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.