Jawaharlal Nehru Technological University Hyderabad B.Tech II MATHEMATICS-IV Previous Question Paper (October - 2020 - R16)

A function f(z) = u(x, y) + iv(x, y) is analytic at a point if it is single-valued and differentiable in a neighborhood of that point. The necessary conditions are that the partial derivatives u_x, u_y, v_x, and v_y exist, are continuous, and satisfy the Cauchy-Riemann equations: u_x = v_y and u_y = -v_x.

For f(z) = |xy|, we have u = |xy| and v = 0. At the origin (0,0):

The Cauchy-Riemann equations 0=0 and 0=-0 are satisfied at the origin. However, for f(z) to be analytic, the derivative must exist. The limit of the difference quotient is:

Along the path y = mx, the limit becomes:

While the limit exists and is 0, f(z) is not analytic because it fails to satisfy the Cauchy-Riemann equations in any neighborhood of the origin (e.g., for x, y > 0, u_x = y, which is not 0 for y ≠ 0), meaning it is not differentiable in a neighborhood of the origin.

Given v = log(x² + y²) + x - 2y, we find the partial derivatives:

Using the Cauchy-Riemann equations u_x = v_y and u_y = -v_x:

Using the Milne-Thomson method, f'(z) = u_x + iv_x. Substituting x = z and y = 0:

Integrating with respect to z:

Thus, the analytic function is f(z) = (-2 + i)z + 2i log(z) + C.