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 .
