Articles of complex analysis

Singular points

Find all the (isolated) singular points of $f$, classify them, and find the residue of $f$ at each singular point. $$f(z) = \frac{z^{1/2}}{z^2 + 1}$$ I think I have $3$ singularities at $z=0,-i$ and $i$ but am unsure about what to do next and what type of singularities they are. I don’t fully understand singularities […]

On the Riemann mapping theorem

Let’s take the family of analytic one to one functions, $f:G\to \mathbb{C}$ (with $G\neq \mathbb{C}$ a region and $z_0\in G$ a fixed point) such that $|f|<1$, $f(z_0)=0$ and $f'(z_0)$ is a real positive number. One question is to find all the regions $G\neq \mathbb{C}$ such that the previous family is non empty. Clearly, thanks to […]

Integral $\int_0^{2\pi}\frac{dx}{2+\cos{x}}$

How do I integrate this? $$\int_0^{2\pi}\frac{dx}{2+\cos{x}}, x\in\mathbb{R}$$ I know the substitution method from real analysis, $t=\tan{\frac{x}{2}}$, but since this problem is in a set of problems about complex integration, I thought there must be another (easier?) way. I tried computing the poles in the complex plane and got $$\text{Re}(z_0)=\pi+2\pi k, k\in\mathbb{Z}; \text{Im}(z_0)=-\log (2\pm\sqrt{3})$$ but what […]

Holomorphic extension of a function to $\mathbb{C}^n$

I am stuck at the following question : Let $f$ be a holomorphic function on $\mathbb{C}^n \setminus \{(z_1, z_2, \ldots, z_n) | z_1=z_2=0\}$. Show that $f$ can be extended to a holomorphic function on $\mathbb{C}^n$. I think that I have to somehow use the Riemann extension theorem here. But I am not being able to […]

Evaluation of $\int_0^\infty \frac{x^2}{1+x^5} \mathrm{d} x$ by contour integration

Consider the following integral: $$\int_{0}^{\infty} \frac{x^{2}}{1+x^{5}} \mathrm{d} x \>.$$ I did the following: Since $-1$ is a pole on the real axis, I took $z_{1}=e^{3\pi/5}$ then constructed an arc between $Rz_{1}$ and $R$ : $f(e^{2\pi i /5}z)=f(z)$ it follows that : $(1-e^{2pi i/5})\int_{\alpha}f(z)dz + \int_{\beta}f(z)dz = 2\pi i \text{ Res } z_{2} f $ , […]

What are some general strategies to build measure preserving real-analytic diffeomorphisms?

One could prove the following theorem in the smooth setting: Theorem Let $(M,m)$ be a $d$ dimensional $C^\infty$ manifold with smooth volume $m$. Let $\{F_i\}_{i=1}^k$ and $\{G_i\}_{i=1}^k$ be two systems of disjoint open subsets, satisfying $m(F_i)=m(G_i)$. Assume $\bar{F}_i$ and $\bar{G}_i$ are diffeomorphic to the $d$ dimensional closed ball in $\mathbb{R}^m$ for $i=1,\ldots k$. Then, given […]

$z\exp(z)$ surjectivity with the Little Picard Theorem

I would like to prove the surjectivity of this function : \begin{align*} f\colon\mathbb{C}&\to\mathbb{C}\\ z&\mapsto z\exp(z) \end{align*} You can use the Little Picard Theorem: If a function $f\colon\mathbb{C}\to\mathbb{C}$ is entire and non-constant, then the set of values that $f(z)$ assumes is either the whole complex plane or the plane minus a single point. Thanks.

If $U$ is connected, any two sections $U \to \mathfrak S$ either coincide or have disjoint images (Is my proof correct?)

I tried proving the following statement by Ahlfors, page 287: If $U$ is connected and $\varphi,\psi: U \to \Gamma(U,\mathfrak S)$, then either $\varphi$ and $\psi$ are identical, or the images $\varphi(U)$ and $\psi(U)$ are disjoint. Indeed, the sets with $\varphi – \psi = 0$ and $\varphi- \psi \neq 0$ are both open. Here $\mathfrak S$ […]

How do I solve this complex integration problem?.

I want to find the value of $$I=\int_{|z|=r}\frac{|dz|}{|z-z_0|^4},$$ where $|z_0|\neq r>0$.

Order of $\frac{f}{g}$

An entire function is of finite order $\rho$ if $$\rho = \inf \{\lambda \geq 0 \ | \ \exists A, B > 0 \ s.t. \ |f(z)|\leq Ae^{B|z|^{\lambda}} \forall z \in \mathbb{C} \}$$ Prove that if $f$ and $g$ are entire functions of finite order $\rho$, and $\frac{f}{g}$ is entire,then $\frac{f}{g}$ is of order $\leq […]