We study the differential equation $$y(x) y'(x) = -4x \tag{E}$$
Let $m$ be the maximal solution determined in question II.B.3).
II.D.1) Show that the solution $m$ is expandable as a power series in a neighborhood of 0. Calculate this expansion and specify its radius of convergence. II.D.2) Deduce the power series expansions of all maximal solutions of $(E)$; specify the radii of convergence of these power series.

Let $f$ be the function defined on $\mathbb { R } _ { + } ^ { * }$ by $f ( x ) = \frac { 1 } { ( 1 - \alpha ) x ^ { \alpha - 1 } }$. By applying to $f$ the Taylor formula with integral remainder at order 2, show that for every $k \in \mathbb { N } ^ { * } , f ( k + 1 ) - f ( k ) = \frac { 1 } { k ^ { \alpha } } - \frac { \alpha } { 2 } \frac { 1 } { k ^ { \alpha + 1 } } + A _ { k }$ where $A _ { k }$ is a real number satisfying $0 \leqslant A _ { k } \leqslant \frac { \alpha ( \alpha + 1 ) } { 2 k ^ { \alpha + 2 } }$.

Let $f$ be the function defined on $\mathbb { R } _ { + } ^ { * }$ by $f ( x ) = \frac { 1 } { ( 1 - \alpha ) x ^ { \alpha - 1 } }$, where $\alpha$ is a real number strictly greater than 1. We fix a non-zero natural integer $p$ and denote by $g = a _ { 0 } f + a _ { 1 } f ^ { \prime } + \cdots + a _ { 2 p - 1 } f ^ { ( 2 p - 1 ) }$. For every $k \in \mathbb { N } ^ { * }$, we set $R ( k ) = g ( k + 1 ) - g ( k ) - f ^ { \prime } ( k )$ so that $g ( k + 1 ) - g ( k ) = f ^ { \prime } ( k ) + R ( k )$.
By applying to $g$ the Taylor formula with integral remainder at order $2 p$, show that there exists a real number $A$ such that for every $k \in \mathbb { N } ^ { * } , | R ( k ) | \leqslant A k ^ { - ( 2 p + \alpha ) }$.

Let $g$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}$. We fix an element $a = (a_1, a_2, \ldots, a_n)$ of $\mathbb{R}^n$. Let $\varphi$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\varphi(t) = g(ta) = g(ta_1, ta_2, \ldots, ta_n)$$ Deduce that in a neighbourhood of 0 $$g(ta) = g(0) + t\left(a_1 \mathrm{D}_1 g(0) + a_2 \mathrm{D}_2 g(0) + \cdots + a_n \mathrm{D}_n g(0)\right) + \mathrm{o}(t)$$

We recall that $\phi$ is defined on $] - 1 , + \infty [$ by $\phi ( s ) = s - \ln ( 1 + s )$, and that two power series $\sum _ { k \geqslant 1 } b _ { k } q ^ { k }$ and $\sum _ { k \geqslant 1 } c _ { k } q ^ { k }$ with strictly positive radius of convergence, $b_1 > 0$, $c_1 < 0$, satisfy $\phi \left( \sum _ { k = 1 } ^ { \infty } b _ { k } q ^ { k } \right) = \phi \left( \sum _ { k = 1 } ^ { \infty } c _ { k } q ^ { k } \right) = q ^ { 2 }$ for $q$ near 0 in $[0,+\infty[$.
Calculate $b _ { 1 } , b _ { 2 } , b _ { 3 }$ and $c _ { 1 } , c _ { 2 }$ and $c _ { 3 }$. Deduce the following asymptotic expansions when $q \rightarrow 0 , q > 0$, for the functions $\phi _ { - } ^ { - 1 }$ and $\phi _ { + } ^ { - 1 }$ as well as their derivatives: $$\begin{array} { l l } \phi _ { + } ^ { - 1 } ( q ) = \sqrt { 2 q } + \frac { 2 q } { 3 } + \frac { q ^ { 3 / 2 } } { 9 \sqrt { 2 } } + o \left( q ^ { 3 / 2 } \right) , & \phi _ { - } ^ { - 1 } ( q ) = - \sqrt { 2 q } + \frac { 2 q } { 3 } - \frac { q ^ { 3 / 2 } } { 9 \sqrt { 2 } } + o \left( q ^ { 3 / 2 } \right) \\ \left( \phi _ { + } ^ { - 1 } \right) ^ { \prime } ( q ) = \frac { 1 } { \sqrt { 2 q } } + \frac { 2 } { 3 } + \frac { \sqrt { q } } { 6 \sqrt { 2 } } + o ( \sqrt { q } ) , & \left( \phi _ { - } ^ { - 1 } \right) ^ { \prime } ( q ) = - \frac { 1 } { \sqrt { 2 q } } + \frac { 2 } { 3 } - \frac { \sqrt { q } } { 6 \sqrt { 2 } } + o ( \sqrt { q } ) \end{array}$$

Using the results of the previous questions, deduce that $$\Gamma ( y ) = e ^ { - y } y ^ { y } \left( \frac { 2 \pi } { y } \right) ^ { 1 / 2 } \left( 1 + \frac { 1 } { 12 y } + o \left( \frac { 1 } { y } \right) \right) \quad \text { when } y \rightarrow + \infty .$$

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$. Show that in a neighbourhood of $x = 0$, the function $F$ can be written in the form
$$F(x) = \sum_{n=0}^{+\infty} c_{n} \frac{(\mathrm{i}x)^{n}}{n!} \tag{S}$$
where $c_{n}$ is the value of Gamma at a point to be specified. Express $c_{n}$ in terms of $n$ and $c_{0}$.
What is the radius of convergence of the power series appearing on the right-hand side of $(S)$?

