# Tagged Questions

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### The normalisation map is a bi-Lipschitz map?

Let $X$ a reduced analytic space, $n: W \rightarrow X$ the normalisation map, $W$ the normalisation of $X$ and $S$ the singular set of $X$. When we restrict $n$ to $W\setminus n^{-1}(S)$, we know that ...
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### Determine the nature of singularities and calculate the residue of $f(z)=\frac{e^z-\mathrm{sin}z-1}{z^5+z^3}$

$$f(z)=\frac{e^z-\mathrm{sin}z-1}{z^5+z^3},\;\;\;\;\;\;\; \mathrm{Res}[f(z),0]$$ I am having trouble determining the nature of singularities. This is what I managed to do: ...
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### Types of singularities of a function

How can I determine the type of a singularity of a function $$f(z)={e^{1/z} \over z-1}+{\pi z \over 2\sin(\pi z)}$$ at $z_0=0$? I don't see an easy way to represent it using Laurent series, neither ...
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Consider the function $$f(z) = {1 \over (z-i)(z+i)}$$ with a Laurent series expansion at $z_0=i$ on a domain $\;\Omega=\left\{z\in \mathbb{C}:2\lt\left|z-i\right|\right\}$ $$\begin{eqnarray}f(z)={1 ... 1answer 54 views ### non-removable singularities of a function are essential singularities of the composition function Let f be a non-constant enrire function on \mathbb{C} such that f(z+i)=f(z) for all z. Let U be an open subset of C and z_0\in U. Let g:U\setminus\{z_0\}\longrightarrow\mathbb{C} be ... 1answer 23 views ### Question about non essential singulariy When reading Ahlfor's Complex Analysis book, I came across the notion of non essential singularity. I know that for a function f(z) an element a\in\mathbb C is a non essential singularity iff ... 1answer 43 views ### Singularity Classification Suppose that I have the following function:$$f(z)=\sin\left(\frac{1}{\sin(\frac{1}{z})}\right)$$If I'm trying to characterize singularities, I know that singularities will be found whenever ... 1answer 98 views ### Removable singularities of a holomorphic function So, I'm a little confused about removable singularities. Consider the function below:$$f(z)=\frac{1}{(1+z^2)^{2/3}}$$Obviously, we have isolated singularities at the points z = \pm i. ... 1answer 66 views ### Removable singularities for continuous functions Let f: D - K \rightarrow \mathbb{C} be holomorphic, where D is a planar domain and K is a compact subset of D. Suppose that f extends continuously to all of D. On which conditions on K ... 2answers 50 views ### Classifying singularity Having trouble classifying a singularity... f(z)=$$z^2-1\over z^6+2z^5+z^4$with$z_0=0$and$z_0=-1$The$z_0=0$is pretty simple, just need to put$z^4$in evidence. But$z_0=-1$I can't seem to ... 1answer 36 views ### Finding the types of singularities of$f(z)=\frac{1}{z\cdot (e^z -1 )}$I am getting trouble to find the types of singularities of $$f(z)=\frac{1}{z\cdot (e^z -1 )}$$ What I tried to do is:$z=0z=2\pi k i$for$z=2\pi k i$its in order 1, but for the first one I ... 1answer 61 views ### Laurent series of an analytic function divided by$z$This is a probably basic question about Laurent series. Say$g(z)$is an analytic function, that$g(0) = 0$, and$f(z) = g(z)/z$. My textbook says$z = 0$is a removable singularity of$f(z)$. A ... 0answers 63 views ### How to analyze the asymptotic properties of this function? Let the function $$f(\mathbf{r})=\int_{\Omega }e^{i\mathbf{k} \cdot \mathbf{r}}d^2\mathbf{k}$$, where$\mathbf{k} ,\mathbf{r}\in\mathbb{R}^2$, and$\Omega \subset \mathbb{R}^2$is some finite region ... 2answers 146 views ### Are the convergence radii circles of a Laurent-series always caused by isolated singularities? Laurent series $$f(z) := \sum_{n=-\infty}^\infty a_n (z-c)^n$$ converge for$r<|z-c|<R$where $$r = \limsup_{n\to\infty}|a_{-n}|^{\frac1n}, \\\frac1R = \limsup_{n\to\infty}|a_n|^{\frac1n}.$$ ... 