MAT334-2018F > Term Test 1

TT1 Problem 3 (noon)

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Victor Ivrii:
(a) Show that $v(x,y)= xe^x \sin (y) +y e^x\cos(y)$ is a harmonic function.

(b) Find the harmonic conjugate function $u(x,y)$.

(c) Consider $u(x,y)+iv(x,y)$ and write it as a function $f(z)$ of $z=x+iy$.

Ye Jin:
(a) WTS $v_{xx} + v_{yy}=0$
$v_x = e^xsiny+xe^xsiny+ye^xcosy$, $v_{xx}=e^xsiny+e^xsiny+xe^xsiny+ye^xcosy$
$v_y =xe^xcosy+e^xcosy-ye^xsiny$, $v_{yy}=-xe^xsiny-e^xsiny-e^xsiny-ye^xcosy$
so, $v_{xx} + v_{yy}=0$

(b) Since v is harmonic, then it is analytic.
$v_x=u_y, -v_y=u_x$
$u=\int v_x dy=\int e^xsiny+xe^xsiny+ye^xcosy dy$
$= -e^xcosy-xe^xcosy+e^x(ysiny+cosy)+h(x)$
$u_x=-e^xcosy-e^xcosy-xe^xcosy+e^x(ysiny+cosy)+h^{'}(x)$
$=-e^xcosy-xe^xcosy+e^xysiny+h^{'}x$
so,$-xe^xcosy-e^xcosy+ye^xsiny=-e^xcosy-xe^xcosy+e^xysiny+h^{'}(x)$
so, $h^{'}(x)=0$
h(x)=c
$u=-xe^xcosy+e^xysiny+c$

(c) $f(z)=u+iv= -xe^xcosy+e^xysiny+c+ixe^xsiny+iye^xcosy$
$=-e^xcosy(x-iy)+e^xsiny(y+ix)+c$
$=-e^xcos(y)\bar{z}+e^xsin(y)i\bar{z}+c$
$=\bar{z}e^x(cosy+isiny)+c$
$=\bar{z}e^{Rez}e^{Imz}+c$
$=\bar{z}e^z+c$

Meng Wu:
For part$(b)$:
CR-equation is:
$$\frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}; \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$

Chae Young Oh:
I think Ye Jin's CR equations are correct, because we want to find the harmonic conjugate of v (so vx=uy,−vy=ux) , not the harmonic conjugate of u (in that case ux=vy, uy=-vx).

Meng Wu:
.