Proving analycity of two complex functions
We have :
a) $x^2-y^2+2ixy$
and
b) $x^2+y^2+y-2ix$
By Cauchy-Riemann equations, $u_x=v_y$ and $-v_x=u_y$ we obtain:
a) $u_x= 2x$, $v_y=2ix$, $-v_x=-2iy$, $u_y=-2y$
b) $u_x= 2x$, $v_y=0$, $-v_x=2i$, $u_y=2y+1$
Clearly, $u_x\ne v_y$ and $-v_x\ne u_y$ for both cases.
Still a) should be analytic.
Why?
Solution 1:
$u$ and $v$ are the real and imaginary parts of the functions, respectively. So, for (a), $$ u(x,y) = x^2 - y^2$$ $$ v(x,y) = 2xy$$ So $u_x = 2x, v_y = 2x$, and $u_y = -2y, -v_x = -2y$.
I'll leave the second one for you to try again.