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 approaches . The absolute value and the argument of along the imaginary axis. The left graphic shows and the right graphic shows . Because is purely imaginary on the imaginary axis, 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 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 . The absolute value and the argument of on the unit circle. The left graphic shows and the right graphic shows .
