📘 complex analysis

1. **Résoudre dans \(\mathbb{C}\) l'équation : \(z^2 - 2iz + 2(1 - 2i) = 0\)** - Énoncé : Trouver les racines complexes de l'équation quadratique donnée.

1. **Problem Statement:** Evaluate the contour integral $$\oint_C \frac{\cos x}{x^2 + 1} \, dx$$ where $C$ is the circle $|x|=2$ using Cauchy's residue theorem. 2. **Formula and Th

1. **Problem statement:** Evaluate the contour integral $$\int_C \frac{\cos x}{x^2 + 1} \, dx$$ where $C$ is the circle $|x|=2$, using Cauchy's residue theorem. 2. **Recall Cauchy'

1. **Problem Statement:** Find the bilinear transformation (also called a Möbius transformation) that maps points $z=1, i, -1$ in the $z$-plane to points $w=2, i, -2$ in the $w$-pl

1. **Problem Statement:** Describe and graph the locus represented by the inequality $$1 < |z + i| \leq 2$$ where $$z = x + iy$$ is a complex number. 2. **Rewrite the expression:**

1. **State and prove Cauchy's Residue Theorem:** Cauchy's Residue Theorem states that if a function $f(z)$ is analytic inside and on a simple closed contour $C$ except for a finite

1. **Problem statement:** Find the residue of the function $$f(z) = \frac{z e}{(z-1)^3}$$ at its poles. 2. **Identify the poles:** The function has a pole at $$z=1$$ of order 3 bec

1. Énoncé du problème : Soit $z = a + ib$ un nombre complexe et $Z = \frac{7 + z}{-3 + iz}$. On cherche la partie réelle et la partie imaginaire de $Z$. Ensuite, on détermine l'équ

1. Let's start by stating the problem: Understanding what an analytic function is in complex analysis. 2. An analytic function is a complex function that is locally given by a conv

1. **Stating the problem:** We are given the expression $$A = e^{i\phi}(e^{i\theta_1} - 1) + e^{i(\theta_1 + \theta_2)}(e^{i\theta_3} - 1)$$ and need to find its complex conjugate,

1. **Problem Statement:** Find the complex conjugate $A^*$ of the expression $$A = e^{i\phi}(e^{i\theta_1} - 1) + e^{i(\theta_1 + \theta_2)}(e^{i\theta_3} - 1).$$ 2. **Recall the r

1. **Stating the problem:** Simplify the expression $$A = e^{i\phi} \left( e^{i\theta_1} - 1 \right) + e^{i(\theta_1 + \theta_2 + \theta_3)} - e^{i(\theta_1 + \theta_2)}$$ and unde

1. The problem is to understand the Zeta function, specifically the Riemann Zeta function $\zeta(s)$, which is defined for complex numbers $s$ with real part greater than 1 by the

1. **State the problem:** Solve the equation $$\left(z - 2\left(\cos\frac{9\pi}{4} + i\sin\frac{9\pi}{4}\right)\right)^4 = -8 + i8\sqrt{3}$$ for the complex number $z$ and represen

1. (a) State four properties of bilinear transformation. 1. A bilinear transformation is a function of the form $$w = \frac{az + b}{cz + d}$$ where $$a,b,c,d$$ are complex numbers

1. **Problem Statement:** (a) State four properties of bilinear transformation.

1. **Stating the problem:** Lerch's Theorem is a result in complex analysis and number theory that relates the Lerch transcendent function to other special functions. 2. **Definiti

1. **Problem Statement:** Find the contour integral $$\int_C f(z) \, dz$$ where $$f(z) = \pi \exp(\pi \overline{z})$$ and $$C$$ is the boundary of the square with vertices $$0, 1,

1. **Énoncé du problème :** Résoudre et analyser l'équation complexe $z^2 - 2i(1 + \sqrt{3})z - 8 = 0$ et étudier les points associés dans le plan complexe. 2. **Calcul de $(\sqrt{

1. The problem asks for the principal value of $\ln i$. 2. Recall that for a complex number $z = re^{i\theta}$, the principal value of the natural logarithm is given by:

1. **State the problem:** We need to find all values of $z$ such that $$e^{iz} = i.$$ 2. **Recall Euler's formula:** $$e^{i\theta} = \cos \theta + i \sin \theta.$$