Complex analysis

Write $z = x + iy$. Which of these functions are holomorphic on all of $\C$ (entire)? Choose all that apply.

Compute $\displaystyle \frac{1}{2\pi i} \oint_{|z| = 2} \frac{3z^2 + 5}{z - 1}\,dz$ (circle traversed once anticlockwise).

For which real constants $a, b$ is $f(x + iy) = x^2 + a y^2 + i\,bxy$ holomorphic on $\C$?

How many zeros (counted with multiplicity) does $p(z) = z^7 - 5z^3 + 1$ have in the open unit disc $|z| < 1$?

Liouville's theorem: a bounded holomorphic function on all of $\C$ is constant. Which example shows that "on all of $\C$" cannot be weakened?

Which Lean statement says that complex conjugation is nowhere complex-differentiable?

Use the residue theorem on the upper semicircle to evaluate $\displaystyle \int_{-\infty}^{\infty} \frac{\cos 2x}{1 + x^2}\,dx$.

Write winding_number(points, p) for a closed polygon given by the list points of complex vertices (the last vertex joins back to the first) and a complex point p not on it. Add up the change in the argument of $z - p$ along each edge — the angle cmath.phase((b - p) / (a - p)) from vertex a to vertex b, which lies in $(-\pi, \pi]$ — and divide the total by $2\pi$. Return the nearest integer. (This is $\frac{1}{2\pi i}\oint \frac{dz}{z - p}$, computed exactly edge by edge.)

A student argues: "$\frac{d}{dz}\log z = \frac1z$, so $\log z$ is a primitive of $\frac1z$ on $\C \setminus \{0\}$, and therefore $\oint_{|z|=1} \frac{dz}{z} = \log(1) - \log(1) = 0$." But the integral is $2\pi i$. What is wrong?

Which statements are true? Choose all that apply.