### Author Topic: Test 2 - E - Q1  (Read 3089 times)

#### yuxuan li

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##### Test 2 - E - Q1
« on: November 06, 2020, 05:09:31 PM »

Question:
(a) Show that $u(x,y)=-8x^{3}+24xy^{2}+4xy$ is a harmonic function.
(b) Find a harmonic conjugate function $v(x,y)$.
(c) Consider $u(x,y) + iv(x,y)$ and write it as a function $f(z)$ of $z=x+iy$.

\begin{align*} &u_x=-24x^{2}+24y^{2}+4y\\ &u_{xx}=-48x\\ &u_y=48xy+4x\\ &u_{yy}=48x\\ \Rightarrow &u_xx+u_yy=-48x+48x=0\\ \Rightarrow &\text{It's harmonic.}\square \end{align*}
\begin{align*} v_x&=-u_y=-48x-4x\\ v_y&=u_x=-24x^{2}+24y^{2}+4y\\ \Rightarrow & v(x,y)=\int{(-48x-4x)}dx+\phi(y)=-(24x^{2}y+2x^{2})+\phi(y)\\ v_y&=-24x^{2}+\phi'(y)\\ \Rightarrow &-24x^{2}+24y^{2}+4y=-24x^{2}+\phi'(y)\\ \Rightarrow &\phi(y)=\int{(24y^{2}+4y)}dy=8y^{3}+2y^{2}+C\\ \Rightarrow &v(x,y)=-24x^{2}y-2x^{2}+8y^{3}+2y^{2}+C\\ \end{align*}
\begin{align*} u(x,y)& + iv(x,y)\\ &=-8x^{3}+24xy^{2}+4xy-24ix^{2}y-2ix^{2}+8iy^{3}+2iy{2}+iC\\ &=-8(x^{3}-iy^{3}-3xy^{2}+3ix^{2}y)-2i(x^{2}-y^{2}+2ixy)+iC\\ &=-8(x+iy)^{3}-2i(x+iy)^{2}+iC\\ &=-8z^{3}-2iz^{2}+iC\\ \Rightarrow & f(z)=-8z^{3}-2iz^{2}+iC \end{align*}