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$. Let $R(x)$ be the real part and $I(x)$ be the imaginary part of $F(x)$.
Determine, in a neighbourhood of 0, the Taylor expansion of $R(x)$ to order 3 and of $I(x)$ to order 4.

Using the results of the previous questions (in particular the integral representation of $I_n$ from question 2, the bounds from question 3, and the Gaussian integral $\int_{-\infty}^{+\infty} e^{-x^{2}/2}\, dx = \sqrt{2\pi}$), deduce Stirling's formula: $$n! \underset{n \rightarrow \infty}{\sim} \sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}.$$

Let $n \in \mathbb{N}^*$, $h = \frac{1}{n+1}$, and $x_i = ih$ for all $i \in \{0, \ldots, n+1\}$. Show that for any function $v \in \mathcal { C } ^ { 4 } ( [ 0,1 ] , \mathbb { R } )$, there exists a constant $C \geq 0$, independent of $n$, such that
$$\forall i \in \{ 1 , \ldots , n \} , \left| v ^ { \prime \prime } \left( x _ { i } \right) - \frac { 1 } { h ^ { 2 } } \left( v \left( x _ { i + 1 } \right) + v \left( x _ { i - 1 } \right) - 2 v \left( x _ { i } \right) \right) \right| \leq C h ^ { 2 }$$

Let $\alpha_n = f^{(n)}(0)$ where $f(x) = \frac{\sin x + 1}{\cos x}$ on $I = ]-\pi/2, \pi/2[$. Let $R$ be the radius of convergence of the power series $\sum_{n \in \mathbb{N}} \frac{\alpha_n}{n!} x^n$ and $g$ its sum. Using Taylor's formula with integral remainder, show $$\forall N \in \mathbb{N}, \forall x \in \left[0, \pi/2\left[, \quad \sum_{n=0}^{N} \frac{\alpha_n}{n!} x^n \leqslant f(x)\right.\right.$$

From now on, $f$ denotes an infinitely differentiable function from $[0,1]$ to $\mathbb{R}$. We assume that there exists a unique point $x_0 \in [0,1]$ where $f'$ vanishes. We also assume that $f''(x_0) > 0$. We are also given an infinitely differentiable function $g : [0,1] \rightarrow \mathbb{R}$.
For all $x \in [x_0, 1]$, we define $$h(x) = \sqrt{|f(x) - f(x_0)|}$$ We admit that the bijection $h : [x_0, 1] \rightarrow [0, h(1)]$ admits an inverse application $h^{-1} : [0, h(1)] \rightarrow [x_0, 1]$ that is infinitely differentiable.
We assume that $x_0 \in ]0,1[$. Show that, as $t \rightarrow +\infty$, $$\int_0^1 g(x) \sin(tf(x)) \mathrm{d}x = g(x_0) \sin\left(tf(x_0) + \frac{\pi}{4}\right) \sqrt{\frac{2\pi}{tf''(x_0)}} + O\left(\frac{1}{t}\right)$$

From now on, $f$ denotes an infinitely differentiable function from $[ 0,1 ]$ to $\mathbb { R }$. We assume that there exists a unique point $x _ { 0 } \in \left[ 0,1 \left[ \right. \right.$ where $f ^ { \prime }$ vanishes. We also assume that $f ^ { \prime \prime } \left( x _ { 0 } \right) > 0$. We are also given an infinitely differentiable function $g : [ 0,1 ] \rightarrow \mathbb { R }$.
For all $x \in \left[ x _ { 0 } , 1 \right]$, we define $$h ( x ) = \sqrt { \left| f ( x ) - f \left( x _ { 0 } \right) \right| }$$ We admit that the bijection $$h : \left\{ \begin{array} { c c c } { \left[ x _ { 0 } , 1 \right] } & \rightarrow & { [ 0 , h ( 1 ) ] } \\ x & \mapsto & h ( x ) \end{array} \right.$$ admits an inverse map $h ^ { - 1 } : [ 0 , h ( 1 ) ] \rightarrow \left[ x _ { 0 } , 1 \right]$ that is infinitely differentiable.
Assume that $\left. x _ { 0 } \in \right] 0,1 [$. Show that, as $t \rightarrow + \infty$, $$\int _ { 0 } ^ { 1 } g ( x ) \sin ( t f ( x ) ) \mathrm { d } x = g \left( x _ { 0 } \right) \sin \left( t f \left( x _ { 0 } \right) + \frac { \pi } { 4 } \right) \sqrt { \frac { 2 \pi } { t f ^ { \prime \prime } \left( x _ { 0 } \right) } } + O \left( \frac { 1 } { t } \right)$$

Conclude that
$$\ln P \left( e ^ { - t } \right) = \frac { \pi ^ { 2 } } { 6 t } + \frac { \ln ( t ) } { 2 } - \frac { \ln ( 2 \pi ) } { 2 } + o ( 1 ) \quad \text { when } t \text { tends to } 0 ^ { + } .$$

Conclude that $$\ln P(e^{-t}) = \frac{\pi^2}{6t} + \frac{\ln(t)}{2} - \frac{\ln(2\pi)}{2} + o(1) \text{ when } t \text{ tends to } 0^+.$$