To find the analytic function $f(z) = u(x, y) + iv(x, y)$ using the Milne-Thompson method, given that:

$u(x, y) + v(x, y) = \frac{\sin(2x)}{\cosh(2y) - \cos(2x)},$

we follow these steps:

1. Express $u + v$ in terms of $x$ and $y$:

$u + v = \frac{\sin(2x)}{\cosh(2y) - \cos(2x)}.$

Here, $u(x, y)$ is the real part and $v(x, y)$ is the imaginary part of $f(z)$.

2. Apply the Milne-Thompson Method:

The Milne-Thompson method utilizes the fact that the Cauchy-Riemann equations must be satisfied for $f(z)$ to be analytic.

In this method, we substitute $z = x + iy$, which gives $f(z) = f(x + iy)$. This leads us to introduce the complex variable transformations $x$ and $y$ using trigonometric and hyperbolic identities.

For $\sin(2x)$ and $\cos(2x)$, we apply the following substitutions based on the properties of complex variables:

• $\sin(2x) \rightarrow \frac{e^{i2z} - e^{-i2z}}{2i}$,
• $\cos(2x) \rightarrow \frac{e^{i2z} + e^{-i2z}}{2}$,
• $\cosh(2y) \rightarrow \frac{e^{2y} + e^{-2y}}{2}$.

3. Find the analytic function $f(z)$:

Now, we substitute $z = x + iy$ into the expression for $u + v$.

Let $z = x + iy$, where:

• $\sin(2x) = \frac{e^{i2z} - e^{-i2z}}{2i}$,
• $\cos(2x) = \frac{e^{i2z} + e^{-i2z}}{2}$,
• $\cosh(2y) = \frac{e^{2y} + e^{-2y}}{2}$.

Thus, the analytic function $f(z)$ is expressed as:

$f(z) = \frac{\sin(2z)}{\cosh(2z) - \cos(2z)}.$

This is the required analytic function $f(z)$.

Summary:

The analytic function corresponding to the given real and imaginary parts, determined using the Milne-Thompson method, is:

$f(z) = \frac{\sin(2z)}{\cosh(2z) - \cos(2z)}.$

Would you like more details on any step, or do you have any questions?

Here are 5 related questions:

1. How do the Cauchy-Riemann equations apply to analytic functions?
2. What are the conditions for a function to be analytic in a domain?
3. Can you explain the Milne-Thompson method in more detail?
4. How can you apply substitutions for trigonometric functions in complex analysis?
5. What role do hyperbolic identities play in finding analytic functions?

Tip: In complex analysis, remember that any analytic function satisfies the Cauchy-Riemann equations, ensuring both real and imaginary parts are harmonically related.