Problem Set 2

1.Let d,q be positive integer and k be an odd positive integer.Find the highest power of 2 which divides {\large S=\sum\limits_{n=1}^{2^{d}k} n^{q} .}

2.Let z_{1},z_{2} be two complex numbers which are linearly independent over \mathbb{R}.Conside the lattice L generated over \mathbb{Z}

L=\{n_{1}z_{1}+n_{2}z_{2}:n_{1},n_{2} \in \mathbb{Z}

Show that the Weirestrass elliptic function{\large \wp(u)=\frac{1}{u^{2}}+\sum\limits_{\omega \in L,\omega \neq 0} (\frac{1}{u-\omega^{2}}-\frac{1}{\omega^{2}}) } is absolutely and uniformly convergent on any compact subset of the complex plane which doesn’t contain any points of L.Also,show that \wp is meromorphic with a double pole only at points in L.

3.The deleted infinite broom is the set of all line segments L_{n} joining (0,0) to (1,\frac{1}{n}) and the point (1,0).Note that the segment between (0,0) and (1,0) through the X-axis is not included.

infinite_broom

Prove that the deleted infinite broom is connected but not path-connected.
The closure of the deleted infinite broom is called the infinite broom.Prove that the infinite broom is path-connected.

4.Show that the roots of Legendre Polynomial in the interval (-1,+1) are distinct.