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Rather than an answer, this is (I believe) an evidence of analyticity boundary along the imaginary axis. Here is a contour plotare plots for the logarithm of the absolute value of $f(x)$:

enter image description here

enter image description here

(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 $f(x)$:

enter image description here

(can be enlarged by clicking twice)

Rather than an answer, this is (I believe) an evidence of analyticity boundary along the imaginary axis. Here are plots for the logarithm of the absolute value of $f(x)$:

enter image description here

enter image description here

(can be enlarged by clicking twice)

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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 $f(x)$:

enter image description here

(can be enlarged by clicking twice)