On the real axis The function along the real axis. is real‐valued for all real arguments. At , the function has a discontinuity. The function along the real axis. The imaginary part vanishes identically on the real axis. The left graphic shows and the right graphic shows . 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 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 complex valued with a piecewise constant real part, and the imaginary part is asymtotically a decreasing function. At the points , the function has logarithmic singularities. The absolute value and the argument of along the imaginary axis. The left graphic shows and the right graphic shows . Because is purely imaginary outside the interval , the argument is constant there. 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 logarithmic singularities. 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 real part is piecewise constant with value . The absolute value and the argument of on the unit circle. The left graphic shows and the right graphic shows .
