📘 complex analysis

1. نصّ المشكلة: مهمة البحث هي إيجاد أصفار دالة زيغما $\, \zeta(s)\, $ باستخدام تقريب مخصص: $$\zeta(s) \approx \sum_{n=1}^{V} \frac{1}{n^s} + \beta(V,s) \cdot V^{1-s}$$

1. **Problem:** Find the absolute value of the conjugate of $$\frac{\sqrt{3} - \sqrt{2} i}{2\sqrt{3} - \sqrt{2} i}$$. **Step 1:** The conjugate of a complex number $$z = x + yi$$ i

1. On commence par rappeler que $\omega = \exp\left(\frac{2i\pi}{n}\right)$ est une racine primitive $n$-ième de l'unité, donc $\omega^n=1$ et les $\omega^k$ pour $k=0,\dots,n-1$ s

1. The problem asks for the argument (angle in the complex plane) of the hyperbolic cosine function, expressed as $\arg(\cosh z)$ for some complex number $z$. 2. Recall the definit

1. The problem asks to identify the locus of the point $z = x + iy$ in the complex plane defined by the equation $$|z-3| + |z+3i| = \text{constant}.$$\n\n2. Write $z = x + iy$, whe

1. **State the problem:** Prove that the function $u = e^{-x}(x \sin y - y \cos y)$ is harmonic and find $v$ such that $f(z) = u + iv$ is analytic. 2. **Check if $u$ is harmonic:**

1. **Démontrer que le triangle ABC est rectangle isocèle** Problème non explicitement détaillé ici, mais rappel : un triangle est rectangle isocèle si et seulement si il a un angle