On the real axis The function along the real axis. is an oscillating periodic function with period that has first‐order poles at . 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. At , the function has an essential singularity and oscillates infinitely often. 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 purely imaginary and vanishes identically. The absolute value and the argument of along the imaginary axis. The left graphic shows and the right graphic shows . Because is vanishes identically on the imaginary axis, the argument is 0 there. On the imaginary axis at infinity The function along the imaginary 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 . 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 .
