Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Justify that $\theta$ and $R$ are well defined.
Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ When $z$ takes successively the values $z_1 = 4$, $z_2 = 2\mathrm{i}$ and $z_3 = 1 - \mathrm{i}\sqrt{3}$, calculate $R(z)$, $\theta(z)$ and $(R(z))^2$.
Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Verify that $\theta(z) \in ]-\pi, \pi[$ and that $R(z) \in \mathcal{P} = \{Z \in \mathbb{C},\, \operatorname{Re}(Z) > 0\}$.
Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Draw on a figure the circle $\mathcal{C}$ with center $O$ and radius $|z|$ and the points $M$ with affixe $z$ and $B$ with affixe $-|z|$. By considering well-chosen angles, show that $$\theta(z) = \operatorname{Arg}(z) = 2\operatorname{Arg}(z + |z|)$$ where $\operatorname{Arg}(z)$ denotes the principal determination of the argument of the complex number $z$.
Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Determine $[R(z)]^2$, $\theta \circ R(z)$ and $|z|^{1/2}\mathrm{e}^{\mathrm{i}\theta(z)/2}$ as functions of $z$, $R(z)$ and $\theta(z)$.
Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Solve using $R$ the equation $Z^2 = z$, with unknown $Z \in \mathbb{C}$.
Let $z$ be a complex number, with real part $x$ and imaginary part $y$, such that $(x,y) \notin \mathbb{R}^{-} \times \{0\}$. We denote $$\theta(z) = 2\arctan\left(\frac{y}{x + \sqrt{x^2 + y^2}}\right) \quad \text{and} \quad R(z) = \frac{z + |z|}{\sqrt{2(\operatorname{Re}(z) + |z|)}}$$ Deduce that $R$ is a bijection from $\mathbb{C} \setminus \mathbb{R}^{-}$ to $\mathcal{P}$. Specify its inverse bijection.
Let $a$ and $b$ be two complex numbers such that $(a,b) \neq (0,0)$. We say that a complex sequence $U = (u_n)_{n \in \mathbb{N}}$ satisfies the recurrence relation $(E_{a,b})$ if $$\forall n \in \mathbb{N}, \quad u_{n+2} = 2a u_{n+1} + b u_n$$ We assume that $a^2 + b \neq 0$. We denote $d = R(a^2 + b)$. We call $W$ the sequence $W = ((a+d)^n)_{n \in \mathbb{N}}$ and $W'$ the sequence $W' = ((a-d)^n)_{n \in \mathbb{N}}$. Show that $U$ satisfies $E_{a,b}$ if and only if $U \in \operatorname{Vect}(W, W')$. Determine $U$ satisfying $E_{a,b}$ and the initial conditions $u_0 = 0$ and $u_1 = 1$, as a function of $d$, $W$ and $W'$.
Let $a$ and $b$ be two complex numbers such that $(a,b) \neq (0,0)$. We say that a complex sequence $U = (u_n)_{n \in \mathbb{N}}$ satisfies the recurrence relation $(E_{a,b})$ if $$\forall n \in \mathbb{N}, \quad u_{n+2} = 2a u_{n+1} + b u_n$$ We assume that $a^2 + b = 0$ and $a \neq 0$. We denote $W$ and $W'$ the sequences $W = (a^n)_{n \in \mathbb{N}}$ and $W' = (na^n)_{n \in \mathbb{N}}$. Show that $U$ satisfies $E_{a,b}$ if and only if $U \in \operatorname{Vect}(W, W')$. Determine $U$ satisfying $E_{a,b}$ and the initial conditions $u_0 = 0$ and $u_1 = 1$, as a function of $a$, $W$ and $W'$.
