Are the following functions analytic? f(z) -
 izz

To determine if the function \( f(z) = iz\bar{z} \) is analytic, we need to check whether it satisfies the Cauchy-Riemann equations, which are necessary (but not sufficient) conditions for a function to be analytic.

### Step 1: Express \( f(z) \) in terms of \( z = x + iy \)
Let \( z = x + iy \), where \( x \) and \( y \) are real variables, and \( i \) is the imaginary unit. The complex conjugate \( \bar{z} \) is given by:

\[
\bar{z} = x - iy
\]

Thus, the function \( f(z) = iz\bar{z} \) can be written as:

\[
f(z) = i(x + iy)(x - iy) = i(x^2 + y^2)
\]

### Step 2: Separate \( f(z) \) into its real and imaginary parts
Let \( f(z) = u(x, y) + iv(x, y) \), where \( u(x, y) \) and \( v(x, y) \) are the real and imaginary parts of the function, respectively.

Expanding the expression:

\[
f(z) = i(x^2 + y^2)
\]

This gives:

\[
u(x, y) = -y^2 + x^2, \quad v(x, y) = 0
\]

### Step 3: Apply the Cauchy-Riemann equations
The Cauchy-Riemann equations are:

\[
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
\]

Compute the partial derivatives:

\[
\frac{\partial u}{\partial x} = \frac{\partial (x^2 - y^2)}{\partial x} = 2x, \quad \frac{\partial u}{\partial y} = \frac{\partial (x^2 - y^2)}{\partial y} = -2y
\]

\[
\frac{\partial v}{\partial x} = \frac{\partial 0}{\partial x} = 0, \quad \frac{\partial v}{\partial y} = \frac{\partial 0}{\partial y} = 0
\]

### Step 4: Check if the Cauchy-Riemann equations hold
- \(\frac{\partial u}{\partial x} = 2x\) and \(\frac{\partial v}{\partial y} = 0\): These are not equal unless \( x = 0 \).
- \(\frac{\partial u}{\partial y} = -2y\) and \(\frac{\partial v}{\partial x} = 0\): These are not equal unless \( y = 0 \).

Since the Cauchy-Riemann equations do not hold everywhere (they only hold on the x-axis and y-axis individually), the function \( f(z) = iz\bar{z} \) is **not analytic** in any open region of the complex plane.

Would you like more details or have any further questions?

### Related Questions:
1. What are the Cauchy-Riemann equations, and why are they important?
2. How do you determine if a function is analytic in a specific region?
3. What is the difference between holomorphic and analytic functions?
4. Can you explain why a function might satisfy the Cauchy-Riemann equations but still not be analytic?
5. What is the geometric interpretation of an analytic function in the complex plane?

### Tip:
When testing whether a function is analytic, always ensure the Cauchy-Riemann equations hold in an open region, not just at isolated points.

Explore the analyticity of the function f(z) = iz\bar{z} through a detailed analysis of the Cauchy-Riemann equations. Understand why this function is not analytic in any open region of the complex plane. Suitable for advanced undergraduate students and researchers.

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