JNTU B.Tech II MATHEMATICS-IV Previous Question Paper (March - 2021 - R16)

Given the real part u(x, y) = e^x(x cos y - y sin y). To find the analytic function f(z) = u + iv, we use the Milne-Thomson method. First, calculate the partial derivatives:

Let φ₁(x, y) = u_x and φ₂(x, y) = u_y. Setting x = z and y = 0:

The analytic function is given by f(z) = ∫(φ₁(z, 0) - iφ₂(z, 0)) dz = ∫(z + 1)e^z dz. Integrating by parts:

If u is harmonic, it satisfies the Laplace equation u_xx + u_yy = 0. If v is a harmonic conjugate of u, the Cauchy-Riemann equations hold: u_x = v_y and u_y = -v_x.

To show that u is a harmonic conjugate of v, we must show that v and -u satisfy the Cauchy-Riemann equations, specifically v_x = (-u)_y and v_y = -(-u)_x, which simplifies to v_x = -u_y and v_y = u_x.

From the definition of harmonic conjugate, we have u_x = v_y and u_y = -v_x. Rearranging these gives:

These are exactly the Cauchy-Riemann equations for the function g(z) = v + i(-u). Since u and v are harmonic and satisfy these equations, -u is the harmonic conjugate of v.