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.