Question

For a given \[ u(x,y)=x^3-3xy^2+3x^2-3y^2+1, \] the analytic function \[ f(z)=u+iv \] is

• A. \(z^3-3z^2+c\)
• B. \(z^3+3z^2+c\)
• C. \(3z^2-z^3+c\)
• D. \(z^2-2z^3+c\)

Hint:

Useful identities: \[ \boxed{ \begin{aligned} \Re(z^2)&=x^2-y^2, \Re(z^3)&=x^3-3xy^2. \end{aligned} } \] These are frequently used to identify analytic functions.

Answer 1

The Correct Option is B

Step 1: Recall standard real parts. Since \[ z=x+iy, \] we have \[ \Re(z^3)=x^3-3xy^2, \] and \[ \Re(3z^2)=3(x^2-y^2). \]

Step 2: Compare with the given function. Adding, \[ \Re(z^3+3z^2) = x^3-3xy^2+3x^2-3y^2, \] which matches the given real part except for the constant \(1\). Hence, \[ f(z)=z^3+3z^2+c, \] where \(c\) is an arbitrary constant. Therefore, \[ \boxed{(B)} \] is the correct answer.