1answer 51 views ### Growth rate of Taylor convergents near pole For any fixed$z_0\in\mathbb{C}\setminus \{0\}$and$\beta\in\mathbb{R}^{+}$, prove that $$\left.T_n\left(\log^{\beta}z;z_0\right)\right|_{z=0}\sim\log^{\beta} n$$ Note: I observed that this holds ... 3answers 1k views ### Type of singularity of$\log z$at$z=0$What type of singularity is$z=0$for$\log z$(any branch)? What is the Laurent series for$\log z$centered at 0, if exist? If the Laurent series has the form$\sum_{k=-\infty}^{\infty} a_kx^k$, ... 2answers 338 views ### Proving that a function has a removable singularity at infinity I'm having trouble with the following exercise from Ahlfors' text (not homework) "If$f(z)$is analytic in a neighborhood of$\infty$and if$z^{-1} \Re f(z) \to0$as$z \to \infty$, show that ... 2answers 211 views ### Singularities of$f(z)=z/\cos(z)$Regarding complex functions (in complex variables), I was wondering why the function$g(z)= \cos(z)$has a singularity at$z = \infty$but$f(z)= \dfrac{z}{\cos(z)}$does not. I am a bit confused ... 1answer 517 views ### Is there a classification of isolated essential singularities? In the thread Why do we categorize all other (iso.) singularities as "essential"?, here is one of the questions that was asked: Do we not care about essential singularities to classify ... 1answer 72 views ### Removable Singularity with bound on derivative Here is a question from a practice exam: Suppose$g(z)$is a holomorphic function everywhere except the origin. Also suppose $$|g'(z)|\leq \frac{1}{|z|^{3/2}} \quad \text{ for } 0<|z|\leq 1$$ ... 1answer 64 views ### Do singularities always appear on all Riemann sheets? Consider a function $$f(z) = \frac{\ln z}{z^2+1}.$$ Besides the branching point$z=0$, the function also has singularities at$z = \pm i$. This singularities should appear on all Riemann sheets. Is ... 1answer 136 views ### questions about singularities and complex functions having poles of order k , proofs and examples 1)i need an example of a non isolated singularity 2) also i need an entire function which assumes every complex value but the number 1+2i and i want to know the way in order to solve some other ... 2answers 558 views ### Contour integral$\int_{|z|=1}\exp(1/z)\sin(1/z)dz$Evaluate the contour integral $$\int_{|z|=1}\exp(1/z)\sin(1/z)\,dz$$ along the circle$|z|=1$counterclockwise once. The singularities are$\dfrac1{\pi k},k\in\mathbb{Z}$plus the limit point$0$. So ... 2answers 305 views ### Removable singularity and laurent series How to show$z=\pm\pi$is a removable singularity for$\frac1{\sin z}+\frac{2z}{z^2-\pi^2}$? I tried to compute the Laurent series, specifically the coefficients for$1/z,1/z^2,...$since if we can ... 1answer 95 views ### Does$\frac {z^5}{\sin z^2-z^2}$have a non-isolated singularity at$0$? Does$\frac {z^5}{\sin z^2-z^2}$have a non-isolated singularity at$0$? If so, is it not meaningful to discuss its residue at$0$? 2answers 302 views ### What is$\lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$for$z \in \mathbb C$? What is$\lim\limits_{z \to 0} |z \cdot \sin(\frac{1}{z})|$for$z \in \mathbb C^*$? I need it to determine the type of the singularity at$z = 0$. 1answer 166 views ### Prove that$f(z)$can not be a polynomial Suppose$f(z)$and$g(z)$are entire functions and that$f(z)$is not constant. If$|f(z)| < |g(z)|$for all$z \in \mathbb C, $prove that$f(z)$can not be a polynomial. I was thinking what I ... 3answers 421 views ### Singularity - Removable or Pole? For the complex-valued function $$f(z) = \left(\frac{\sin 3z}{z^2}-\frac{3}{z}\right)$$ classify the singularity at$z=0$and calculate its residue. Attempt at Solution Rewriting$f(z) = ...
The Casorati Weierstrass Theorem: Let $f:X\subseteq \mathbb{C}\to \mathbb{C}$ have an essential singularity at $w\in \mathbb{C}$. Then, \forall \epsilon,\delta>0,\zeta\in ...
### Singularities of $e^{z - \frac{1}{z}}$
I believe $e^{z - \frac{1}{z}}$ has essential singularities at $z = 0$ and $z = \infty$ (in both cases because of a $\frac{1}{z}$ in the exponent) but I'm having a hard time proving this. How can one ...