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Suppose $f: \mathbb C \times (-1,1) \to \mathbb C$ is a smooth function that satisfies $f(0,t)=1$ for all $t\in (-1,1)$. Assume that for any $k\in \mathbb N$, any $z \in \mathbb C$ and any $t \in (-1,...

I am interested in the standard name for the following weak form of $k$-differentiability. Definition. A function $f:\mathbb R\to\mathbb R$ is called Taylor $k$-differentiable at a point $x_0$ if ...

Suppose I have a multivariate function $f$ from $\mathbb{C}^d$ to $\mathbb{C}$ that accepts a Taylor expension of the form $$f(\mathbf x) = \sum\limits_{\mathbf k \in \mathbb N^d} a_{\mathbf k} \...