The funtion is identically zero

Let $f$ be an entire function such that

$$\left| f(z) \right| \leq \log \left( 1 + \left| z \right| \right) \quad \text{forall} \; z \in \mathbb{C}$$

Show that $f(z)=0 $ forall $z \in \mathbb{C}$.

Solution

On an entire function

Let $n \in \mathbb{N}$ and let $f$ be an entire function. Prove that for any arbitrary positive numbers $a, b$ it holds that:

$$\frac{\bigintsss_{0}^{2\pi} e^{-i n t}f \left ( z+a e^{it} \right ) \, {\rm d}t}{\bigintsss_{0}^{2\pi} e^{-i n t} f\left ( z + b e^{it} \right ) \, {\rm d}t} = \left ( \frac{a}{b} \right )^n$$

Solution

Work along an oriented curve

Prove that the work

$$\mathcal{W}=- \oint \limits_{\gamma} \frac{(x, y, z)}{\left ( x^2+y^2+z^2 \right )^{3/2}} \cdot \, {\rm d}(x, y, z)$$

produced along a $\mathcal{C}^1$ oriented curve $\gamma$ of $\mathbb{R}^3 \setminus \{(0, 0, 0) \}$ depends only on the distances of starting and ending point of $\gamma$ about the origin.

Solution

On known formulae

1. Evaluate the area of the disk of center $(0, 0)$ and radius $R>0$
2. Let $f$ be a continuous function such that $f(z) \geq 0 , \; z \in [0, R]$. Prove , using the previous question as well as Cavalieri's principal that the volume $\mathcal{M}=\{(x,y, z) \in \mathbb{R}^3 : z \in [0, R] \mid x^2+y^2 \leq f^2(z) \}$ produced by an entire rotation by the graph of $f$ (which is a curve on the $xy$ plane) around the $z$ axis is equal to:
   
   $$\mathcal{V}\left ( \mathcal{M} \right )= \pi \int_{0}^{R}f^2(z) \, {\rm d}z$$


3.  If the function of the previous question is continuously differentiable , then prove that the area of the surface:
   
   $$\mathbb{S}=\left \{ \left ( f(z)\cos \varphi, f(z) \sin \varphi, z \right ) \in \mathbb{R}^3 : z \in [0, R], \varphi \in [0, 2\pi] \right \}$$
   
   produced by an entire rotation by the graph of $f$ around the $z$ axis is equal to:
   
   $$\mathcal{A}\left ( \mathbb{S} \right ) = 2\pi \int_{0}^{R} f(z) \sqrt{1+\big(f'(z)\big)^2} \, {\rm d}z$$

Solution

Even permutation

Let $\alpha$ and $\beta$ be elements of $\mathcal{S}_n$. Prove that ${\alpha}^{-1}{\beta}^{-1}\alpha \beta$ is an even permutation.

Solution

Probably a beauty and a beast

Here is something that arose while playing around with a class of particular integrals.

Let $\displaystyle a_n= \int_0^1 \left( e^x - 1 - x - \frac{x^2}{2} - \cdots - \frac{x^n}{n!} \right) \, {\rm d}x$. Evaluate the series

$$\mathcal{S}=\sum_{n=0}^{\infty} a_n$$

Solution

On functions

Here are some examples on functions with "strange" properties.

Give an example of a function that:

1. is only continuous at a single point.
2. is continuous at isolated points.

Does there exist a function that:

1. is continuous only on the rationals?
2. is continuous only on the irrationals?

Explain your answer giving a brief explanation. 

Solution

On a determinant

Let $p$ be a prime number and let $\omega$ be a primitive $p$-th root of unity. Define:

$$\mathcal{V} = \det \begin{pmatrix}
1 &1  &1  &\cdots  &1 \\
 1& \omega &\omega^2  &\cdots  &\omega^{p-1} \\
1 &\omega^2  &\left ( \omega^2 \right )^2  &\cdots  &\left ( \omega^2 \right )^{p-1} \\
 \vdots&\vdots  &\vdots  &\ddots  & \vdots\\
 1& \omega^{p-1} &\left ( \omega^{p-1} \right )^2  &\cdots  & \left ( \omega^{p-1} \right )^{p-1}
\end{pmatrix}$$

Evaluate the rational number $\mathcal{V}^2$.

Solution

Double alternating sum

Let $\displaystyle c_n= \sum_{k=1}^{\infty} \frac{(-1)^{k-1}}{k+n}$. Evaluate the sum:

$$\mathcal{S}=\sum_{n=1}^{\infty} \frac{c_n}{n}$$

Solution

A beautiful double sum

Prove that:

$$ \sum_{n=1}^{\infty}\left [ \frac{1}{n} \sum_{m=0}^{\infty} \frac{1}{(2m+1)^{2n}} - \log \left ( 1 + \frac{1}{n} \right ) \right ]$$

Solution