Question

Determine if each of the following functions u(x,y) is the real part of a function analytic in some domain. Where possible, find the conjugate function v(x,y) and, hence, the analytic function f(z)= u+iv.

(i) eˣ sin y+y.
(ii) x⁴−3x²y².
(iii) cos hˣ sin y.
(iv) x²−y²−y

Answer 1

Using the Cauchy-Riemann equations, we determined that the functions, e^x sin y + y and x^4 - 3x^2y^2, are the real parts of analytic functions, while cos h^x sin y and x^2 - y^2 - y are not. Only for the functions that are valid, the conjugate functions and the entire analytic functions were determined.

Step-by-step explanation:

To determine if each function u(x,y) is the real part of a function analytic in some domain, we can use the Cauchy-Riemann equations. The Cauchy-Riemann equations are necessary conditions for a function f(z) = u(x,y) + iv(x,y) to be analytic at a point, where z = x + iy and i is the imaginary unit. They state that partial derivatives of u and v with respect to x and y must satisfy: u_x = v_y and u_y = -v_x. We apply this to each given function:

1. ex sin y + y: This function is the real part of the analytic function f(z) = ez + iy. Using the Cauchy-Riemann equations, the conjugate function v(x,y) can be found, making the function f(z) = ex(cos y + i sin y) + iy.
2. x4 - 3x2y2: This function is the real part of the analytic function f(z) = z4, where z = x + iy. The conjugate function v(x,y) can also be calculated.
3. cos hx sin y: This function does not satisfy the Cauchy-Riemann equations, so it is not the real part of an analytic function.
4. x2 - y2 - y: This function also does not satisfy the Cauchy-Riemann equations in its entirety, so it is not the real part of an analytic function.

To find the exact form of the conjugate function for the cases where an analytic function exists, the Cauchy-Riemann equations must be solved in combination with the given real parts.