grandes-ecoles 2018 Q26

grandes-ecoles · France · centrale-maths2__official Sequences and series, recurrence and convergence Series convergence and power series analysis

A function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands as a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i}y)^n$$ Show that $f$ is of class $\mathcal{C}^1$ on $D(0,R)$ and that its partial derivatives expand as power series on $D(0,R)$. What can we deduce about the function $f$?

A function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands as a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that
$$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i}y)^n$$
Show that $f$ is of class $\mathcal{C}^1$ on $D(0,R)$ and that its partial derivatives expand as power series on $D(0,R)$. What can we deduce about the function $f$?

Paper Questions

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