In this question, we return to the case of II.B.3 (i.e., $\alpha = 3$, $p = 3$, $n = 100$). Knowing that $6 ! a _ { 6 } = \frac { 1 } { 42 }$, recover that the error $\left| S ( 3 ) - \widetilde { S } _ { 100,4 } ( 3 ) \right|$ is bounded by an expression of order $10 ^ { - 17 }$.

For every natural integer $p \geqslant 1$ and every real number $x$, we set $\widetilde { A } _ { p } ( x ) = A _ { p } \left( \frac { x } { 2 \pi } - \left[ \frac { x } { 2 \pi } \right] \right)$. Show that $\widetilde { A } _ { p }$ is $2 \pi$-periodic and piecewise continuous.

For every natural integer $p \geqslant 1$ and every real number $x$, we set $\widetilde { A } _ { p } ( x ) = A _ { p } \left( \frac { x } { 2 \pi } - \left[ \frac { x } { 2 \pi } \right] \right)$. Using question III.B.1, determine the Fourier coefficients of $\widetilde { A } _ { p }$: $$\widehat { A } _ { p } ( n ) = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \widetilde { A } _ { p } ( t ) e ^ { - i n x } d x$$

For every natural integer $p \geqslant 1$ and every real number $x$, we set $\widetilde { A } _ { p } ( x ) = A _ { p } \left( \frac { x } { 2 \pi } - \left[ \frac { x } { 2 \pi } \right] \right)$. Study the convergence of the Fourier series of $\widetilde { A } _ { p }$.

For every natural integer $p \geqslant 1$ and every real number $x$, we set $\widetilde { A } _ { p } ( x ) = A _ { p } \left( \frac { x } { 2 \pi } - \left[ \frac { x } { 2 \pi } \right] \right)$. For $p \in \mathbb { N } ^ { * }$, deduce that $a _ { 2 p } = A _ { 2 p } ( 0 ) = ( - 1 ) ^ { p + 1 } \frac { S ( 2 p ) } { 2 ^ { 2 p - 1 } \pi ^ { 2 p } }$.

Show that, for all integers $n , p \geqslant 1$ $$\left| \frac { a _ { 2 p + 2 } f ^ { ( 2 p + 2 ) } ( n ) } { a _ { 2 p } f ^ { ( 2 p ) } ( n ) } \right| = \frac { ( \alpha + 2 p ) ( \alpha + 2 p - 1 ) S ( 2 p + 2 ) } { 4 n ^ { 2 } \pi ^ { 2 } S ( 2 p ) }$$ where $f$ is defined on $\mathbb { R } _ { + } ^ { * }$ by $f ( x ) = \frac { 1 } { ( 1 - \alpha ) x ^ { \alpha - 1 } }$.

The function $h$ is defined on $\mathbb{R}$ by $$h(u) = u - [u] - 1/2$$ Let $\varphi$ be the application defined for all $x > 0$ by: $$\varphi(x) = \int_{0}^{+\infty} \frac{h(u)}{u+x} du$$ By repeating the integration by parts from question IV.C, prove that the application $\varphi$ is of class $\mathcal{C}^{1}$ on $\mathbb{R}_{+}^{*}$ and that for all $x > 0$, $$\varphi^{\prime}(x) = -\int_{0}^{+\infty} \frac{h(u)}{(u+x)^{2}} du$$

We fix $x > 0$ and for all natural integers $n$, we define $F_{n}(x)$ by: $$F_{n}(x) = \ln\left(\frac{n! \, n^{x+1}}{(x+1)(x+2) \ldots (x+n+1)}\right)$$ Show that for all natural integers $i$: $$\int_{x+i}^{x+i+1} \ln t \, dt = \ln(x+i) - \int_{i}^{i+1} \frac{u-i-1}{u+x} du$$

We fix $x > 0$ and for all natural integers $n$, we define $F_{n}(x)$ by: $$F_{n}(x) = \ln\left(\frac{n! \, n^{x+1}}{(x+1)(x+2) \ldots (x+n+1)}\right)$$ Deduce that: $$F_{n}(x) = G_{n}(x) - \int_{0}^{n+1} \frac{h(u)}{u+x} du$$ where $$G_{n}(x) = \ln n! + (x+1)\ln n - \left(x+n+\frac{3}{2}\right)\ln(x+n+1) + n+1 + \left(x+\frac{1}{2}\right)\ln x$$

We fix $x > 0$ and for all natural integers $n$, we define $F_{n}(x)$ by: $$F_{n}(x) = \ln\left(\frac{n! \, n^{x+1}}{(x+1)(x+2) \ldots (x+n+1)}\right)$$ and $$G_{n}(x) = \ln n! + (x+1)\ln n - \left(x+n+\frac{3}{2}\right)\ln(x+n+1) + n+1 + \left(x+\frac{1}{2}\right)\ln x$$
V.C.1) Using Stirling's formula, show that: $$\lim_{n \rightarrow +\infty} G_{n}(x) = \left(x+\frac{1}{2}\right)\ln x - x + \ln\sqrt{2\pi}$$
V.C.2) Deduce that: $$\ln \Gamma(x+1) = \left(x+\frac{1}{2}\right)\ln x - x + \ln\sqrt{2\pi} - \int_{0}^{+\infty} \frac{h(u)}{u+x} du$$

Using the identity $$\ln \Gamma(x+1) = \left(x+\frac{1}{2}\right)\ln x - x + \ln\sqrt{2\pi} - \int_{0}^{+\infty} \frac{h(u)}{u+x} du$$ show that for all strictly positive real $x$, $$\frac{\Gamma^{\prime}(x+1)}{\Gamma(x+1)} = \ln x + \frac{1}{2x} + \int_{0}^{+\infty} \frac{h(u)}{(u+x)^{2}} du$$

Deduce that $$\int _ { 0 } ^ { 2 \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) d \theta = \begin{cases} 4 \pi \ln ( | x | ) & \text { if } | x | > 1 \\ 0 & \text { if } | x | < 1 \end{cases}$$

Deduce $\int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$ for $x \in \mathbb { R } \backslash \{ - 1,1 \}$.

