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A harmonic function is analytic if it satisfies the Laplace equation. If $u(x,y)=2x^2-2y^2+4xy$ is a harmonic function, then its conjugate harmonic function $v(x,y)$ is

  1. $4xy-2x^2+2y^2+ \text{constant}$
  2. $4y^2-4xy + \text{constant}$
  3. $2x^2-2y^2+ xy + \text{constant}$
  4. $-4xy+2y^2-2x^2+ \text{constant}$
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