On the real axis The function along the real axis. is real‐valued in the interval . The function along the real axis. The left graphic shows and the right graphic shows . is piecewise constant along the real axis. At , the function has logarithmic singularities. 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 in the interval . 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 . For , 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 absolute value of the imaginary part is asymptotically an increasing function. The absolute value and the argument of along the imaginary axis. The left graphic shows and the right graphic shows . 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 . Interestingly, the imaginary part is piecewise constant with values on the unit circle. The absolute value and the argument of on the unit circle. The left graphic shows and the right graphic shows .
