We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by $$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$ Justify that $\theta$ is of class $C ^ { \infty }$ on $\mathbb { R } ^ { + * }$ and demonstrate that, for all $n \in \mathbb { N } ^ { * }$, there exists $P _ { n } \in \mathbb { C } [ X ]$ such that $$\forall x \in ] 0 , + \infty \left[ \quad \theta ^ { ( n ) } ( x ) = \frac { P _ { n } ( \ln x ) } { x ^ { n } } \theta ( x ) \right.$$
We define the function $\theta : \mathbb { R } \rightarrow \mathbb { C }$ by
$$\begin{cases} \theta ( x ) = 0 & \text { if } x \leqslant 0 \\ \theta ( x ) = \exp \left( - \frac { \ln ^ { 2 } x } { 4 \pi ^ { 2 } } + \mathrm { i } \frac { \ln x } { 2 \pi } \right) & \text { if } x > 0 \end{cases}$$
Justify that $\theta$ is of class $C ^ { \infty }$ on $\mathbb { R } ^ { + * }$ and demonstrate that, for all $n \in \mathbb { N } ^ { * }$, there exists $P _ { n } \in \mathbb { C } [ X ]$ such that
$$\forall x \in ] 0 , + \infty \left[ \quad \theta ^ { ( n ) } ( x ) = \frac { P _ { n } ( \ln x ) } { x ^ { n } } \theta ( x ) \right.$$