grandes-ecoles 2018 Q31

grandes-ecoles · France · centrale-maths2__official Taylor series Prove smoothness or power series expandability of a function
Show that if $g$ belongs to $\mathcal{H}(D(0,R))$ then there exists a function $H$ that expands as a power series on $D(0,R)$ such that $g$ is the real part of $H$.
One may consider a power series that is a primitive of the power series associated with the function $h$ from the previous question. 
Show that if $g$ belongs to $\mathcal{H}(D(0,R))$ then there exists a function $H$ that expands as a power series on $D(0,R)$ such that $g$ is the real part of $H$.

One may consider a power series that is a primitive of the power series associated with the function $h$ from the previous question.
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