To find the analytic function \( f(z) \) where \( f(z) = z^3 \), we can follow a step-by-step approach to verify that \( f(z) \) is analytic.

### 1. **Definition of Analytic Function**:
An analytic function is one that satisfies the Cauchy-Riemann equations in a given domain, and whose partial derivatives are continuous in that domain.

### 2. **Function in terms of \( z = x + iy \)**:
We start by expressing \( f(z) \) in terms of its real and imaginary components. Let \( z = x + iy \), where \( x \) and \( y \) are real numbers and \( i \) is the imaginary unit.

The function \( f(z) = z^3 \) becomes:
\[
f(z) = (x + iy)^3
\]

Expanding this:
\[
f(z) = x^3 + 3x^2(iy) + 3x(iy)^2 + (iy)^3
\]
\[
f(z) = x^3 + 3x^2iy - 3xy^2 - iy^3
\]
\[
f(z) = (x^3 - 3xy^2) + i(3x^2y - y^3)
\]

Thus, we can write \( f(z) \) as:
\[
f(z) = u(x, y) + iv(x, y)
\]
where:
- \( u(x, y) = x^3 - 3xy^2 \) (real part),
- \( v(x, y) = 3x^2y - y^3 \) (imaginary part).

### 3. **Cauchy-Riemann Equations**:
For \( f(z) \) to be analytic, the real part \( u(x, y) \) and the imaginary part \( v(x, y) \) must satisfy the Cauchy-Riemann equations:
\[
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
\]

#### First equation:
\[
\frac{\partial u}{\partial x} = 3x^2 - 3y^2, \quad \frac{\partial v}{\partial y} = 3x^2 - 3y^2
\]
Both are equal, so the first Cauchy-Riemann equation is satisfied.

#### Second equation:
\[
\frac{\partial u}{\partial y} = -6xy, \quad \frac{\partial v}{\partial x} = 6xy
\]
Since \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \), the second Cauchy-Riemann equation is also satisfied.

### 4. **Conclusion**:
Since both Cauchy-Riemann equations are satisfied and the partial derivatives are continuous, \( f(z) = z^3 \) is an analytic function.

Would you like further details or have any questions?

Here are five related questions to explore further:

1. What are the Cauchy-Riemann equations in polar coordinates?
2. Can \( z^n \) (for any \( n \)) be analytic for all \( n \)?
3. How does the analyticity of a function relate to complex differentiability?
4. What are the physical interpretations of analytic functions in fluid dynamics?
5. Can you generalize this result to other powers of \( z \) and test analyticity?

**Tip:** An analytic function is also holomorphic, meaning it is differentiable everywhere in its domain.

Explore how to verify the analyticity of the function f(z) = z^3 by applying the Cauchy-Riemann equations. This step-by-step guide covers complex analysis concepts such as real and imaginary components, partial derivatives, and the verification of Cauchy-Riemann conditions. Suitable for undergraduate students studying complex analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation