Poisson integral

Evaluate the integral:

$$\mathcal{J} = \int_0^\pi \frac{{\rm d}x}{1-2a \cos x + a^2} \quad , \quad \left| a \right| <1$$

Solution

Pseudo sum

Let $\alpha, \beta $ be positive irrational numbers such that $\displaystyle \frac{1}{\alpha} + \frac{1}{\beta}=1$. Evaluate the (pseudo) sum:


$$\sum_{n=1}^{\infty} \left(\frac{1}{\lfloor n\alpha\rfloor^2}+\frac{1}{\lfloor n\beta\rfloor^2}\right)$$

Solution

The complex sequence does not converge uniformly

Prove that there does not exist a sequence $\{ p_n(z)\}_{n \in \mathbb{N}}$ of complex polynomials such that $p_n(z) \rightarrow \frac{1}{z}$ uniformly on $\mathcal{C}_R=\{ z \in \mathbb{C} \mid \left| z \right| = R\}$.

Solution

A double Putnam 2016 series

Evaluate the series:

$$\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\sum_{n=0}^{\infty}\frac{1}{k2^n+1}$$

(Putnam 2016)

Solution

The group is abelian

Let $\mathcal{G}$ be a finite group such that $\left ( \left | \mathcal{G} \right |  , 3 \right ) =1$. If for the elements $a, \beta \in \mathcal{G}$ holds that

$$\left ( a \beta \right )^3 = a^3 \beta^3$$

then prove that $\mathcal{G}$ is abelian.

Solution

Not Lebesgue integrable function

Let $x \in \mathbb{R}$. Given the series:

\begin{equation} \sum_{n=2}^{\infty} \frac{\sin nx}{\ln n} \end{equation}

1. Prove that $(1)$ converges forall $x \in \mathbb{R}$.
2. Prove that $(1)$ is not a Fourier series of a Lebesgue integrable function.

Solution

Identity matrix

Let $A \in \mathcal{M}_3 \left( \mathbb{R} \right)$ such that $\det A =1$ and ${\rm tr} (A)= {\rm tr} (A^{-1})=0$. Prove that $A^3=\mathbb{I}_{3 \times 3}$.

Solution

Linear map and trace

Let $f:\mathbb{F}^{n \times n} \rightarrow \mathbb{F}$ be a linear map such that $f\left ( AB \right ) = f \left ( BA \right )$  forall $A, B \in \mathbb{F}^{n \times n}$. Prove that there exists a $\kappa \in \mathbb{F}$ such that $f\left ( A \right ) = \kappa \;{\rm tr} \left ( A \right )$ forall $A \in \mathbb{F}^{n \times n}$.

Solution

$\mathbb{R}^2 \rightarrow \mathbb{R}$

Prove that there does not exist an $1-1$ and continuous mapping from $\mathbb{R}^2$ to $\mathbb{R}$.

Solution

A zero determinant

Let $A \in \mathcal{M}_n \left( \mathbb{C} \right)$ with $n \geq 2$ such that

$$\det \left ( A+X \right )=\det A + \det X$$

for every matrix $X \in \mathcal{M}_n \left( \mathbb{C} \right)$. Prove that $A=\mathbb{O}$.

Solution