1
vote
0answers
17 views

Maximum intervals of a solution and singularities [closed]

Let $X$ be a vector field of $C^1$ calsse in $\Delta \subseteq \mathbb{R}^n$. Prove that if $\varphi(t)$ is a trajectory of $X$ defined maximum range $(\omega_-,\omega_+)$ with: $$\lim_{t \rightarrow ...
1
vote
1answer
35 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: $z=0$ $z=2\pi k i$ for $z=2\pi k i$ its in order 1, but for the first one I ...
5
votes
2answers
597 views

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 ...