Let's go through the problems one by one.

Problem 1:

You are asked to show that the given functions $u(x, y)$ are harmonic in $\mathbb{R}^2$, i.e.,

$\Delta u(x, y) = \frac{\partial^2 u}{\partial x^2}(x, y) + \frac{\partial^2 u}{\partial y^2}(x, y) = 0$

For each of the following functions:

(a) $u(x, y) = e^{-y} \cos x$

• Compute $\frac{\partial u}{\partial x}$ and $\frac{\partial^2 u}{\partial x^2}$.
• Compute $\frac{\partial u}{\partial y}$ and $\frac{\partial^2 u}{\partial y^2}$.
• Add the second partial derivatives and verify if the sum is zero.

(b) $u(x, y) = xy + 3x^2y - y^3$

• Repeat the same process as above: find the second partial derivatives with respect to $x$ and $y$, and check if they sum to zero.

(c) $u(x, y) = \frac{-y}{x^2 + y^2}$

• This function involves a quotient, so apply the quotient rule when finding the partial derivatives.
• Verify if the Laplacian sums to zero.

Additionally, you are asked to find the harmonic conjugate $v(x, y)$ for each function. Recall that if $f(z) = u(x, y) + iv(x, y)$ is analytic in $\mathbb{C}$, then $u$ and $v$ must satisfy the Cauchy-Riemann equations:

$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

For each $u(x, y)$, solve these equations to find $v(x, y)$.

Problem 2:

You are asked to solve Laplace's equation in polar coordinates:

$\frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} = 0$

(a)

You need to derive Laplace's equation in polar coordinates and show it matches the provided form.

(b)

Show that $u(r, \theta) = \log |z| = \log r$ is harmonic on the punctured plane $\mathbb{C} \setminus \{0\}$, by substituting this into the polar Laplace equation and verifying it holds.

(c)

You need to show that $\log |z|$ has no conjugate harmonic function on $\mathbb{C} \setminus \{0\}$, but does on the slit plane $\mathbb{C} \setminus (-\infty, 0]$. This requires analyzing the behavior of $\log z$ and its branches.

Problem 3:

Evaluate the following integrals over the unit circle $\gamma$ with the counterclockwise orientation:

(a) $\int_{\gamma} z \, dz$

This integral can be evaluated using the parametrization of the unit circle, $z = e^{i\theta}$, where $\theta \in [0, 2\pi]$.

(b) $\int_{\gamma} \overline{z} \, dz$

Here, $\overline{z} = \frac{1}{z}$ on the unit circle, so rewrite the integral using this expression and solve.

I will now proceed to solve each part. Do you have any specific part you'd like to focus on first?