3
votes
2answers
72 views

How to calculate $\int_{\partial B_2(0)}\frac{2z^2+7z+11}{z^3+4z^2-z-4}\;dz$?

I want to calculate $$\displaystyle\int_{\partial B_2(0)}\underbrace{\frac{2z^2+7z+11}{z^3+4z^2-z-4}}_{=:f(z)}\;dz\tag{0}$$ Partial fraction decomposition yields ...
3
votes
1answer
62 views

If $f$ has pole of order $m$, then $\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$

Statement: Let $$f(z):=\sum_{k=-\infty}^\infty a_kz^k$$ have a pole of order $m$ at $z_0$. Then $$\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$$ ...
1
vote
1answer
36 views

$\int _{C^{+}(0,3)} \frac {dz}{2-\sin z}$

$$\int _{C^{+}(0,3)} \frac {dz}{2-\sin z},$$ $z$ is complex. I have no idea how to solve $2-\sin z$. I will be really grateful for any help
0
votes
0answers
47 views

A question on particular functions in $L^\infty$

Let D be the open unit disk in the complex plane, $\mu$ be the arc length measure, and $f, g\in L^{\infty}(\partial D,\mu)$ satisfying the following equations: $$ \int_{\partial D} ...
0
votes
1answer
85 views

an amazing complex integral along unit circle

Let $f$ be an entire function on $\mathbb{C}$. Let $z_0\in\mathbb{C}$. Let $C=\{z\in\mathbb{C}\mid |z-z_0|=1\}$. Suppose $f(z)\neq f(z_0)$ for any $|z-z_0|\leq 1$, $z\neq z_0$. Given $f'(z_0)=2$, ...
0
votes
1answer
90 views

Integral of holomorphic function which tends to $0$

Let $R > 0 $, $z \in \mathbb{C}, \ f : D(z,R) \rightarrow \mathbb{C} $. $Re(f) \ $ and $Im(f) \ $ are $C^{1} $ on $D(z,R) \ $. Then f is complex differentiable in $z$ if and only if $$ \lim_{r ...
1
vote
1answer
136 views

$\int_{0}^{+\infty}\frac{\sin x}{x^{k}(1+x^{2})}dx \ $ via residue calculus

I want to evaluate with calculus of residues $$\int_{0}^{+\infty}\frac{\sin x}{x^{k}(1+x^{2})}dx \ $$ $ k \in \mathbb{N}, k \geq 1$ If $k = 1$ we have $$\int_{0}^{+\infty}\frac{\sin ...
1
vote
1answer
85 views

Dirac Delta — Symmetry

I had a curiosity question rise up in the middle of the night regarding the behavior of the Dirac Delta. Because it's not a function per-se, I am not sure how a concept like "integration" symmetry ...
1
vote
3answers
114 views

Calculating the integral expression $\int_0^{\infty}\lambda^{t-1}e^{-\lambda z}d\lambda$ for complex-valued z

The problem is computing the integral expression $$f(t)=\int_0^{\infty}\lambda^{t-1}e^{-\lambda z}d\lambda$$ where z is a complex variable with $Re(z)> 0$ and t is a real variable. Is it correct to ...
1
vote
0answers
104 views

Complex integration from zero to infinity at different directions

The problem is computing the integral expression $$f(t)=\int_0^{\infty}\lambda^{t-1}e^{-\lambda z}d\lambda$$ where z is a complex variable with $Re(z)\ge 0$. Is it correct to substitute $w=\lambda z$ ...
0
votes
1answer
83 views

A taylor series for an integral with a singularity

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a smooth function with a single root we call $y_*$. Then define \begin{equation} F_{\delta}:=\int^{y_*+ \delta/2}_{y_*- \delta/2} 1/f(y)dy ...
2
votes
1answer
173 views

Definite integral involving hyperbolic cosine

I have had no experience so far with hyperbolic functions so any help will be appreciated. This is on the chapter of complex integration but I would especially appreciate it if you could turn this ...
3
votes
2answers
148 views

Computation of a certain integral

I would like to compute the following integral. This is for a complex analysis course but I managed to around some other integrals using real analysis methodologies. Hopefully one might be able to do ...