = (0, 0):

  • When (x,y)(0,0)(x, y) \neq (0, 0), x2+y20\sqrt{x^2 + y^2} \neq 0. However, at (0,0)(0, 0), f(z)f(z) is undefined because z=0|z| = 0 in the denominator. We analyze the C-R equations separately under the assumption u(x,y)u(x, y) and v(x,y)v(x, y) are set to zero at (0,0)(0, 0).

The partial derivatives at the origin: ux=uy=vx=vy=0.\frac{\partial u}{\partial x} = \frac{\partial u}{\partial y} = \frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} = 0.

Thus, the C-R equations: ux=vy,uy=vx,\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}, are satisfied trivially at (0,0)(0, 0).


Step 3: Check if f(0)f'(0) exists

The derivative of f(z)f(z) is given by: f(z)=limh0f(h)f(0)h.f'(z) = \lim_{h \to 0} \frac{f(h) - f(0)}{h}.

At z=0z = 0, f(0)f(0) is undefined because z=0|z| = 0 leads to a division by zero in f(z)f(z). Even if we define f(0)=0f(0) = 0 artificially, the value of f(h)f(h) depends on the direction in which h0h \to 0.

To see this, consider z=reiθz = re^{i\theta} in polar coordinates. Then: f(z)=zz=eiθ.f(z) = \frac{\overline{z}}{|z|} = e^{-i\theta}. As z0z \to 0 along different angles θ\theta, the value of f(z)f(z) does not converge to a unique value. Therefore, f(0)f'(0) does not exist.


Conclusion

  1. The Cauchy-Riemann equations are satisfied at the origin (0,0)(0, 0), but this is trivial since uu and vv are zero at that point.
  2. The derivative f(0)f'(0) does not exist because f(z)f(z) is undefined at z=0z = 0, and the directional limit depends on θ\theta.

Would you like me to expand on any part of the proof?

Here are related questions to explore:

  1. What is the significance of the Cauchy-Riemann equations in complex analysis?
  2. How is the derivative of a complex function defined differently from real functions?
  3. Can a function be differentiable at a point without satisfying the Cauchy-Riemann equations?
  4. Why is the directionality of the limit crucial for determining f(0)f'(0)?
  5. How can the concept of holomorphic functions be applied here?

Tip: When analyzing functions involving division by z|z|, always check the behavior at z=0z = 0 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