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To solve this problem, we need to find the analytic function \( w(z) = u(x, y) + iv(x, y) \), where the real part \( u(x, y) \) is given, and \( w(0) = i \). This problem can be approached using the **Cauchy-Riemann equations** to find the imaginary part \( v(x, y) \) and then determine the full analytic function.

### Given:
- \( u(x, y) = y^3 - 3x^2 y + x^2 - y^2 \)
- \( w(0) = i \) implies that at \( z = 0 \), \( w(0) = u(0, 0) + iv(0, 0) = i \).

#### Step 1: Use the Cauchy-Riemann equations.

For \( w(z) = u(x, y) + iv(x, y) \) to be analytic, the Cauchy-Riemann equations must hold:

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

First, we compute the partial derivatives of \( u(x, y) \).

\[
\frac{\partial u}{\partial x} = -6xy + 2x
\]
\[
\frac{\partial u}{\partial y} = 3y^2 - 3x^2 - 2y
\]

#### Step 2: Use Cauchy-Riemann equations to find \( v(x, y) \).

From \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \):

\[
\frac{\partial v}{\partial y} = -6xy + 2x
\]

Now integrate with respect to \( y \):

\[
v(x, y) = -3xy^2 + 2xy + f(x)
\]

Next, use \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \):

\[
\frac{\partial v}{\partial x} = -3y^2 + 2y + f'(x)
\]

Equating this to \( -\frac{\partial u}{\partial y} \), we have:

\[
- (3y^2 - 3x^2 - 2y) = -3y^2 + 2y + f'(x)
\]

Simplifying:

\[
3x^2 = f'(x)
\]

Integrating with respect to \( x \):

\[
f(x) = x^3 + C
\]

Thus, \( v(x, y) = -3xy^2 + 2xy + x^3 + C \).

#### Step 3: Determine the constant \( C \).

We know that \( w(0) = i \). At \( z = 0 \) (i.e., \( x = 0 \) and \( y = 0 \)):

\[
w(0) = u(0, 0) + iv(0, 0) = i
\]

Since \( u(0, 0) = 0 \) and \( v(0, 0) = C \), we have:

\[
C = 1
\]

Thus, \( v(x, y) = -3xy^2 + 2xy + x^3 + 1 \).

#### Step 4: Write the analytic function.

The analytic function \( w(z) \) is:

\[
w(z) = u(x, y) + iv(x, y) = \left( y^3 - 3x^2 y + x^2 - y^2 \right) + i \left( -3xy^2 + 2xy + x^3 + 1 \right)
\]

Or in terms of \( z = x + iy \), this is the final analytic function.

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Do you have any questions or need further details?

Here are 5 related questions:
1. What are the Cauchy-Riemann equations and how are they used to determine analyticity?
2. How do you integrate functions involving multiple variables like in this problem?
3. What is the significance of the condition \( w(0) = i \)?
4. How can the real and imaginary parts of an analytic function be interpreted geometrically?
5. Can the form of an analytic function always be uniquely determined from its real part?

**Tip**: When solving for analytic functions, always start by verifying the Cauchy-Riemann equations before attempting to integrate for the imaginary part.

Намерете аналитична функция w(z) = u + iv, w(0) = i, ако u(x,y) = y^3 - 3x^2y + x^2 - y^2.

This problem explores how to find the analytic function w(z) when given the real part u(x, y) = y^3 - 3x^2y + x^2 - y^2, using the Cauchy-Riemann equations. The step-by-step solution includes finding the imaginary part v(x, y) and constructing the full analytic function w(z). Suitable for undergraduate students studying complex analysis.

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