Proof

Since the real and imaginary parts of any analytic function are harmonic functions, and all harmonic functions are solutions of the biharmonic equation, it is only necessary to show that the general Westergaard function F(z) is analytic to verify it satisfies the biharmonic equation.

The Cauchy-Riemann conditions imply that F(z) is analytic iff

  • {partial Re F}/{partial y} = -{partial Im F}/{partial x} = -Im F prime, and
  • {partial Im F}/{partial y} = {partial Re F}/{partial x} = Re F prime.

Now, F(z) takes on only real values, so Re F(z) = F(z) and Im F(z) =0, which reduce the Cauchy-Riemann conditions to,

  • {partial Re F}/{partial y} = {partial F}/{partial y} = 0, and
  • 0 = {partial Re F}/{partial x} = Re F prime = {partial F}/{partial x} = F prime.

The Cauchy-Riemann conditions for analytic functions Y and Z imply

  • {partial Re Z}/{partial y} = -{partial Im Z}/{partial x} = -Im Z prime,
  • {partial Im Z}/{partial y} = {partial Re Z}/{partial x} = Re Z prime,
  • {partial Re Y}/{partial y} = -{partial Im Y}/{partial x} = -Im Y prime, and
  • {partial Im Y}/{partial y} = {partial Re Y}/{partial x} = Re Y prime.

The first two Cauchy-Riemann conditions show that, for the general Westergaard function F(z),

  • {partial F}/{partial x} = Re hat{Z} + y(Im Z + Im Y), and
  • {partial F}/{partial y} = Im hat{Y} + y(Re Z + Re Y).

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