In complex analysis, I am using $$H(x)=e^{-e^{-Rx}}$$ as an analytic approximation of the step function(for large R).
This function seems to be entire since it only has exponents(I did not prove this). However, I noticed that it may have poles at negative infinity.
So my questions are:
- Are there any poles/singularities?
- If so, what are the residues?
- R is also the radius of my circular contour centered at zero. R will approach infinity. Are the singularities included in the contour?