Exercise 2 (Candidates who have followed the specialization course)
We assume the following results are known:

• the composition of two plane similarities is a plane similarity;
• the inverse transformation of a plane similarity is a plane similarity;
• a plane similarity that leaves three non-collinear points of the plane invariant is the identity of the plane.

1. Let A, B and C be three non-collinear points in the plane and $s$ and $s'$ be two similarities of the plane such that $s(\mathrm{A}) = s'(\mathrm{A})$, $s(\mathrm{B}) = s'(\mathrm{B})$ and $s(\mathrm{C}) = s'(\mathrm{C})$. Show that $s = s'$.
2. The complex plane is referred to the orthonormal frame $(\mathrm{O}, \vec{u}, \vec{v})$. We are given the points A with affix $2$, E with affix $1 + \mathrm{i}$, F with affix $2 + \mathrm{i}$ and G with affix $3 + \mathrm{i}$. a. Calculate the lengths of the sides of the triangles OAG and OEF. Deduce that these triangles are similar. b. Show that OEF is the image of OAG by an indirect similarity $S$, by determining the complex form of $S$. c. Let $h$ be the homothety with centre O and ratio $\frac{1}{\sqrt{2}}$. We set $\mathrm{A}' = h(\mathrm{A})$ and $\mathrm{G}' = h(\mathrm{G})$, and we call I the midpoint of $[\mathrm{EA}']$. We denote by $\sigma$ the orthogonal symmetry with axis (OI). Show that $S = \sigma \circ h$.

The complex plane is given an orthonormal coordinate system ( $\mathrm { O } , \vec { u } , \vec { v }$ ). To every point $M$ with affixe $z$ in the plane, we associate the point $M ^ { \prime }$ with affixe $z ^ { \prime }$ defined by:
$$z ^ { \prime } = z ^ { 2 } + 4 z + 3 .$$

1. A point $M$ is called invariant when it coincides with the associated point $M ^ { \prime }$.
   Prove that there exist two invariant points. Give the affixe of each of these points in algebraic form, then in exponential form.
2. Let A be the point with affixe $\frac { - 3 - \mathrm { i } \sqrt { 3 } } { 2 }$ and B the point with affixe $\frac { - 3 + \mathrm { i } \sqrt { 3 } } { 2 }$.
   Show that OAB is an equilateral triangle.
3. Determine the set $\mathcal { E }$ of points $M$ with affixe $z = x + \mathrm { i } y$ where $x$ and $y$ are real, such that the associated point $M ^ { \prime }$ lies on the real axis.
4. In the complex plane, represent the points A and B as well as the set $\mathcal { E }$.

We place ourselves in the complex plane with coordinate system $(O ; \vec { u } , \vec { v })$. Let $f$ be the transformation that associates to any non-zero complex number $z$ the complex number $f ( z )$ defined by:
$$f ( z ) = z + \frac { 1 } { z }$$
We denote by $M$ the point with affixe $z$ and $M ^ { \prime }$ the point with affixe $f ( z )$.

1. We call A the point with affixe $a = - \frac { \sqrt { 2 } } { 2 } + \mathrm { i } \frac { \sqrt { 2 } } { 2 }$. a. Determine the exponential form of $a$. b. Determine the algebraic form of $f ( a )$.
2. Solve, in the set of complex numbers, the equation $f ( z ) = 1$.
3. Let $M$ be a point with affixe $z$ on the circle $\mathscr { C }$ with center O and radius 1. a. Justify that the affixe $z$ can be written in the form $z = \mathrm { e } ^ { \mathrm { i } \theta }$ with $\theta$ a real number. b. Show that $f ( z )$ is a real number.
4. Describe and represent the set of points $M$ with affixe $z$ such that $f ( z )$ is a real number.

