Rather than an answer, this is (I believe) an evidence of analyticity boundary along the imaginary axis. Here is a contour plot for the logarithm of the absolute value of $f(x)$:
(can be enlarged by clicking twice)
Rather than an answer, this is (I believe) an evidence of analyticity boundary along the imaginary axis. Here is a contour plot for the logarithm of the absolute value of $f(x)$:
(can be enlarged by clicking twice)