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.

Explore how analytic functions relate to Cauchy-Riemann equations. Analyze why two different equations for complex functions, f(z) and g(z), are not identical despite sharing underlying conditions. This article delves into the implications of these equations in complex analysis.

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