Deduce that, for all $x \in ] - 1,1 [$, we have $\int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta = 0$.
Deduce the value of $\int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$ in the case $| x | > 1$.

Show that the improper integral $\int _ { 0 } ^ { \pi / 2 } \ln ( \cos \theta ) d \theta$ converges.

Show that $\int _ { 0 } ^ { \pi } \ln ( \sin \theta ) \mathrm { d } \theta = 2 \int _ { 0 } ^ { \pi / 2 } \ln ( \sin \theta ) \mathrm { d } \theta = 2 \int _ { 0 } ^ { \pi / 2 } \ln ( \cos \theta ) \mathrm { d } \theta$.

Deduce that $\int _ { 0 } ^ { \pi } \ln ( \sin \theta ) \mathrm { d } \theta = - \pi \ln 2$.

Deduce that $\int _ { 0 } ^ { \pi } \ln ( 2 - 2 \cos \theta ) \mathrm { d } \theta = \int _ { 0 } ^ { \pi } \ln ( 2 + 2 \cos \theta ) \mathrm { d } \theta = 0$.

Let $f$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$.
Show that $f$ is differentiable on $\mathbb { R } \backslash \{ - 1,1 \}$ and that $$\forall x \in \mathbb { R } \backslash \{ - 1,1 \} \quad f ^ { \prime } ( x ) = \int _ { 0 } ^ { \pi } \frac { 2 x - 2 \cos \theta } { x ^ { 2 } - 2 x \cos \theta + 1 } \mathrm { ~d} \theta$$

Let $f$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$.
Deduce that $\forall x \in \mathbb { R } \backslash \{ - 1,1 \}$ $$f ^ { \prime } ( x ) = 4 \int _ { 0 } ^ { + \infty } \frac { ( x + 1 ) t ^ { 2 } + ( x - 1 ) } { \left( ( x + 1 ) ^ { 2 } t ^ { 2 } + ( x - 1 ) ^ { 2 } \right) \left( t ^ { 2 } + 1 \right) } \mathrm { d } t$$

Let $f$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$.
Deduce that $$f ( x ) = \begin{cases} 2 \pi \ln ( | x | ) & \text { if } | x | > 1 \\ 0 & \text { if } | x | < 1 \end{cases}$$
One will first determine coefficients $A$ and $B$ as functions of $x$ such that $\frac { ( x + 1 ) T + ( x - 1 ) } { \left( ( x + 1 ) ^ { 2 } T + ( x - 1 ) ^ { 2 } \right) ( T + 1 ) } = \frac { A } { ( x + 1 ) ^ { 2 } T + ( x - 1 ) ^ { 2 } } + \frac { B } { T + 1 }$ for all $T \in \mathbb { R }$ such that these fractions are defined.

Let $f$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$.
Show that $f$ is continuous on $\mathbb { R }$ and that $f ( 1 ) = f ( - 1 ) = 0$.
One may show that $\forall x \in \mathbb { R } , x ^ { 2 } - 2 x \cos \theta + 1 \geqslant \sin ^ { 2 } \theta$ and use the dominated convergence theorem.

Show that $\forall x \in \mathbb { R } \backslash \{ - 1,1 \}$ $$\int _ { 0 } ^ { 2 \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta = \lim _ { n \rightarrow + \infty } \left( \frac { 2 \pi } { n } \sum _ { k = 1 } ^ { n } \ln \left( x ^ { 2 } - 2 x \cos \frac { 2 k \pi } { n } + 1 \right) \right)$$

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.

Let $M \in \mathbb{R}_+^* \cup \{+\infty\}$ and $f : {]-\infty, M[} \rightarrow \mathbb{R}$ be a continuous function such that $$\forall (x, y) \in {\left]-\infty, \frac{M}{2}\right[}^2, \quad 2f(x+y) = f(2x) + f(2y) \tag{IV.1}$$
Let $\alpha$ be a number strictly less than $\frac{M}{2}$ and $F$ be the antiderivative of $f$ vanishing at $\alpha$. Show that for all $x$ and $y$ in $]-\infty, \frac{M}{2}[$, with $y \neq \alpha$, we have: $$f(2x) = 2\frac{F(x+y) - F(x+\alpha) - \frac{1}{4}F(2y) + \frac{1}{4}F(2\alpha)}{y - \alpha}$$

We model the density of tissues by an unknown function $f$ zero outside the zone to be studied. Assuming that each incident X-ray beam is carried by an affine line $\Delta$, and denoting by $I_e$ and $I_s$ its intensity measured on either side of the targeted zone: $$\ln\left(\frac{I_e}{I_s}\right) = \int_\Delta f$$
Propose a rigorous definition of the right-hand side of this equation in the case where $\Delta = \Delta\left(q, \vec{u}_\theta\right)$.

