📘 complex analysis

1. Problem statement: Evaluate the contour integrals given for various circles in the complex plane. 2. For ca) \(\oint_{|z|=1.5} \frac{dz}{z^2 - 2z} = \oint_{|z|=1.5} \frac{dz}{z(

1. 問題の確認： 条件(i)と(ii)を満たす複素数$z$を求める。$z=\pm L$は明らかに解であるが、それ以外の解を求める。

1. The problem asks to show that the non-trivial zeros of the analytic continuation of the function $f(s) = \sum_{n=1}^\infty n^{-s}$, originally defined for $\mathrm{Re}(s) > 1$,

1. The problem asks to show that the non-trivial zeros of the analytic continuation of the function $f(s) = \sum_{n=1}^\infty n^{-s}$, defined initially for $\mathrm{Re}(s) > 1$, l

1. **State the problem:** We are given a complex number $z$ with modulus $8$ and argument $\frac{2\pi}{3}$. We need to find $z^{1/z}$ in Cartesian form. 2. **Express $z$ in polar f

1. **Déterminer les solutions complexes de :** **a) Résoudre $z^3 + i = 0$**

1. Euler's formula is a fundamental equation in complex analysis that establishes a deep relationship between trigonometric functions and the exponential function. 2. The formula s

1. The problem is to evaluate the contour integral $$\oint \frac{\sin z}{(z+i)^3} \, dz$$ around a closed contour enclosing the singularity at $z = -i$. 2. Notice that the integran

1. The problem asks to determine the values of $\ln(-1 - i)$ and its principal value. 2. Recall that for a complex number $z = re^{i\theta}$, the complex logarithm is given by:

1. Find the harmonic conjugate of $u(x,y) = x^3 - 3xy^2$. The harmonic conjugate $v$ satisfies the Cauchy-Riemann equations:

1. Find the harmonic conjugate of $u(x,y) = x^3 - 3xy^2$. The harmonic conjugate $v$ satisfies the Cauchy-Riemann equations:

1. **State the problem:** We are given the function $$\psi(x,y) = 3x^2 y + 2x^2 - y^3 - 2y^2$$.

1. **State the problem:** We need to sketch the region in the complex plane (z-plane) defined by the equation $$|z+3i| + |z-3i| = 5$$. 2. **Interpret the equation:** Let $z = x + y

1. **State Cauchy’s integral theorem:** Cauchy’s integral theorem states that if a function $f(z)$ is analytic (holomorphic) inside and on a simple closed contour $C$, then the con

1. **State Cauchy’s integral theorem:** Cauchy’s integral theorem states that if a function $f(z)$ is analytic (holomorphic) inside and on a simple closed contour $C$, then the int

1. Problem: Evaluate $$\lim_{z \to \pi i} e^{z^2}$$. Step 1: Substitute $z = \pi i$.

1. **Problem 8:** Prove continuity of \( f(z) = \frac{x^3(1+i) - y^3(1-i)}{x^2 + y^2} \) for \( z \neq 0 \) and \( f(0) = 0 \), and check Cauchy-Riemann equations at origin. Step 1

1. Δίνεται η παράγωγος της συνάρτησης $$f(z)$$ που παρατίθεται ως $$f'(z)= f(z)\left[- \frac{1}{z-1} - \frac{1}{z-a} - \frac{1}{z-b}\right]$$.\n\n2. Αρχικά, για να επιβεβαιώσουμε τ

1. Evaluate $$\int_C z \, dz$$ where $$C$$ is the line segment from $$z=0$$ to $$z=1+i$$. Parameterize the line segment: $$z(t) = t(1+i)$$ for $$t \in [0,1]$$.

1. Exercice 4 : Montrer que la transformation $T:z \mapsto z' = 2iz + 2 + i$ admet un point invariant $A$ d'affixe $a$. On cherche $a$ tel que $a' = a$, donc $$a = 2ia + 2 + i.$$

1. We are asked to find the residue of the function $f(z) = \frac{1}{\sin z}$ at its singularity points. 2. The function $\sin z$ has zeros at $z = n\pi$ for all integers $n$.