We denote $B$ the function defined on $\mathbb { R } ^ { + * }$ by $B ( x ) = \int _ { 0 } ^ { 1 } ( \ln ( 1 - t ) ) ^ { 2 } t ^ { x - 1 } \mathrm {~d} t$.
Give without justification an expression, using an integral, of $B ^ { ( p ) } ( x )$, for every natural integer $p$ and every real $x > 0$.

We denote $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ for $r \geqslant 2$, and $B ( x ) = \int _ { 0 } ^ { 1 } ( \ln ( 1 - t ) ) ^ { 2 } t ^ { x - 1 } \mathrm {~d} t$. We have shown that $S _ { r } = \frac { ( - 1 ) ^ { r } } { 2 ( r - 2 ) ! } \int _ { 0 } ^ { 1 } \frac { ( \ln t ) ^ { r - 2 } ( \ln ( 1 - t ) ) ^ { 2 } } { t } \mathrm {~d} t$.
Deduce that for every integer $r \geqslant 2 , S _ { r } = \frac { ( - 1 ) ^ { r } } { 2 ( r - 2 ) ! } \lim _ { x \rightarrow 0 ^ { + } } B ^ { ( r - 2 ) } ( x )$.

We denote $S _ { r } = \sum _ { n = 1 } ^ { + \infty } \frac { H _ { n } } { ( n + 1 ) ^ { r } }$ for $r \geqslant 2$, $\zeta ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { 1 } { n ^ { x } }$ for $x > 1$, and $B ( x ) = \int _ { 0 } ^ { 1 } ( \ln ( 1 - t ) ) ^ { 2 } t ^ { x - 1 } \mathrm {~d} t$. We have $S _ { r } = \frac { ( - 1 ) ^ { r } } { 2 ( r - 2 ) ! } \lim _ { x \rightarrow 0 ^ { + } } B ^ { ( r - 2 ) } ( x )$.
Find again the value of $S _ { 2 }$ already calculated in I.F.3.

Show that $\Gamma$ is well defined and that for all $y > 0 , y \Gamma ( y ) = \Gamma ( y + 1 )$. Deduce that, for all $n \in \mathbb { N } , \Gamma ( n + 1 ) = n !$.
Recall that $\Gamma : ] 0 , + \infty [ \rightarrow \mathbb { R }$ is defined by $\Gamma ( y ) = \int _ { 0 } ^ { \infty } e ^ { - t } t ^ { y - 1 } d t$ and that $\Gamma \left( \frac { 1 } { 2 } \right) = \sqrt { \pi }$.

Show that for all $y > 0$, we have $\Gamma ( y ) = y ^ { - 1 } \int _ { 0 } ^ { + \infty } e ^ { - t } t ^ { y } d t$, then that
$$\Gamma ( y ) = e ^ { - y } y ^ { y } \int _ { - 1 } ^ { + \infty } e ^ { - y \phi ( s ) } d s$$
where $\phi$ is the function defined on $] - 1 , + \infty [$ by $\phi ( s ) = s - \ln ( 1 + s )$.
Recall that $\Gamma : ] 0 , + \infty [ \rightarrow \mathbb { R }$ is defined by $\Gamma ( y ) = \int _ { 0 } ^ { \infty } e ^ { - t } t ^ { y - 1 } d t$.

We consider a function $f : ] 0 , + \infty [ \rightarrow \mathbb { R }$ continuous piecewise satisfying the two following properties: (a) there exist an integer $K \geqslant 0$ and a real $C > 0$ such that $| f ( t ) | \leqslant C t ^ { K }$ on $[ 1 , + \infty [$, (b) there exist an integer $N \geqslant 0$, two reals $\lambda > 0$ and $\mu > 0$ and reals $a _ { 0 } , \ldots , a _ { N }$ such that $$f ( t ) = \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu } + o \left( t ^ { ( N + \lambda - \mu ) / \mu } \right) \quad \text { when } t \rightarrow 0 .$$ We denote $\rho _ { N } ( t ) = f ( t ) - \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu }$ the remainder of the asymptotic expansion of $f$.
We fix $\delta > 0$ and $\alpha \in \mathbb { R }$. Show that for all $x > 0$, the function $t \mapsto e ^ { - t / x } t ^ { \alpha }$ is integrable on $[ \delta , + \infty [$ and that for all $n \in \mathbb { N }$, we have: $$\int _ { \delta } ^ { + \infty } e ^ { - t / x } t ^ { \alpha } d t = o \left( x ^ { n } \right) \quad \text { when } x \rightarrow 0 ^ { + }$$ Deduce that for all $n \in \mathbb { N }$, $$\int _ { \delta } ^ { + \infty } e ^ { - t / x } \rho _ { N } ( t ) d t = o \left( x ^ { n } \right) \quad \text { when } x \rightarrow 0 ^ { + }$$

