show that the fuction f(z) = z bar is not analytic anywhere in the complex plane

To show that the function \( f(z) = \bar{z} \) (the complex conjugate of \( z \)) is **not analytic** anywhere in the complex plane, we need to verify whether it satisfies the **Cauchy-Riemann equations**, which are necessary conditions for a function to be analytic.

### 1. Writing \( z \) in terms of real and imaginary parts

Let \( z = x + iy \), where:
- \( x = \Re(z) \) is the real part of \( z \),
- \( y = \Im(z) \) is the imaginary part of \( z \).

The complex conjugate of \( z \) is given by:
\[
f(z) = \bar{z} = x - iy
\]

Now, express \( f(z) \) as a function of \( x \) and \( y \):
\[
f(z) = u(x, y) + iv(x, y)
\]
where:
- \( u(x, y) = x \) is the real part of \( f(z) \),
- \( v(x, y) = -y \) is the imaginary part of \( f(z) \).

### 2. Applying the Cauchy-Riemann equations

For \( f(z) \) to be analytic, the Cauchy-Riemann equations must hold:
\[
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
\]

Let’s compute the partial derivatives:

- \( u(x, y) = x \) and \( v(x, y) = -y \),
- \( \frac{\partial u}{\partial x} = 1 \),
- \( \frac{\partial u}{\partial y} = 0 \),
- \( \frac{\partial v}{\partial x} = 0 \),
- \( \frac{\partial v}{\partial y} = -1 \).

Now, check the Cauchy-Riemann equations:

1. \( \frac{\partial u}{\partial x} = 1 \) and \( \frac{\partial v}{\partial y} = -1 \): these are not equal, so the first Cauchy-Riemann equation is **not satisfied**.
2. \( \frac{\partial u}{\partial y} = 0 \) and \( -\frac{\partial v}{\partial x} = 0 \): this is satisfied, but the first equation is already violated.

### 3. Conclusion

Since the Cauchy-Riemann equations are not fully satisfied, the function \( f(z) = \bar{z} \) is **not analytic** at any point in the complex plane.

This means \( f(z) \) is **not differentiable** anywhere, so it cannot be analytic anywhere in the complex plane.

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Would you like to explore more details or have further questions?

Here are 5 related questions:
1. What are the necessary and sufficient conditions for a function to be analytic?
2. How do we define differentiability for complex functions?
3. Can a function be analytic if it fails the Cauchy-Riemann equations at only one point?
4. What are some examples of functions that are analytic everywhere in the complex plane?
5. How does the complex conjugate operation affect differentiability?

**Tip:** Analytic functions are also called **holomorphic** when they are differentiable in a neighborhood of every point in a domain.

show that the function f(z) = z bar is not analytic anywhere in the complex plane

Explore why the function f(z) = z bar, the complex conjugate of z, is not analytic anywhere in the complex plane. This solution utilizes the Cauchy-Riemann equations to demonstrate the failure of analyticity, making it a key topic in undergraduate-level complex analysis.

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