We model the density of tissues by an unknown function $f$ zero outside the zone to be studied. Assuming that each incident X-ray beam is carried by an affine line $\Delta$, and denoting by $I_e$ and $I_s$ its intensity measured on either side of the targeted zone: $$\ln\left(\frac{I_e}{I_s}\right) = \int_\Delta f$$
Explain how the Radon inversion formula allows us in principle to know the density of tissues in the radiographed zone.

We denote $\varphi$ the function defined by: $$\begin{cases} \varphi(x) = 0 & \text{if } |x| \geqslant 1 \\ \varphi(x) = \exp\left(-\frac{x^2}{1-x^2}\right) & \text{if } |x| < 1 \end{cases}$$
a) Show that $\int_{\mathbb{R}} \varphi(t) \mathrm{d}t$ is a strictly positive real number. b) For every real number $x$, we set $\theta(x) = \frac{\varphi(x)}{\int_{\mathbb{R}} \varphi(t) \mathrm{d}t}$ and, for every non-zero natural number $n$, $\rho_n(x) = n\theta(nx)$.
Show that $$\forall n \in \mathbb{N}^* \quad \int_{\mathbb{R}} \rho_n(x) \mathrm{d}x = 1$$

For every function $f$ belonging to $\mathcal{F}_{sr}$ and every non-zero natural number $n$, we set $$\left(f * \rho_n\right)(x) = \int_{\mathbb{R}} f(t) \rho_n(x-t) \mathrm{d}t$$
Let $I$ be the function that equals 1 on the interval $[-1,1]$, and 0 elsewhere. For $n \in \mathbb{N}^*$, we set $I_n(x) = I * \rho_n(x)$. a) For $n \in \mathbb{N}^*$ and $x \in \mathbb{R}$, express $I_n(x)$ in terms of $\varphi$. b) For $n \in \mathbb{N}^*$, show that $I_n$ belongs to $\mathcal{D}$ and study its variations. c) Sketch the graphs of $I_2$ and $I_3$. d) Show that the sequence of functions $(I_n)$ converges pointwise to a function $J$ which we shall determine. Show that $J$ and $I$ are equal except on a finite set of points. e) Does the sequence of functions $(I_n)$ converge uniformly to $J$?

We call a distribution on $\mathcal{D}$ any linear map $T : \mathcal{D} \rightarrow \mathbb{R}$ which satisfies $$\forall \varphi \in \mathcal{D}, \forall (\varphi_n)_{n \in \mathbb{N}} \in \mathcal{D}^{\mathbb{N}} \quad \varphi_n \xrightarrow{\mathcal{D}} \varphi \Longrightarrow T(\varphi_n) \rightarrow T(\varphi)$$
Show that if $f \in \mathcal{F}_{sr}$ then the map $T_f$ defined by $$\forall \varphi \in \mathcal{D} \quad T_f(\varphi) = \int_{-\infty}^{+\infty} f(x) \varphi(x) \mathrm{d}x$$ defines a distribution on $\mathcal{D}$.

Let $U$ be the function defined by $$\begin{cases} U(x) = 1 & \text{if } x \geqslant 0 \\ U(x) = 0 & \text{if } x < 0 \end{cases}$$ Justify that $U$ defines a distribution on $\mathcal{D}$.

