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We are given the function \( f(z) = (z + i)^3 \), and we are asked to express it in terms of the transformation \( z = re^{i\theta} \), where \( r \) is the modulus of \( z \), and \( \theta \) is the argument (angle) of \( z \).

### Step 1: Express \( z \) in polar form
Using the transformation \( z = re^{i\theta} \), we substitute this into the function:
\[
f(z) = (z + i)^3 = \left( re^{i\theta} + i \right)^3
\]

### Step 2: Combine \( re^{i\theta} \) and \( i \)
We can rewrite \( i \) in polar form as \( i = e^{i\frac{\pi}{2}} \). Now, the expression becomes:
\[
z + i = re^{i\theta} + e^{i\frac{\pi}{2}}
\]

Factor out \( e^{i\theta} \) from both terms:
\[
z + i = e^{i\theta} \left( r + e^{i\left(\frac{\pi}{2} - \theta \right)} \right)
\]

### Step 3: Cube the expression
Now, cube the result:
\[
f(z) = \left( e^{i\theta} \left( r + e^{i\left(\frac{\pi}{2} - \theta \right)} \right) \right)^3
\]

This expands to:
\[
f(z) = e^{i3\theta} \left( r + e^{i\left(\frac{\pi}{2} - \theta \right)} \right)^3
\]

### Final expression
Thus, the function \( f(z) \) in terms of polar coordinates \( z = re^{i\theta} \) is:
\[
f(z) = e^{i3\theta} \left( r + e^{i\left(\frac{\pi}{2} - \theta \right)} \right)^3
\]

Would you like a detailed explanation of the cubing process or any further clarifications?

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Here are some related questions:
1. How is \( z = re^{i\theta} \) derived from Cartesian coordinates?
2. How do we expand complex expressions when cubing?
3. What is the geometric interpretation of cubing a complex number?
4. How does the modulus \( r \) affect the transformation of \( z \)?
5. Can this function be applied to any general power \( (z + i)^n \)?

**Tip**: The polar form of complex numbers is very useful in simplifying expressions involving powers and multiplication!

f(z) = (z + i)^3 en términos de la transformación z = re^{i\theta}

Learn how to express the complex function f(z) = (z + i)^3 in terms of polar coordinates using z = re^{iθ}. This solution explores polar form conversion, properties of exponentiation in complex numbers, and cubing complex expressions.

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