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Math Help - given u, find v, the corresponding harmonic function of u.

  1. #1
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    given u, find v, the corresponding harmonic function of u.

    Can someone check my working here? thanks

    form the corresponding analytic function f(z)= u+iv
    u=4xy^3 - 4x^3y+x
     u_x=4y^3-12x^2y+1=v_y

    v=\int4y^3-12x^2y+1\cdot dx
    v=y^4-6y^2x^2+y+f(x)

    v_x=-12xy^2 + f'(x) = -u_y= -12xy^2 + 4x^3
    f'(x)= 4x^3
    f(x)=x^4 + c
     v=y^4 - 6y^2x^2 + y + x^4 +c

     f(z) = 4xy^3 - 4x^3y+x + i(y^4 - 6y^2x^2 + y + x^4 +c)
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    You answer is correct. If you want to check it just verify the Cauchy Riemann equations hold. (They do)
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  3. #3
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    Opalg's Avatar
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    You might just want to conclude by saying that f(z) = iz^4+z+ic.
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