Let $a$ be a real number. a) Show that the map $\delta_a$ which associates to every $\varphi \in \mathcal{D}$ the value $\varphi(a)$ is a distribution. b) Using the sequence of functions $(\varphi_n)_{n \in \mathbb{N}^*}$ of elements of $\mathcal{D}$ defined by $$\forall t \in \mathbb{R}, \varphi_n(t) = \begin{cases} \exp\left(\frac{(t-a)^2}{(t-a+1/n)(t-a-1/n)}\right) & \text{if } t \in ]a-1/n, a+1/n[ \\ 0 & \text{otherwise} \end{cases}$$ show that $\forall f \in \mathcal{F}_{sr}, T_f \neq \delta_a$.

If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$ Let $U$ be the function defined by $$\begin{cases} U(x) = 1 & \text{if } x \geqslant 0 \\ U(x) = 0 & \text{if } x < 0 \end{cases}$$ Show that $T_U' = \delta_0$.

If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$
We consider the map $T$ which associates to every function $\varphi$ of $\mathcal{D}$ the real number $T(\varphi)$ defined by $$T(\varphi) = \int_{-1}^{0} t\varphi(t) \mathrm{d}t + \int_{0}^{+\infty} \varphi(t) \mathrm{d}t$$
a) Show that $T$ is a regular distribution. b) Calculate the derivative of this distribution.

If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$ If $f$ is an element of $\mathcal{F}_{sr}$ and if $a$ is a real number, we set $$\lim_{x \rightarrow a^-} f(x) = f(a^-) \quad \text{and} \quad \lim_{x \rightarrow a^+} f(x) = f(a^+)$$ The difference $f(a^+) - f(a^-)$, called the jump at $a$, is denoted $\sigma(a)$. a) Let $a_1, \ldots, a_p$ be real numbers such that $a_1 < \ldots < a_p$. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a piecewise $\mathcal{C}^1$ function. We further assume that $f$ is continuous on $]-\infty, a_1[ \cup ]a_1, a_2[ \cup \ldots \cup ]a_p, +\infty[$. Show that $$T_f' = T_{f'} + \sum_{i=1}^{p} \sigma(a_i) \delta_{a_i}$$ b) Recover by this method the results of questions II.B.3 and II.B.4.b.

We say that the sequence of distributions $(T_n)_{n \in \mathbb{N}}$ converges to the distribution $T$ if $$\forall \varphi \in \mathcal{D}, \lim_{n \rightarrow \infty} T_n(\varphi) = T(\varphi)$$
For $n$ a non-zero natural number, we consider the function $U_n$ zero on the negative reals, affine on the interval $[0, 1/n]$, equal to 1 for reals greater than $1/n$ and continuous on $\mathbb{R}$. a) Show that the sequence of regular distributions $(T_{U_n})_{n \in \mathbb{N}}$ converges to $T_U$. b) Show that $$\forall \varphi \in \mathcal{D} \quad T_{U_n}'(\varphi) = \int_0^{1/n} n\varphi(t) \mathrm{d}t$$ c) Deduce that the distribution $T_{U_n}'$ is regular and give a function $V_n$ such that $T_{V_n} = T_{U_n}'$. d) Sketch $V_n$ for $n = 1, 2, 4$. e) Show that if the sequence of distributions $(T_n)_{n \in \mathbb{N}}$ converges to the distribution $T$, then $(T_n')_{n \in \mathbb{N}}$ converges to $T'$. f) What is the limit of $T_{U_n}'$ as $n$ tends to infinity?

We say that the sequence of distributions $(T_n)_{n \in \mathbb{N}}$ converges to the distribution $T$ if $$\forall \varphi \in \mathcal{D}, \lim_{n \rightarrow \infty} T_n(\varphi) = T(\varphi)$$
For every non-zero natural number $n$, we consider the functions $$\begin{cases} f_n(x) = \dfrac{n}{1 + n^2 x^2} & \\ g_n(x) = nx^n & \text{if } x \in [0,1] \text{ and zero elsewhere} \\ h_n(x) = n^2 \sin nx & \text{if } x \in [-\pi/n, \pi/n] \text{ and zero elsewhere} \end{cases}$$
a) Verify that they belong to $\mathcal{F}_{sr}$. b) Study the variations of the functions $f_n, g_n$ and $h_n$ then sketch their graphs for $n = 1$ and $n = 2$. c) Study the convergence of the sequences of distributions $(T_{f_n}), (T_{g_n})$ and $(T_{h_n})$.