Suppose u(x, y) and v(x, y) are the real and imaginary parts of an analytic
function f . Can g(z) = v(x, y)+ iu(x, y) be an analytic function?

Yes, the function \( g(z) = v(x, y) + i u(x, y) \) can also be an analytic function, provided that \( u(x, y) \) and \( v(x, y) \) satisfy the Cauchy-Riemann equations.

### Explanation:

Let \( f(z) = u(x, y) + i v(x, y) \) be an analytic function, where \( z = x + iy \), and \( u(x, y) \) and \( v(x, y) \) are the real and imaginary parts of \( f(z) \), respectively.

For \( f(z) \) to be analytic, the Cauchy-Riemann equations must hold:

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

Now consider the function \( g(z) = v(x, y) + i u(x, y) \). For \( g(z) \) to be analytic, the real part \( v(x, y) \) and the imaginary part \( u(x, y) \) must satisfy the Cauchy-Riemann equations in the same form, but with the roles of \( u \) and \( v \) swapped:

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

If \( u(x, y) \) and \( v(x, y) \) satisfy the original Cauchy-Riemann equations for \( f(z) \), then these new equations for \( g(z) \) are also satisfied because:

1. \( \frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y} \)
2. \( \frac{\partial v}{\partial y} = \frac{\partial u}{\partial x} \)

Thus, \( g(z) \) can indeed be an analytic function.

Would you like a detailed derivation of the Cauchy-Riemann equations or further clarification on any part of this? 

Here are some related questions:

1. What are the necessary conditions for a function to be analytic?
2. Can you provide examples of functions that are analytic everywhere?
3. How do the Cauchy-Riemann equations relate to harmonic functions?
4. What is the significance of the Cauchy-Riemann equations in complex analysis?
5. How would the result change if \( u(x, y) \) and \( v(x, y) \) were swapped initially?

**Tip:** Remember that the Cauchy-Riemann equations are not only necessary but also sufficient conditions for differentiability in complex analysis, provided the partial derivatives are continuous.

Understand how the function g(z) = v(x, y) + iu(x, y) can be analytic by satisfying the Cauchy-Riemann equations. Learn the necessary conditions for analyticity in complex analysis with detailed explanations and examples.

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