Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define $\widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta )$ and $\widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$ on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$. For $n \in \mathbb { Z }$, let $$\forall r \in \mathbb { R } _ { + } ^ { * } , \quad c _ { n , f } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { f } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta, \quad c _ { n , g } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { g } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta$$
Show that $c _ { n , f }$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R } _ { + } ^ { * }$ and satisfies $$\forall r \in \mathbb { R } _ { + } ^ { * } , \quad \left( c _ { n , f } \right) ^ { \prime } ( r ) = \frac { i n } { r } c _ { n , g } ( r )$$