To show whether the function $f(z) = \frac{\overline{z}}{|z|}$ satisfies the Cauchy-Riemann (C-R) equations at the origin, let's proceed step by step.

Step 1: Express $f(z)$ in terms of $z = x + iy$

Let $z = x + iy$, where $x$ and $y$ are real numbers. Then: $\overline{z} = x - iy \quad \text{and} \quad |z| = \sqrt{x^2 + y^2}.$

Thus, the function can be rewritten as: $f(z) = \frac{\overline{z}}{|z|} = \frac{x - iy}{\sqrt{x^2 + y^2}}.$

In terms of real and imaginary components, we have: $f(z) = u(x, y) + iv(x, y),$ where $u(x, y) = \frac{x}{\sqrt{x^2 + y^2}}, \quad v(x, y) = -\frac{y}{\sqrt{x^2 + y^2}}.$

Step 2: C-R Equations

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

We need to compute the partial derivatives of $u(x, y)$ and $v(x, y)$ and evaluate these equations at the origin ($x = 0, y = 0$).

Step 3: Compute Partial Derivatives

Using $u(x, y) = \frac{x}{\sqrt{x^2 + y^2}}$ and $v(x, y) = -\frac{y}{\sqrt{x^2 + y^2}}$:

Partial derivatives of $u(x, y)$:

$\frac{\partial u}{\partial x} = \frac{\partial}{\partial x} \left( \frac{x}{\sqrt{x^2 + y^2}} \right) = \frac{\sqrt{x^2 + y^2} - x \cdot \frac{2x}{2\sqrt{x^2 + y^2}}}{x^2 + y^2}.$

Simplify: $\frac{\partial u}{\partial x} = \frac{y^2}{(x^2 + y^2)^{3/2}}.$

$\frac{\partial u}{\partial y} = \frac{\partial}{\partial y} \left( \frac{x}{\sqrt{x^2 + y^2}} \right) = \frac{-x \cdot y}{(x^2 + y^2)^{3/2}}.$

Partial derivatives of $v(x, y)$:

$\frac{\partial v}{\partial x} = \frac{\partial}{\partial x} \left( -\frac{y}{\sqrt{x^2 + y^2}} \right) = \frac{y \cdot x}{(x^2 + y^2)^{3/2}}.$

$\frac{\partial v}{\partial y} = \frac{\partial}{\partial y} \left( -\frac{y}{\sqrt{x^2 + y^2}} \right) = \frac{-\sqrt{x^2 + y^2} + y \cdot \frac{2y}{2\sqrt{x^2 + y^2}}}{x^2 + y^2}.$

Simplify: $\frac{\partial v}{\partial y} = -\frac{x^2}{(x^2 + y^2)^{3/2}}.$

Step 4: Evaluate at the Origin ($x = 0, y = 0$)

At $x = 0$ and $y = 0$, the terms $\sqrt{x^2 + y^2}$ in the denominator make the partial derivatives undefined because $|z| = 0$ at the origin.

Thus, the function $f(z)$ is not differentiable at the origin, and the C-R equations cannot be satisfied.

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Conclusion:

The function $f(z) = \frac{\overline{z}}{|z|}$ does not satisfy the Cauchy-Riemann equations at the origin because the partial derivatives become undefined at that point.

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Related Questions:

1. What are the Cauchy-Riemann equations, and why are they significant?
2. How do we determine if a function is differentiable in the complex plane?
3. Can $f(z)$ be differentiable anywhere other than the origin? Why or why not?
4. What is the geometric interpretation of $f(z) = \frac{\overline{z}}{|z|}$?
5. How does the modulus of $z$ affect differentiability in complex functions?

Tip: Always check the domain of the function and ensure the partial derivatives are well-defined before applying the Cauchy-Riemann equations.