We denote $V_n(z) = U_{n+1}(z, -1)$ for all $z \in \mathbb{C}$ and $n \in \mathbb{N}$, where $U(a,b) = (U_n(a,b))_{n \in \mathbb{N}}$ is the unique sequence satisfying $E_{a,b}$ with initial conditions $U_0(a,b) = 0$ and $U_1(a,b) = 1$. Explicitly write $V_1(z)$, $V_2(z)$ and $V_3(z)$ and determine their roots in $\mathbb{C}$.
We denote $V_n(z) = U_{n+1}(z, -1)$ for all $z \in \mathbb{C}$ and $n \in \mathbb{N}$, where $U(a,b) = (U_n(a,b))_{n \in \mathbb{N}}$ is the unique sequence satisfying $E_{a,b}$ with initial conditions $U_0(a,b) = 0$ and $U_1(a,b) = 1$. Show that, for all $z \in \mathbb{C}$ and $n \in \mathbb{N}$, we have $$V_n(z) = \sum_{j=0}^{\lfloor n/2 \rfloor} \binom{n-j}{j} (2z)^{n-2j} (-1)^j$$ One may proceed by induction.
Let $z \in \mathbb{C}$. We denote $C_z$ (respectively $\Omega_z$) the set of points in the plane with complex affixe $Z$ such that $|Z(Z-2z)| = 1$ (respectively $|Z(Z-2z)| < 1$). In this question we assume that $z$ is a real number denoted $a$. We work in the orthonormal frame $\mathcal{R}'$ with center $O'$ with affixe $a$, obtained from $\mathcal{R}$ by translation. Show that an equation of the curve $C_a$ in ``polar coordinates $(\rho, \theta)$'' in the frame $\mathcal{R}'$ is $$\left(\rho^2 + a^2\right)^2 - 4a^2 \rho^2 \cos^2\theta = 1$$
Let $z \in \mathbb{C}$. We denote $C_z$ (respectively $\Omega_z$) the set of points in the plane with complex affixe $Z$ such that $|Z(Z-2z)| = 1$ (respectively $|Z(Z-2z)| < 1$). In this question we assume that $z$ is a real number denoted $a$. The curve $C_a$ in polar coordinates $(\rho, \theta)$ in the frame $\mathcal{R}'$ satisfies $$\left(\rho^2 + a^2\right)^2 - 4a^2 \rho^2 \cos^2\theta = 1$$ Simplify this equation when $a = 1$. Study and sketch the shape of the curve $C_1$.
Let $z \in \mathbb{C}$. We denote $\Omega_z$ the set of points in the plane with complex affixe $Z$ such that $|Z(Z-2z)| < 1$. Justify that $\Omega_z$ is a bounded subset of the plane. Is it open? closed? compact?
Let $z \in \mathbb{C}$. We denote $\Omega_z$ the set of points in the plane with complex affixe $Z$ such that $|Z(Z-2z)| < 1$. Justify that the origin $O$ is an interior point of $\Omega_z$.
We use the notation $R$ introduced in part I. Let $z \in \mathbb{C}$ such that $z^2 \neq 1$. We denote $$r = \left|R\left(z^2 - 1\right)\right|, \quad s = \left|z + R\left(z^2 - 1\right)\right|, \quad t = \left|z - R\left(z^2 - 1\right)\right|, \quad h = \max(s,t)$$ We also denote $V_n(z) = U_{n+1}(z,-1)$ for all $z \in \mathbb{C}$ and $n \in \mathbb{N}$. Prove that, for all $n \in \mathbb{N}$, $$\left|V_n(z)\right| \leqslant \frac{h^{n+1}}{r}$$
We use the notation $R$ introduced in part I and $V_n(z) = U_{n+1}(z,-1)$. Let $z \in \mathbb{C}$ such that $z^2 \neq 1$, with $r$, $s$, $t$, $h$ as defined in II.C.1. What can be said about the radius of convergence of the power series $Z \mapsto \sum_{n=0}^{+\infty} V_n(z) Z^n$? We denote $g_z$ its sum.
