Question

Let f(Z) = u(x, y) + i(v(x, y)) be an analytical function. If u = 5x + 2xy then v is equal to _________, where c is a constant.
1. x² + y² - 5x + c
2. x² - y² - 5xy + c
3. x² + y² + 5xy + c
4. y² - x² + 5y + c

Answer 1

Correct Answer - Option 4 : y² - x² + 5y + c

Concept:

Let f(Z) = u(x, y) + i(v(x, y)) be an analytical function.

If the real part u(x, y) of analytic function f(z) is given then to find imaginary part v(x, y) of f(z) we can use the following procedure:

1). Find u_x and u_y

2). Consider \( dv= \frac{{\partial v}}{{\partial x}}dx + \frac{{\partial v}}{{\partial y}}dy\)

3). dv = v_x dx + v_y dy = -u_y dx + u_x dy.

​4). \(v = \;\smallint \left( { - {u_y}} \right)dx + \;\smallint \left( {{u_x}} \right)dy + c\)

Calculation:

Given:

u = 5x + 2xy

u_x = 5 + 2 × y, u_y = 2 × x.

dv = v_x dx + v_y dy = -u_y dx + u_x dy = (-2 × x) dx + (5 + 2 × y) dy.

Now by integrating we get:

\(v = \;\smallint \left( { - {u_y}} \right)dx + \;\smallint \left( {{u_x}} \right)dy \)

\( v = \;\smallint \left( { - 2x} \right)dx + \;\smallint \left( {5 + 2y} \right)dy + c\)

y2 - x2 + 5y + c