Exercise 2
The complex plane is equipped with a direct orthonormal coordinate system $(\mathrm{O}; \vec{u}; \vec{v})$ with unit 2 cm. We call $f$ the function that, to any point $M$, distinct from point O and with affixe a complex number $z$, associates the point $M'$ with affixe $z'$ such that $$z' = -\frac{1}{z}$$
1. Consider the points A and B with affixes respectively $z_{\mathrm{A}} = -1 + \mathrm{i}$ and $z_{\mathrm{B}} = \frac{1}{2}\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$.
a. Determine the algebraic form of the affixe of point $\mathrm{A}'$ image of point A by the function $f$.
b. Determine the exponential form of the affixe of point $\mathrm{B}'$ image of point B by the function $f$.
c. On your paper, place the points $\mathrm{A}, \mathrm{B}, \mathrm{A}'$ and $\mathrm{B}'$ in the direct orthonormal coordinate system $(\mathrm{O}; \vec{u}; \vec{v})$. For points B and $\mathrm{B}'$, construction lines should be left visible.
2. Let $r$ be a strictly positive real number and $\theta$ a real number. Consider the complex number $z$ defined by $z = r\mathrm{e}^{\mathrm{i}\theta}$.
a. Show that $z' = \frac{1}{r}\mathrm{e}^{\mathrm{i}(\pi - \theta)}$.
b. Is it true that if a point $M$, distinct from O, belongs to the disk with center O and radius 1 without belonging to the circle with center O and radius 1, then its image $M'$ by the function $f$ is outside this disk? Justify.
3. Let the circle $\Gamma$ with center K with affixe $z_{\mathrm{K}} = -\frac{1}{2}$ and radius $\frac{1}{2}$.
a. Show that a Cartesian equation of the circle $\Gamma$ is $x^2 + x + y^2 = 0$.
b. Let $z = x + \mathrm{i}y$ with $x$ and $y$ not both zero. Determine the algebraic form of $z'$ as a function of $x$ and $y$.
c. Let $M$ be a point, distinct from O, on the circle $\Gamma$. Show that the image $M'$ of point $M$ by the function $f$ belongs to the line with equation $x = 1$.

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.

Problem 3
Consider a mapping $w = f ( z )$ of a domain $D$ on the complex $z$ plane to a domain $\Delta$ on the complex $w$ plane. Points on the complex $z$ and $w$ planes correspond to complex numbers $z = x + i y$ and $w = u + i v$, respectively. Here, $x , y$, $u$ and $v$ are real numbers, and $i$ is the imaginary unit.
I. Let $w = \sin z$.

1. Express $u$ and $v$ as functions of $x$ and $y$, respectively.
2. Suppose the domain $D _ { 1 } = \left\{ ( x , y ) \left\lvert \, 0 \leq x \leq \frac { \pi } { 2 } \right. , y \geq 0 \right\}$ on the $z$ plane is transformed to a domain on the $w$ plane. Show the transformed domain on the $w$ plane by drawing the transformed images corresponding to the three half-lines: $x = 0 , x = \frac { \pi } { 2 }$ and $x = c$ at $y \geq 0$ on the $z$ plane. Here, $c$ is a real constant on $0 < c < \frac { \pi } { 2 }$.

II. If a real function $g ( x , y )$ has continuous first and second partial derivatives and satisfies Laplace's equation $\frac { \partial ^ { 2 } g } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } g } { \partial y ^ { 2 } } = 0$ in a domain $\Omega$ on a plane, $g ( x , y )$ is said to be harmonic in $\Omega$.
Suppose that a function $f ( z ) = u ( x , y ) + i v ( x , y )$ is holomorphic in $D$ on the $z$ plane:

1. Show both $u ( x , y )$ and $v ( x , y )$ are harmonic in $D$ on the $z$ plane.
2. Suppose a function $h ( u , v )$ is harmonic in $\Delta$ on the $w$ plane, show a function $H ( x , y ) = h ( u ( x , y ) , v ( x , y ) )$ is harmonic in $D$ on the $z$ plane.

III. Suppose a function $h ( u , v )$ is harmonic in the domain $\Delta _ { 1 } = \{ ( u , v ) \mid u \geq 0 , v \geq 0 \}$ on the $w$ plane and satisfies the following boundary conditions:
$$\begin{aligned} & h ( 0 , v ) = 0 \quad ( v \geq 0 ) \\ & h ( u , 0 ) = 1 \quad ( u \geq 1 ) \\ & \frac { \partial h } { \partial v } ( u , 0 ) = 0 \quad ( 0 \leq u \leq 1 ) \end{aligned}$$

1. Let $z = \arcsin w$ and $H ( x , y ) = h ( u , v )$. Find the boundary conditions for $H ( x , y )$ corresponding to Equations (1), (2) and (3). Use the principal values of inverse trigonometric functions.
2. Find the function $H ( x , y )$ which satisfies the boundary conditions obtained in Question III.1.
3. Find $h ( u , 0 )$ on the interval $0 \leq u \leq 1$.