We use the notation $V_n(z) = U_{n+1}(z,-1)$ and $g_z$ the sum of the power series $\sum_{n=0}^{+\infty} V_n(z) Z^n$. When it makes sense, calculate $\left(1 - 2zZ + Z^2\right) g_z(Z)$.
Let $z \in \mathbb{C}$. We denote $\Omega_z$ the set of points in the plane with complex affixe $Z$ such that $|Z(Z-2z)| < 1$, and $V_n(z) = U_{n+1}(z,-1)$. Show that there exists a non-empty open disk $\Delta$ with center $O$ included in $\Omega_z$ such that $$\forall Z \in \Delta, \quad \frac{1}{1 - 2zZ + Z^2} = \sum_{n=0}^{+\infty} V_n(z) Z^n = \sum_{p=0}^{+\infty} \left(Z^p(2z - Z)^p\right)$$
Let $z \in \mathbb{C}$. Consider the function of the real variable $x$ $$G_z : x \mapsto \sum_{p=0}^{+\infty} \left(x^p(2z - x)^p\right)$$ Deduce (from II.C.5) that $G_z$ admits a Taylor expansion to any order at 0. We denote it $$G_z(x) = \sum_{k=0}^{n} a_k x^k + o\left(x^n\right) \quad x \to 0$$ Determine the coefficients $a_k$ for $k \in \mathbb{N}$.
We denote $\alpha$ a real number such that $\alpha > -1/2$, $E$ the $\mathbb{R}$-vector space of functions of class $\mathcal{C}^\infty$ on $[-1,1]$ with real values, and $$S_\alpha(f,g) = \int_{-1}^{1} f(t)g(t)\left(1-t^2\right)^{\alpha - \frac{1}{2}} \mathrm{~d}t$$ Verify that $S_\alpha$ is an inner product on $E$.
We denote $\alpha > -1/2$, $E$ the $\mathbb{R}$-vector space of functions of class $\mathcal{C}^\infty$ on $[-1,1]$ with real values, and $$\varphi_\alpha(y) : t \mapsto \left(1-t^2\right)y''(t) - (2\alpha+1)t\,y'(t)$$ Justify that $\varphi_\alpha$ is an endomorphism of $E$. Is it injective?
We denote $\alpha > -1/2$, $E$ the $\mathbb{R}$-vector space of functions of class $\mathcal{C}^\infty$ on $[-1,1]$ with real values, $$\varphi_\alpha(y) : t \mapsto \left(1-t^2\right)y''(t) - (2\alpha+1)t\,y'(t)$$ and $$S_\alpha(f,g) = \int_{-1}^{1} f(t)g(t)\left(1-t^2\right)^{\alpha - \frac{1}{2}} \mathrm{~d}t$$ Show that $$\forall (f,g) \in E^2, \quad S_\alpha\left(\varphi_\alpha(f), g\right) = S_\alpha\left(f, \varphi_\alpha(g)\right)$$ One may calculate the derivative of $t \mapsto \left(1-t^2\right)^{\alpha+\frac{1}{2}} f'(t)$.
We denote $\alpha > -1/2$, $F_n$ the vector subspace of $E$ of polynomial functions of degree less than or equal to $n$ (where $n \in \mathbb{N}$), and $$\varphi_\alpha(y) : t \mapsto \left(1-t^2\right)y''(t) - (2\alpha+1)t\,y'(t)$$ Justify that $\varphi_\alpha$ induces on $F_n$ an endomorphism and that this induced endomorphism (still denoted $\varphi_\alpha$) is diagonalizable.
We denote $\alpha > -1/2$, $F_n$ the vector subspace of $E$ of polynomial functions of degree less than or equal to $n$ (where $n \in \mathbb{N}$), and $$\varphi_\alpha(y) : t \mapsto \left(1-t^2\right)y''(t) - (2\alpha+1)t\,y'(t)$$ Show that there exists a basis of $F_n$ consisting of eigenvectors of $\varphi_\alpha$ of pairwise distinct degrees.
