How do I verify that the function $u(x,y)=x^2-y^2-y$ is harmonic
Verify that the function $u(x,y)=x^2-y^2-y$ is harmonic
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$\begingroup$
sequences-and-series
harmonic-functions
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3What's the definition of a function being harmonic? Check that your $u$ satisfies it. – 2012-11-10
3 Answers
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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 $u_{xx}+u_{yy}=2(a+b)\implies b=-a$