Deduce that $$\int _ { 0 } ^ { \pi / 2 } \ln ( \sin \theta ) d \theta = - \pi \frac { \ln 2 } { 2 }$$
Then recover the result from question III.B.6.

We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
a) Show that for $x \in \mathbb{R}_{+}^{*}$, $$T_{-1}(x) \leq \int_{0}^{1} e^{-1/u^{2}} \frac{du}{u} + \int_{1}^{\infty} e^{-xu} \frac{du}{u}$$ Deduce that $T_{-1}(x) \leq 2$ for $x \geq 1$ and that $$T_{-1}(x) \leq 2 + \int_{x}^{1} e^{-w} \frac{dw}{w} \leq 2 - \ln x$$ if $0 < x \leq 1$.
b) Let $L \in [0,1]$, and $\rho \in C([0,L])$. We set $$[F(\rho)](x) = \int_{0}^{L} \rho(y) T_{-1}(|x-y|) dy$$ Show that $[F(\rho)](x)$ is well-defined for $x \in [0,L]$ and that $$\|F(\rho)\|_{\infty} \leq (4L + 2L|\ln L|) \|\rho\|_{\infty}.$$

Let $\varepsilon$ and $r$ be fixed such that $0 < \varepsilon < r$. With the change of variables $q = r\cos\theta$, establish that $$\int_\varepsilon^r \frac{\mathrm{d}q}{q^2\sqrt{r^2-q^2}} = \frac{\sqrt{r^2-\varepsilon^2}}{r^2\varepsilon}$$

For every pair of natural integers $( p , q )$ and for every $\varepsilon \in ] 0,1 [$, we denote $$I _ { p , q } = \int _ { 0 } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t \quad \text { and } \quad I _ { p , q } ^ { \varepsilon } = \int _ { \varepsilon } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t$$
Deduce that we have $\forall p \in \mathbb { N } , \forall q \in \mathbb { N } ^ { * } , \quad I _ { p , q } = - \frac { q } { p + 1 } I _ { p , q - 1 }$.

For every pair of natural integers $( p , q )$ and for every $\varepsilon \in ] 0,1 [$, we denote $$I _ { p , q } = \int _ { 0 } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t \quad \text { and } \quad I _ { p , q } ^ { \varepsilon } = \int _ { \varepsilon } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t$$
Deduce an expression for $I _ { p , q }$ as a function of the integers $p$ and $q$.

Let $r$ be a non-zero natural integer and $f$ a function expandable as a power series on $] - 1,1 [$. We assume that for every $x$ in $] - 1,1 \left[ , f ( x ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n } \right.$ and that $\sum _ { n \geqslant 0 } \frac { a _ { n } } { ( n + 1 ) ^ { r } }$ converges absolutely.
Show that $\int _ { 0 } ^ { 1 } ( \ln t ) ^ { r - 1 } f ( t ) \mathrm { d } t = ( - 1 ) ^ { r - 1 } ( r - 1 ) ! \sum _ { n = 0 } ^ { + \infty } \frac { a _ { n } } { ( n + 1 ) ^ { r } }$.

Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k }$ and $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ when the series converges.
Deduce from the previous questions that for every integer $r \geqslant 2$, $$S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } } = \frac { ( - 1 ) ^ { r } } { ( r - 1 ) ! } \int _ { 0 } ^ { 1 } ( \ln t ) ^ { r - 1 } \frac { \ln ( 1 - t ) } { 1 - t } \mathrm {~d} t$$

Throughout the problem, we denote for every integer $n \geqslant 1$, $H _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { k }$ and $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ when the series converges.
Establish that we then have $S _ { r } = \frac { ( - 1 ) ^ { r } } { 2 ( r - 2 ) ! } \int _ { 0 } ^ { 1 } \frac { ( \ln t ) ^ { r - 2 } ( \ln ( 1 - t ) ) ^ { 2 } } { t } \mathrm {~d} t$.

Let $x > 0$. Show that $t \mapsto t ^ { x - 1 } e ^ { - t }$ is integrable on $] 0 , + \infty [$.

Throughout the rest of the problem, we denote $\Gamma$ the function defined on $\mathbb { R } ^ { + * }$ by $\Gamma ( x ) = \int _ { 0 } ^ { + \infty } t ^ { x - 1 } e ^ { - t } \mathrm {~d} t$. We admit that $\Gamma$ is of class $\mathcal { C } ^ { \infty }$ on its domain of definition, takes strictly positive values and satisfies, for every real $x > 0$, the relation $\Gamma ( x + 1 ) = x \Gamma ( x )$.
Let $x$ and $\alpha$ be two strictly positive real numbers. Justify the existence of $\int _ { 0 } ^ { + \infty } t ^ { x - 1 } e ^ { - \alpha t } \mathrm {~d} t$ and give its value as a function of $\Gamma ( x )$ and $\alpha ^ { x }$.

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$.
Justify the existence of $\beta ( x , y )$ for $x > 0$ and $y > 0$.

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$.
Show that for all real $x > 0$ and $y > 0 , \beta ( x , y ) = \beta ( y , x )$.