You're correct; the equations derived for $g(z) = v(x, y) + i u(x, y)$ are not exactly the same as the original Cauchy-Riemann equations for $f(z) = u(x, y) + i v(x, y)$. Let's clarify this with more precision.

Analysis:

For the function $f(z) = u(x, y) + i v(x, y)$ to be analytic, the Cauchy-Riemann equations must be satisfied:

$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

Now, consider the function $g(z) = v(x, y) + i u(x, y)$. For $g(z)$ to be analytic, its real part $v(x, y)$ and imaginary part $u(x, y)$ must satisfy the Cauchy-Riemann equations:

$\frac{\partial v}{\partial x} = \frac{\partial u}{\partial y}, \quad \frac{\partial v}{\partial y} = -\frac{\partial u}{\partial x}$

Let's compare these with the original equations:

1. The equation $\frac{\partial v}{\partial x} = \frac{\partial u}{\partial y}$ for $g(z)$ is the same as $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$ for $f(z)$ with the roles of $u$ and $v$ swapped and a change in sign.

2. The equation $\frac{\partial v}{\partial y} = -\frac{\partial u}{\partial x}$ for $g(z)$ is the same as $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ for $f(z)$ with the roles of $u$ and $v$ swapped.

Conclusion:

Because the Cauchy-Riemann equations for $g(z)$ involve a sign change and a swap of roles between $u$ and $v$, $g(z)$ can be analytic if and only if $f(z)$ is analytic and the sign change and role swap are consistent with the Cauchy-Riemann conditions. Specifically, $g(z)$ is analytic if $u(x, y)$ and $v(x, y)$ satisfy the Cauchy-Riemann equations for $f(z)$ in such a way that when you swap the functions, the equations for $g(z)$ hold.

So while the equations aren't exactly the same in form, they are consistent with the Cauchy-Riemann conditions being met under the right circumstances.

Would you like to explore this further, perhaps by working through an example?

Here are some related questions:

1. What are the consequences if $u(x, y)$ and $v(x, y)$ do not satisfy the Cauchy-Riemann equations for $g(z)$?
2. Can we construct a non-analytic function by reversing the roles of $u$ and $v$?
3. How does the analyticity of $g(z)$ relate to the harmonicity of $u(x, y)$ and $v(x, y)$?
4. Are there specific examples where both $f(z)$ and $g(z)$ are analytic?
5. How would you interpret $g(z)$ geometrically if it is analytic?

Tip: Always remember that in complex analysis, small sign changes or swaps in equations can have significant implications for whether a function is analytic.