Kramers–Kronig, Bode, and the meaning of zero

The implications of causality are captured by the Kramers–Kronig relations between the real and imaginary parts of a linear response function. In 1937, Bode derived a similar relation between the magnitude (response gain) and the phase. Although the Kramers–Kronig relations are an equality, the Bode’s relation is effectively an inequality. This difference is explained using elementary examples and is traced back to delays in the flow of information within the system formed by the physical object and the measurement apparatus.