$e^{1/z}$ has an essential singularity at $z=0$, which by the Casorati–Weierstrass theorem implies that for every complex number $W$ there is a sequence $z_k\to 0$ with $e^{1/z_k}\to W$. Is there some elegant way to see this directly without proving Casorati–Weierstrass first? (I know that the pr...

Let $\mathfrak {g}$ be a Lie bialgebra. Then $\mathfrak {g}^{\ast}$ is also a Lie bialgebra which is dual to $\mathfrak {g}.$ Let the brackets on $\mathfrak {g}$ and $\mathfrak {g}^{\ast}$ be denoted by $b$ and $b'$ respectively. Then how to define the coadjoint action of $\mathfrak {g}^{\ast}$ o...

I am wondering how one can prove that for an arbitrary odd natural number $n$ and an arbitrary natural number $a$ the power-sum $S_{(a,n)}=1^n+2^n+\ldots +a^n $ is divisible by $S_{(a,1)}=1+2+\ldots + a$ in the most elementary way (possible) which comes to your mind. By an elementary way I mean t...