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On the real axis The function  along the real axis.  is real‐valued for all real arguments. At  , the function  has a discontinuity. The function  along the real axis. The imaginary part vanishes identically on the real axis. The left graphic shows  and the right graphic shows  . 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 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 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 complex valued with a piecewise constant real part, and the imaginary part is asymtotically a decreasing function. At the points  , the function  has logarithmic singularities. The absolute value and the argument of  along the imaginary axis. The left graphic shows  and the right graphic shows  . Because  is purely imaginary outside the interval  , the argument is constant there. 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 logarithmic singularities. 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  . Interestingly, the real part is piecewise constant with value  . The absolute value and the argument of  on the unit circle. The left graphic shows  and the right graphic shows  .
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