That was in some sense intentional -- I prefer exp(i μ k) exp(-k² σ² / (2 n)) for the clarity of the expression in Fourier space. That issue is not avoided --

That was in some sense intentional -- I prefer exp(i μ k) exp(-k² σ² / (2 n)) for the clarity of the expression in Fourier space. That issue is not avoided -- it becomes ln(cosh(k)), which still only has a finite radius of convergence due to branch points at ± i π/2, and a complicated branch-sheet structure. It is not dissimilar from 1/(1 + x^2) -- a perfectly smooth ordinary real function on the real line, very pleasant to work with, but the (real!) Taylor series at 0, 1 - x^2 + x^4 - x^6 + ... does not converge outside of the interval (-1, 1) because it must go to infinity at ± i on the complex plane.