Show that the partial derivatives of a harmonic function are harmonic, too.

Difficulty: 2.

Which ones of the following functions are harmonic?

For each harmonic function \(\displaystyle f\) from the list, provide a holomorphic function those real part is \(\displaystyle f\), and the harmonic conjugate of \(\displaystyle f\).

\(\displaystyle (x,y)\mapsto x; \quad (x,y)\mapsto x^2; \quad x^2+y^2; \quad x^2-y^2; \quad \log(x^2+y^2) \quad \frac{1-x^2-y^2}{1-2x+x^2+y^2} \)

Difficulty: 3.

Which ones of the following functions are harmonic?

For each harmonic function \(\displaystyle f\) from the list, provide a holomorphic function those real part is \(\displaystyle f\), and the harmonic conjugate of \(\displaystyle f\).

\(\displaystyle \quad (x,y)\mapsto y; \quad xy; \quad x^3y+xy^3; \quad x^3y-xy^3 \)

Difficulty: 3.

Near the equator, the bottom of the Pacific Ocean got punctured when the destroyer "Kazincbarcika" drilled into it. On the surface a stable, non-swirling waterspout was formed. What function \(\displaystyle f(r)\) can describe the depth in distance \(\displaystyle r\) from the axis?

Difficulty: 4. Hint is provided for this problem.

Let \(\displaystyle n\) be a positive integer and let \(\displaystyle -1<a<1\).

\(\displaystyle \int_{-\pi}^\pi \frac{\cos(nt)}{1-2a\cos t+a^2} \dt =? \qquad \int_{-\pi}^\pi \frac{\sin(nt)}{1-2a\cos t+a^2} \dt =? \)

Difficulty: 4. Hint is provided for this problem.

Show that a real polynomial on two variables is harmonic if and only if it is the real part of a complex polynomial.

Difficulty: 5.

The function \(\displaystyle u\) is harmonic inside the unit disk and continuous along the set \(\displaystyle \overline{B}(0,1)\setminus\{1\}\), and \(\displaystyle u=0\) at the points of the unit circle. Does it follow that \(\displaystyle u\equiv0\)?

Difficulty: 7. Solution is available for this problem.

Construct a kernel function \(\displaystyle \varphi:\R\times(0,\infty)\times\R\to(0,\infty)\) with the following property: whenever a function \(\displaystyle h(x,y)\) is harmonic and bounded in the interior of the upper half-plane and it is continuous on the closed half-plane then

\(\displaystyle h(x,y) = \int_{-\infty}^\infty h(t,0) \, \varphi(x,y,t) \dt \)

holds for every \(\displaystyle y>0\).

Difficulty: 9.