Find the analytic function f(z) = u + iv where u + v =
sin2x
cosh2y−cos2x
,

using Milne-Thompson method

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.

Find the analytic function f(z) = u + iv where u + v = sin(2x) / (cosh(2y) - cos(2x)), using the Milne-Thompson method.

Discover how to find the analytic function f(z) = u + iv where u + v = sin(2x) / (cosh(2y) - cos(2x)) using the Milne-Thompson method. Explore complex analysis concepts, the application of Cauchy-Riemann equations, and transformations for trigonometric and hyperbolic functions.

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