As said in a paper I am reading on p 2677 in the text directly above FIG3, this should be a standard result about Fourier transforms of analytic functions. In the paper the authors use these methodes to investigate the intermittent behaviour of turbulent flows kicking in at high frequencies.
But I am not familiar with this, so can somebody explain to me how exactly these singularities in the complex time plane are related to the high frequency behaviour of a function?