3 problems

Let $D$ be a polydisc, let $p\in\mathbb{N}^{\times}$, and let $\upsilon_j(\boldsymbol{x})$ for $0\leqslant j\leqslant p$ be holomorphic on $D\cap(\{0\}\times\mathbb{C}^n)$, with…

Consider the ramified initial-value problem … … where $c_1,c_2\in\mathbb{C}^{\times}$, $c_3\in\mathbb{C}$, and $u_0$ is holomorphic near $0$. For each root $\tau$ of…

Consider the ramified initial-value problem … … with $N_0>2$, $c_1,c_2\in\mathbb{C}^{\times}$, and $c_j\in\mathbb{C}$ for $j\geqslant 3$. For a root $\tau$ of $\tau^2-u_x(0,0)=0$,…