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On the real axis The function  along the real axis.  is real‐valued along the real axis that approaches  as  . The absolute value and the argument of  along the real axis. The left graphic shows  and the right graphic shows  . The argument is piecewise constant. On the real axis at infinity The function  along the real axis. The left graphic shows  and the right graphic shows  . The imaginary part vanishes identically along the real axis. The absolute value and the argument of  along the imaginary axis. The left graphic shows  and the right graphic shows  . The argument is piecewise constant. On the imaginary axes The real part and the imaginary part of  along the imaginary axis. The left graphic shows  and the right graphic shows  . Along the imaginary axis,  is purely imaginary and the imaginary part is oscillating and periodic with poles at  . The absolute value and the argument of  along the imaginary axis. The left graphic shows  and the right graphic shows  . The argument is piecewise constant. On the imaginary axis at infinity The function  along the imaginary axis. The left graphic shows  and the right graphic shows  . At  , the function  has an essential singularity. The absolute value and the argument of  along the imaginary axis. The left graphic shows  and the right graphic shows  . On the unit circle The real part and the imaginary part of  on the unit circle. The left graphic shows  and the right graphic shows  . The absolute value and the argument of  on the unit circle. The left graphic shows  and the right graphic shows  .
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