2D plots along the axes and unit circle

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 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 .