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.