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.

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