Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I can't figure out this question:

For what values of the constants A and B is the polynomial function $F(x,y) = (-5)x^5 + Ax^{3}y^{2} + Bxy^{4}$ harmonic in the whole $xy$-plane?

$A=?$ $B=?$

share|cite|improve this question
So, where did the computations of the Laplacian led you? – Davide Giraudo Dec 9 '12 at 20:16
Do you know the definition of a harmonic function? Do you know how to calculate partial derivatives? – Old John Dec 9 '12 at 20:25

(Just to save this question from going unanswered for ever)

For $F$ to be harmonic we need $$\frac{\partial^2 F}{\partial x^2} + \frac{\partial^2 F}{\partial y^2} = 0$$ throughout the plane.

Since $\frac{\partial^2 F}{\partial x^2} = -100x^3 + 6Axy^2$ and $\frac{\partial^2 F}{\partial y^2} = 2Ax^3 + 12Bxy^2$, this means that we need the function $-100x^3 + 6Axy^2 + 2Ax^3 + 12Bxy^2$ to be identically zero throughout the plane.

This implies that $2A-100=0$ and $12B+6B=0$, giving $a=50$ and $B=-25$, and thus the harmonic function is $F(x,y) = -5x^5 + 50x^3y^2 - 25xy^4.$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.