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If a real function $f:ℝ→ℝ$ is twice differentiable at a point $x$, then the first derivative must be continuous at $x$, and assuming $f′(x)>0$, then there exist $δ>0$ such that $f′(y)>0 $ for all $y∈( …

Let $f:ℝ→ℝ$ be a real analytic function with infinitely many isolated zeros. Let us define the function: $$h(s₁,s₂,...,s_{r+1})=\prod_{k=1}^{r+1}f^{(k+1)}(\left(1-2\prod_{j=1}^{k}s_{j}\right)$$ Also, …

Let $f:\mathbb{R} \to \mathbb{R}$ be a real analytic function. Assume that $f$ has simple trivial zeros at each nonpositive integer. Then, all the $k$-th derivatives $f^{(k)}$ of $f$ have necessarily …

Let $f:ℝ²→ℝ$ be an arbitrary harmonic function. A level curve in two dimensions is a curve on which the value of a function $f(x,y)$ is a constant. My question is: Find sufficient and necessary condit …

Let us consider the following conjecture: Conjecture: There are no integer solutions of the equation $$x^{y-z}z^{x-y}=y^{x-z}$$ with $x,y,z$ distinct positive integers greater than or equal to $2$. …

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. The modularity theorem implies that $L(C,s)$ is the $L$-functi …