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How do I verify that the function $u(x,y)=x^2-y^2-y$ is harmonic

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    What's the definition of a function being harmonic? Check that your $u$ satisfies it.2012-11-10

3 Answers 3

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$u_{xx}=2\;\;,\;u_{yy}=-2\Longrightarrow \Delta u=...?$

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It is the real part of $z^2 +iz$, therefore harmonic.

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Let $u(x,y)=ax^2+2hxy+by^2+2gx+2fy+c=0$

So, $u_x=2ax+2hy+2g, u_{xx}=2a$

Similarly, $u_{yy}=2b$

Using this, we need $u_{xx}+u_{yy}=0$ and $u_{xx}+u_{yy}=2(a+b)\implies b=-a$