We consider a function $f : ] 0 , + \infty [ \rightarrow \mathbb { R }$ continuous piecewise satisfying the two following properties: (a) there exist an integer $K \geqslant 0$ and a real $C > 0$ such that $| f ( t ) | \leqslant C t ^ { K }$ on $[ 1 , + \infty [$, (b) there exist an integer $N \geqslant 0$, two reals $\lambda > 0$ and $\mu > 0$ and reals $a _ { 0 } , \ldots , a _ { N }$ such that $$f ( t ) = \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu } + o \left( t ^ { ( N + \lambda - \mu ) / \mu } \right) \quad \text { when } t \rightarrow 0 .$$ We denote $\rho _ { N } ( t ) = f ( t ) - \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu }$ the remainder of the asymptotic expansion of $f$.
We fix $\varepsilon > 0$. Show the existence of $\delta > 0$ and a constant $C ^ { \prime }$ independent of $\varepsilon$ and $\delta$ such that $$\forall x > 0 , \quad \left| \int _ { 0 } ^ { \delta } e ^ { - t / x } \rho _ { N } ( t ) d t \right| \leqslant C ^ { \prime } \varepsilon x ^ { ( N + \lambda ) / \mu }$$

We consider a function $f : ] 0 , + \infty [ \rightarrow \mathbb { R }$ continuous piecewise satisfying the two following properties: (a) there exist an integer $K \geqslant 0$ and a real $C > 0$ such that $| f ( t ) | \leqslant C t ^ { K }$ on $[ 1 , + \infty [$, (b) there exist an integer $N \geqslant 0$, two reals $\lambda > 0$ and $\mu > 0$ and reals $a _ { 0 } , \ldots , a _ { N }$ such that $$f ( t ) = \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu } + o \left( t ^ { ( N + \lambda - \mu ) / \mu } \right) \quad \text { when } t \rightarrow 0 .$$ We denote $\rho _ { N } ( t ) = f ( t ) - \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu }$ the remainder of the asymptotic expansion of $f$.
Deduce that $$\int _ { 0 } ^ { + \infty } e ^ { - t / x } \rho _ { N } ( t ) d t = o \left( x ^ { ( N + \lambda ) / \mu } \right) \quad \text { when } x \rightarrow 0 ^ { + }$$

We consider a function $f : ] 0 , + \infty [ \rightarrow \mathbb { R }$ continuous piecewise satisfying the two following properties: (a) there exist an integer $K \geqslant 0$ and a real $C > 0$ such that $| f ( t ) | \leqslant C t ^ { K }$ on $[ 1 , + \infty [$, (b) there exist an integer $N \geqslant 0$, two reals $\lambda > 0$ and $\mu > 0$ and reals $a _ { 0 } , \ldots , a _ { N }$ such that $$f ( t ) = \sum _ { k = 0 } ^ { N } a _ { k } t ^ { ( k + \lambda - \mu ) / \mu } + o \left( t ^ { ( N + \lambda - \mu ) / \mu } \right) \quad \text { when } t \rightarrow 0 .$$
We denote $F$ the function defined by: $$F ( x ) = \int _ { 0 } ^ { + \infty } e ^ { - t / x } f ( t ) d t$$
Show that $F$ is well defined on $] 0 , + \infty [$ and that it satisfies the following asymptotic formula: $$F ( x ) = \sum _ { k = 0 } ^ { N } a _ { k } \Gamma \left( \frac { k + \lambda } { \mu } \right) x ^ { ( k + \lambda ) / \mu } + o \left( x ^ { ( N + \lambda ) / \mu } \right) \quad \text { when } x \rightarrow 0 ^ { + } .$$

We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $$F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$$
Show that $F$ is well defined and of class $\mathscr { C } ^ { \infty }$ on $] 0 , + \infty [$.

