📘 complex analysis

1. **Énoncé du problème** : Résoudre dans $\mathbb{C}$ les équations données, exprimer certains nombres complexes sous forme exponentielle, démontrer des égalités trigonométriques,

1. The problem asks to sketch the set $S$ of points in the complex plane satisfying the inequalities for problems 13, 15, 17, 23, 14, 16, 18, and 24. 2. Each problem describes a re

1. **State the problem:** Find the set of points $z = x + yi$ in the complex plane satisfying $\operatorname{Re}(z) < -1$.

1. **State the problem:** Solve the equation $$Z^{i} - 1 = 0$$ where $Z$ is a complex number and $i$ is the imaginary unit. 2. **Rewrite the equation:** We want to find $Z$ such th

1. **State the problem:** We want to evaluate the limit as $R \to \infty$ of the sum over $n \in \mathbb{N}_0$ of the contour integrals over $\partial C_R$ of the function

1. The problem is to understand and draw the set $$A = \{ z \in \mathbb{C} \mid |i + 3| > 4 \wedge \operatorname{Re}(z) \leq |m(2 + i)| \}$$ where $z$ is a complex number. 2. First

1. **State the problem:** We are given a complex number $z = x + it$ and the equation $\arg(z-2) - \arg(z+2) = \frac{\pi}{4}$. We need to find the Cartesian form of this equation.

1. **State the problem:** Solve the equation $$z^{i} - 1 = 0$$ and represent the solution on an Argand diagram. 2. **Rewrite the equation:** We want to find all complex numbers $$z

1. **Problem (a):** Evaluate the integral $$\int_0^{2\pi} \frac{d\theta}{\frac{5}{4} + \sin \theta}.$$\n\n2. **Formula and rules:** For integrals of the form $$\int_0^{2\pi} \frac{

1. **Problem statement:** Find the value of $\frac{15ab}{r^2}$ where the locus of complex numbers $z$ satisfies $$\operatorname{Re}\left(\frac{z-1}{2z+i}\right) + \operatorname{Re}

1. The problem asks to find the residue of the function $$f(z) = \frac{z+2}{(z+1)(z-2)(z-3)}$$ at the point $$z_0 = 2$$. 2. The residue at a simple pole $$z_0$$ of a function $$f(z

1. The problem asks for the residue of the function $$f(z) = \frac{1}{(z-2)(z+2)(z-3)}$$ at the point $$z_0 = 3$$. 2. The residue at a simple pole $$z_0$$ of a function $$f(z) = \f

1. Розглянемо перше завдання: розкласти в ряд Тейлора функцію $$F(z)=\ln(4z^2 - 4z - 8)$$ при $$z_0=3$$. 2. Спочатку спростимо аргумент логарифму:

1. **Problem Statement:** Given the function $f(z) = e^x (\cos x + i \sin y)$ where $z = x + iy$, show that:

1. **Problem statement:** Given points $A=1$, $A'=-1$, $B=i$, $B'=-i$ in the complex plane, and a point $M$ with complex number $z$ different from $0, A, A', B, B'$, define points

1. **Problem statement:** Find the poles or removable singularities of the functions and determine their orders. 2. **Recall:** A pole of order $m$ at $z=z_0$ means the function be

1. **Problem Statement:** We want to understand the concept of homotopy between paths in a region $G \subseteq \mathbb{C}$ and its implications for integrals of holomorphic functio

1. **State the problem:** Calculate the value of $(-1+i)^{2i}$ where $i$ is the imaginary unit. 2. **Recall the formula:** For a complex number $z$ and a complex exponent $w$, we u

1. The problem is to evaluate the function $f(x) = e^{i\pi}$.\n\n2. We use Euler's formula, which states that for any real number $\theta$, $$e^{i\theta} = \cos(\theta) + i\sin(\th

1. The problem is to understand the function $f(x) = e^i$. 2. Here, $e$ is the base of the natural logarithm, approximately 2.71828, and $i$ is the imaginary unit, defined by $i^2

1. مسئله را بیان می‌کنیم: ناحیه همگرایی سری $$\sum_{n=0}^\infty \left(-\frac{r}{z-1}\right)^n \frac{1}{z-1}$$ را بیابید. 2. ابتدا سری را به شکل استاندارد سری هندسی بازنویسی می‌کنیم