We denote $\alpha > -1/2$, $F_n$ the vector subspace of $E$ of polynomial functions of degree less than or equal to $n$ (where $n \in \mathbb{N}$), and $$\varphi_\alpha(y) : t \mapsto \left(1-t^2\right)y''(t) - (2\alpha+1)t\,y'(t)$$ Verify that two eigenvectors of $\varphi_\alpha$ of distinct degrees are associated with distinct eigenvalues. One may be interested in the leading coefficient of a judiciously chosen polynomial.
We denote $\alpha > -1/2$, $F_n$ the vector subspace of $E$ of polynomial functions of degree less than or equal to $n$ (where $n \in \mathbb{N}$), $$\varphi_\alpha(y) : t \mapsto \left(1-t^2\right)y''(t) - (2\alpha+1)t\,y'(t)$$ and $$S_\alpha(f,g) = \int_{-1}^{1} f(t)g(t)\left(1-t^2\right)^{\alpha - \frac{1}{2}} \mathrm{~d}t$$ Justify that two eigenvectors of $\varphi_\alpha$ of distinct degrees are orthogonal (with respect to $S_\alpha$).
We denote $\alpha > -1/2$, $F_n$ the vector subspace of $E$ of polynomial functions of degree less than or equal to $n$ (where $n \in \mathbb{N}$), and $$\varphi_\alpha(y) : t \mapsto \left(1-t^2\right)y''(t) - (2\alpha+1)t\,y'(t)$$ Show that any eigenvector of $\varphi_\alpha$ of degree greater than or equal to 1 vanishes at least once in the interval $]-1,1[$.
We assume $\alpha = 1$. We denote $\|\cdot\|$ the norm associated with $S_1$, and $$\varphi_1(y) : t \mapsto \left(1-t^2\right)y''(t) - 3t\,y'(t)$$ Justify that, for all $k \in \mathbb{N}$, there exists a unique polynomial eigenvector of $\varphi_1$ of degree $k$, of norm 1 and with positive leading coefficient. We denote it $T_k$.
We assume $\alpha = 1$ and use the notation $V_n(z) = U_{n+1}(z,-1)$. Let $t \in ]0,\pi[$. Show that the function $$H_t : x \mapsto \frac{1}{1 - 2x\cos(t) + x^2}$$ is expandable as a power series on $]-1,1[$.
We assume $\alpha = 1$ and use the notation $V_n(z) = U_{n+1}(z,-1)$. Using the expansion of $H_t$ as a power series, deduce that $$\forall n \in \mathbb{N},\, \forall t \in ]0,\pi[, \quad V_n(\cos t) = \frac{\sin((n+1)t)}{\sin t}$$
We assume $\alpha = 1$ and use the notation $V_n(z) = U_{n+1}(z,-1)$, and $$\varphi_1(y) : t \mapsto \left(1-t^2\right)y''(t) - 3t\,y'(t)$$ By differentiating twice the function $t \mapsto (\sin t)\,V_n(\cos t) - \sin((n+1)t)$, show that for all $n \in \mathbb{N}$, $V_n$ is an eigenvector of $\varphi_1$.
We assume $\alpha = 1$. We denote $\|\cdot\|$ the norm associated with $S_1$, $T_k$ the unique polynomial eigenvector of $\varphi_1$ of degree $k$, of norm 1 and with positive leading coefficient, and $V_n(z) = U_{n+1}(z,-1)$. Deduce that, for all $n \in \mathbb{N}$, $V_n$ and $T_n$ are proportional. Explicitly state the proportionality coefficient.
We assume $\alpha = 1$. We denote $T_n$ the unique polynomial eigenvector of $\varphi_1$ of degree $n$, of norm 1 (with respect to $S_1$) and with positive leading coefficient. For $n \in \mathbb{N}^*$, determine the roots of $T_n$.