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Funciones de variable compleja Sabiendo que (x+iy)
Exprese cada función indicada en la forma f(z)=u+iv

The function provided in the image is:

\[
f(z) = 3\bar{z}^2 + 2z
\]

Where \( z = x + iy \), and \( \bar{z} \) is the complex conjugate of \( z \), i.e., \( \bar{z} = x - iy \).

Now, the task is to express this function in the form \( f(z) = u + iv \), where \( u \) and \( v \) are the real and imaginary parts of the function.

### Step 1: Express \( \bar{z}^2 \)
Since \( \bar{z} = x - iy \), we can compute \( \bar{z}^2 \) as:

\[
\bar{z}^2 = (x - iy)^2 = x^2 - 2ixy - y^2
\]

### Step 2: Substitute into \( f(z) \)
Substitute \( \bar{z}^2 \) into the original function:

\[
f(z) = 3(x^2 - 2ixy - y^2) + 2(x + iy)
\]

### Step 3: Expand the terms
Expand the function:

\[
f(z) = 3(x^2 - y^2 - 2ixy) + 2x + 2iy
\]
\[
f(z) = 3(x^2 - y^2) - 6ixy + 2x + 2iy
\]

### Step 4: Collect real and imaginary parts
Now, group the real and imaginary terms:

- Real part: \( 3(x^2 - y^2) + 2x \)
- Imaginary part: \( -6xy + 2y \)

Thus, the function can be expressed as:

\[
f(z) = \left( 3(x^2 - y^2) + 2x \right) + i \left( -6xy + 2y \right)
\]

Therefore, the real part \( u \) and the imaginary part \( v \) are:

- \( u(x, y) = 3(x^2 - y^2) + 2x \)
- \( v(x, y) = -6xy + 2y \)

This is the required expression in the form \( f(z) = u + iv \).

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Would you like more details on this calculation? Here are some related questions to consider:

1. How do you compute higher powers of complex conjugates?
2. Can this function be differentiated in the context of complex analysis?
3. What is the geometric interpretation of \( \bar{z} \) in the complex plane?
4. How does the function behave for purely real or purely imaginary inputs?
5. What role does the complex conjugate play in the function's properties?

**Tip:** When working with complex functions, always consider the real and imaginary parts separately to simplify calculations.

Funciones de variable compleja Sabiendo que (x+iy) Exprese cada función indicada en la forma f(z)=u+iv

Explore how to express the complex function f(z) = 3z̅^2 + 2z, where z = x + iy, in terms of its real and imaginary parts. This step-by-step solution covers key concepts from complex analysis, including complex conjugates and real-imaginary separation.

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