To find the analytic function $f(z) = u + iv$ for the given function $u + v = \frac{\sin^2 x}{\cosh^2 y - \cos^2 x}$, we can employ Milne-Thompson’s Method. This method involves finding the complex potential function $f(z) = u(x, y) + iv(x, y)$, where $u(x, y)$ and $v(x, y)$ are harmonic conjugates. Here’s a step-by-step process:

1. Express the Given Function in Terms of $z = x + iy$:

   Using Milne-Thompson’s Method, we express $f(z)$ as a function of $z$ and $\overline{z}$: $z = x + iy, \quad \overline{z} = x - iy$ where $x = \operatorname{Re}(z)$ and $y = \operatorname{Im}(z)$.

2. Determine $u(x, y)$:

   Since $u + v = \frac{\sin^2 x}{\cosh^2 y - \cos^2 x}$, assume: $u(x, y) = \operatorname{Re} \left( \frac{\sin^2 x}{\cosh^2 y - \cos^2 x} \right)$

3. Formulate the Harmonic Conjugate $v(x, y)$:

   The function $u$ given satisfies the Laplace equation, which is a necessary condition for $u$ to be the real part of an analytic function. To find $v(x, y)$, we need to determine the harmonic conjugate by using the Cauchy-Riemann equations: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

4. Calculate $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$:

   Differentiate $u$ with respect to $x$ and $y$, and solve these partial derivatives to find $v(x, y)$.

5. Construct the Analytic Function $f(z)$:

   Finally, once $v(x, y)$ is determined, the function $f(z)$ can be written as: $f(z) = u(x, y) + iv(x, y)$

Would you like a more detailed breakdown of the steps, specifically the calculations for $v(x, y)$ using the Cauchy-Riemann equations? Here are some related questions that may be of interest:

1. How does Milne-Thompson’s method simplify finding an analytic function?
2. Why is the Laplace equation important in determining $u$ and $v$?
3. What is the significance of harmonic conjugates in complex analysis?
4. Can the Cauchy-Riemann equations be applied to any complex function?
5. How would the result change if $u + v$ were given differently?

Tip: When applying Milne-Thompson’s method, always verify that the function satisfies the Cauchy-Riemann equations to confirm analyticity.