📘 complex analysis

1. **Problem:** Prove that the set of complex numbers $\mathbb{C}$ is a field under the usual operations of addition and multiplication. 2. **Definition:** A field is a set equippe

1. The problem asks if we can define a generalized derivative for an analytic function in general. 2. An analytic function is a complex function that is locally given by a converge

1. The problem asks if we can define a generalized derivative for analytic functions in complex analysis. 2. In complex analysis, an analytic function is one that is complex differ

1. **Problem statement:** Evaluate the integral $$\int_0^{2\pi} \frac{\cos 3\theta}{5 + 4 \cos \theta} d\theta$$ using contour integration. 2. **Formula and approach:** We use the

1. **Problem:** Calculate the contour integral $$\oint_C \frac{z}{(z-3)^3} \, dz$$ where $$C: |z|=1$$. 2. **Step 1: Identify singularities inside the contour.**

1. **Énoncé du problème :** Trouver la similitude plane directe $S$ qui transforme $A$ en $E$ et $D$ en $F$ avec les affixes données.

1. Énoncé du problème : Résoudre dans $\mathbb{C}$ l'équation $$\left(\frac{z - 2i}{z + i}\right)^3 + \left(\frac{z - 2i}{z + i}\right)^2 + \frac{z - 2i}{z + i} + 1 = 0.$$ 2. Poson

1. **Check analyticity of** $f(z) = e^{z^2}$ using Cauchy-Riemann equations. 2. Write $z = x + iy$, then $f(z) = u(x,y) + iv(x,y)$ where

1. The problem is to understand what a Riemann sphere is. 2. The Riemann sphere is a way to extend the complex plane by adding a point at infinity.

1. **Problem Statement:** Show that the function $w = \overline{z}$ (the complex conjugate of $z$) is not differentiable anywhere except possibly at the origin. 2. **Recall the def

1. The problem is to find the value of $\sinh\left(3+\frac{\pi i}{6}\right)$. 2. Recall the formula for the hyperbolic sine of a complex number $z = x + yi$:

1. **Problem Statement:** Given the complex number $z = \cos t + i \sin t$, find the magnitude $|z - 1|$ and express it in terms of $t$. Also, analyze the differential $dz$ and its

1. The problem is to understand the function $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$, known as the Riemann zeta function. 2. This function is defined as an infinite series w

1. **State the problem:** Solve the equation $$z^4 = e^{\frac{2\pi i}{3}}$$ for the complex number $z$. 2. **Formula and rules:** To solve equations of the form $$z^n = w$$ where $

1. **Problem:** Solve the equation in $\mathbb{C}$: $$e^z = 3\sqrt{3} - 3i.$$

1. **State the problem:** Find the real and imaginary parts of $$\left(\sqrt{i}\right)^{\sqrt{i}}$$. 2. **Recall important formulas and rules:**

1. **State the problem:** Find the real and imaginary parts of $$\left(\sqrt{i}\right)^{\sqrt{i}}$$. 2. **Recall the formula and rules:**

1. The problem is to find the value of $$z = \frac{\log(1+i)}{1-i}$$ where $i$ is the imaginary unit with $i^2 = -1$. 2. Recall that $\log$ of a complex number $z = re^{i\theta}$ (

1. **Problem statement:** We analyze the complex number expressions and geometric properties given, including the expression \(\frac{Z - \overline{Z}}{1 + Z \cdot \overline{Z}}\),

1. The problem involves evaluating the infinite sum expression involving complex exponentials and factorials, specifically expressions of the form $$\sum_{k=0}^\infty \frac{1}{k!}$

1. The problem involves evaluating the infinite sum expression involving complex exponentials and factorials, specifically expressions of the form $$\sum_{k=0}^\infty \frac{1}{k!}$