In the paper "The Inverse Function Theorem of Nash and Moser" by Richard S. Hamilton it is claimed that there exists a function $\phi$ such that:
For $n=0,1,2,...$. In fact one is provided. The example is:
My trouble is that the paper said that verifying the integral can be done by methods of contour integration by recognizing this function as the real part of a holomorphic function. The trouble I'm having is that I have no idea how to identify this holomorphic function. Any help is greatly appreciated. Thank you in advance.