We consider the function $F : ] 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $F ( x ) = \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - 1 } d t$.
For $N \in \mathbb { N } ^ { * }$ and $x > 0$, we set $$\begin{aligned} r _ { N } ( x ) & = ( - 1 ) ^ { N } N ! x ^ { N + 1 } e ^ { - 1 / x } \\ S _ { N } ( x ) & = \sum _ { k = 1 } ^ { N } ( - 1 ) ^ { k - 1 } ( k - 1 ) ! x ^ { k } e ^ { - 1 / x } \\ R _ { N } ( x ) & = ( - 1 ) ^ { N } N ! x ^ { N } \int _ { 1 } ^ { + \infty } e ^ { - t / x } t ^ { - ( N + 1 ) } d t \end{aligned}$$
Show that, for all $N \geqslant 1$ and all $x > 0 , F ( x ) = S _ { N } ( x ) + R _ { N } ( x )$.

What is the domain of definition $\mathcal{D}$ of the function $\Gamma$, where for $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$?

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. For all $x \in \mathcal{D}$, express $\Gamma(x+1)$ in terms of $x$ and $\Gamma(x)$.
Deduce from this, for all $x \in \mathcal{D}$ and all $n \in \mathbb{N}^{*}$, an expression for $\Gamma(x+n)$ in terms of $x$, $n$ and $\Gamma(x)$, as well as the value of $\Gamma(n)$ for all $n \geqslant 1$.

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Show the existence of the two integrals $\int_{0}^{+\infty} e^{-t^{2}} \mathrm{~d}t$ and $\int_{0}^{+\infty} e^{-t^{4}} \mathrm{~d}t$ and express them using $\Gamma$.

Let $a$ and $b$ be two real numbers such that $0 < a < b$. Show that, for all $t > 0$ and all $x \in [a, b]$,
$$t^{x} \leqslant \max\left(t^{a}, t^{b}\right) \leqslant t^{a} + t^{b}$$

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Show that $\Gamma$ is of class $\mathcal{C}^{\infty}$ on $\mathcal{D}$.
Let $k \in \mathbb{N}^{*}$ and $x \in \mathcal{D}$. Express $\Gamma^{(k)}(x)$, the $k$-th derivative of $\Gamma$ at point $x$, in the form of an integral.

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Show that $\Gamma^{\prime}$ vanishes at a unique real number $\xi$ whose integer part will be determined.

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Deduce the variations of $\Gamma$ on $\mathcal{D}$. Specify in particular the limits of $\Gamma$ at 0 and at $+\infty$. Also specify the limits of $\Gamma^{\prime}$ at 0 and at $+\infty$. Sketch the graph of $\Gamma$.

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$, where $\mathrm{i}$ denotes the complex number with modulus 1 and argument $\pi/2$.
Show that the function $F : \begin{aligned} & \mathbb{R} \rightarrow \mathbb{C} \\ & x \mapsto F(x) \end{aligned}$ is defined and of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$.
Let $k$ be a non-zero natural number and let $x$ be a real number. Give an integral expression for $F^{(k)}(x)$, the $k$-th derivative of $F$ at $x$. Specify $F(0)$.

We assume $\lambda < 1$. Determine $\lim_{n \rightarrow +\infty} \left((n\lambda)^{-n} \int_{0}^{n\lambda} (n\lambda - t)^{n} \mathrm{e}^{t} \mathrm{~d}t\right)$.

We assume $\lambda < 1$, and $D_{n} = \sum_{k=n+1}^{+\infty} \frac{(n\lambda)^{k}}{k!}$. Using Taylor's formula with integral remainder, deduce an equivalent of $D_{n}$ when $n \rightarrow +\infty$.

For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Show that $f$ is defined and continuous on $[0, +\infty[$ and of class $C^{2}$ on $]0, +\infty[$.

For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Determine the limits of $f$ and $f^{\prime}$ at $+\infty$.

For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Express $f^{\prime\prime}$ on $]0, +\infty[$ using standard functions and deduce that $$\forall x > 0, \quad f^{\prime}(x) = \ln(x) - \frac{1}{2} \ln\left(x^{2} + 1\right)$$

For $x \in \mathbb{R}^{+}$, we define $$f(x) = \int_{0}^{\infty} \frac{1 - \cos t}{t^{2}} \mathrm{e}^{-xt} \mathrm{~d}t$$ Show $$\left\{ \begin{array}{l} \forall x > 0, \quad f(x) = x \ln(x) - \frac{1}{2} x \ln\left(x^{2} + 1\right) - \arctan(x) + \frac{\pi}{2} \\ f(0) = \frac{\pi}{2} \end{array} \right.$$