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I got the answer "no such $v$ exists" for both exercises. However, I am very surprised, since the $u's$ I have been given seem very nice and arbitrary. I would like to know if I solved them correctly, and if so, what underlying reason makes these seemingly nice functions so hostile?

$1) u(z) = x^3$

Using the Cauchy Riemann equations, we need $3x^2 = v_y$ and $0 = v_x$. The first equation gives us $v(z) = 3x^2y + g_1 (y)$, and the second gives us $v(z) = g_2(y)$. There is no $v(z)$ which satisfies these requirements.

$2) u(z) = x^2+y$

Using the Cauchy Riemann equations, we need $2x = v_y$ and $-1 = v_x$. The first equation gives us $v(z) = 2xy + g_1 (y)$, and the second gives us $v(z) = -x + g_2(y)$. There is no $v(z)$ which satisfies these requirements.

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    A function can be the real (or imaginary) part of a holomorphic function iff it is harmonic: $\Delta u(x,y) = 0$. – 2017-02-22

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For such a function to exist on an open set, $u$ must be harmonic: that is, $$ u_{xx}+u_{yy}=0 $$

To see that this is necessary, suppose $u$ and $v$ satisfy the Cauchy-Riemann equations: $$ u_x=v_y\\ u_y=-v_x $$ Differentiating the first equation by $x$ gives $u_{xx}=v_{yx}$; differentiating the second by $y$ gives $u_{yy}=-v_{xy}$. By the symmetry of partial derivatives, it follows that $u_{xx}=-u_{yy}$.

(Such a function $v$ is called a harmonic conjugate of $u$.)

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    So I gather $u_{xx} + u_{yy} = 0$ is a necessary but not sufficient condition to ensure that $f$ is differentiable? – 2017-02-22
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    @Ovi: It's locally sufficient. That is, if $u$ is harmonic, then for every $z$ in its domain there's some open set containing $z$ on which $u$ has a harmonic conjugate. There are times when you can't patch them together to get a global harmonic conjugate for $u$. The classic example of this is $u(z)=\log(|z|)$. It "wants" to have $v(z)=\arg z$ as its harmonic conjugate, but you can't define that consistently on $u$'s entire domain ($\Bbb{C} - \{0\}$). – 2017-02-22
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    Thanks, I think I need to do some reading on harmonic functions. – 2017-02-22