2D plots along the axes and unit circle

On the real axis

The function along the real axis. is real‐valued for all real arguments.

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 absolute value of the imaginary part is asymptotically a decreasing function. At the points , the function has a logarithmic singularity.

The absolute value and the argument of along the imaginary axis. The left graphic shows and the right graphic shows . Because is purely imaginary in the interval , the argument is constant in this interval.

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 a logarithmic 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 . Interestingly, the real part is piecewise constant with the value .

The absolute value and the argument of on the unit circle. The left graphic shows and the right graphic shows .