Control engineering says that the magnification is the gain, but why does the book give the magnification of a system under the first 1 standard and the last 1 standard when analyzing the transfer fun

Control engineering says that the magnification is the gain, but why does the book give the magnification of a system under the first 1 standard and the last 1 standard when analyzing the transfer function? These two gains are different. Usually it is an open-loop transfer function for the system.

The tail 1 equation is in the form of time constant, and the gain at this time is usually called open-loop gain. This form is often used to find steady-state error and draw Bode diagram.

The first formula 1 is in the form of zero pole, and the gain at this time can be called root trajectory gain. This form is often used to draw the root trajectory of the system.

As you said, the two are usually not equal, and there will be a certain correspondence.