I was collecting data on 3 particular functions, to find a relationship between them: Let $f(n)$ be the product of all the nonzero digits of a number so that $P(n)$ is the sum of all $f(n)\le n$ $S(n)$ is simply the sum of all consecutive integers $\le n$, basically $\cfrac { n(n+1) }{ 2 } $ $...

I was collecting data on 3 particular functions, to find a relationship between them: Let $f(n)$ be the product of all the nonzero digits of a number so that $P(n)$ is the sum of all $f(n)\le n$ $S(n)$ is simply the sum of all consecutive integers $\le n$, basically $\cfrac { n(n+1) }{ 2 } $ $...

Let $z$ be a point in the complex plane, and $\gamma$ be a closed curve. Is it possible that $$n(\gamma,z) = \frac{1}{2\pi i}\int_\gamma \frac{dw}{w-z}$$ becomes unbounded? In other words, is it possible to find a curve $\gamma$ such that it winds around a fixed point infinitely many times?

Is there a polynomial function $P(x)$ with real coefficients that has an infinite number of roots? What about if $P(x)$ is the null polynomial, $P(x)=0$ for all x?

Statement Let $f:\mathbb \Omega \to \mathbb C$ be a holomorphic function, $f \neq 0$ ($\Omega$ is a region, i.e., an open, nonempty, connected set). Prove that in every compact subset $K$ of $\Omega$, the set of zeros of $f$ is finite. I've read in Stein's textbook a proof of the statement "Sup...

Suppose $f$ is continuous on the closed unit disc $\overline{\mathbb{D}}$ and is holomorphic on the open unit disc $\mathbb{D}$. Must $f$ have finitely many zeros in $\mathbb{D}$? I know that this is true if $f$ is holomorphic in $\overline{\mathbb{D}}$ (by compactness of the closed unit disc), b...

My book says that the following two theorems are equivalent but i can't seem to show one direction, please help. Liouvilles theorem: if a number $\xi $ is a real algebraic number of degree $n\geq 2 $ then there exists a constant $c>0$ s.t $$| \xi-\frac {p}{q}| < \frac {c}{q^n }$$ has no solution...

It is certainly possible to study all kinds of topics in pure mathematics without any knowledge of physics, because you will always find literature/researchers who are used to explain the key concepts to fellow mathematicians without any knowledge in physics. But here are some examples of useful...