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The document contains a complex variables and linear algebra assignment. Here’s a summary of the questions:

1. **Determine the analytic functions**:
   - (a) Find \( f(z) = u + iv \) where \( u - v = e^{-2xy} \sin(x^2 - y^2) + e^{2xy} \cos(x^2 - y^2) \).
   - (b) Find \( g(z) \) such that \( \Re[g'(z)] = 3x^2 - 4y - 3y^2 \) and \( g(1 + i) = 0 \).

2. **Identify where functions are analytic**:
   - (a) \( f(z) = iz - |z|^2 \)
   - (b) \( f(z) = e^{1/z} \).

3. **Find bilinear transformations**:
   - (a) Map the disk \( |z - 1| > 2 \) in the \( z \)-plane onto \( |w| < 1 \).
   - (b) Find a bilinear transformation such that \( z = i \) is the only fixed point, and \( z = 1 \) maps to \( \infty \).

4. **Find images of regions or curves under mappings**:
   - (a) Map the half-plane \( \Im(z) > 0 \) under \( w = (1 + i)z + 2 \).
   - (b) Map the circle \( |z| = a \), \( a > 1 \), under \( w = z + \frac{1}{z} \), and identify fixed points.

5. **Analyze a function**:
   - (a) Obtain the Taylor series expansion of \( f(z) = \frac{1}{z^2 + (1 + 2i)z + 2i} \) about \( z = 0 \).
   - (b) Find the radius of convergence.

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Would you like help with a specific question or a detailed explanation on one of these problems?

Here are some related questions for further exploration:
1. How do you determine the radius of convergence for a Taylor series?
2. What is the process to find bilinear transformations?
3. How can you check if a function is analytic?
4. What are fixed points in the context of complex transformations?
5. How can you express a complex function as \( f(z) = u(x, y) + iv(x, y) \)?

**Tip**: When working with analytic functions, remember that the Cauchy-Riemann equations play a crucial role in determining analyticity.

The document contains a complex variables and linear algebra assignment with various questions on analytic functions, bilinear transformations, and Taylor series expansion.

This B.Tech engineering assignment covers key concepts in complex variables and linear algebra, including finding analytic functions, bilinear transformations, and Taylor series expansion. Explore detailed solutions for mapping functions and finding the radius of convergence in Taylor expansions.

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