Notes

Chapter 4: Systems Based on Numbers

Section 6: Mathematical Functions


Three sine functions

All zeros of the function Sin[a x] + Sin[b x] lie on the real axis. But for Sin[a x] + Sin[b x] + Sin[c x], there are usually zeros off the real axis (even say for a=1, b=3/2, c=5/3), as shown in the pictures below .

If a, b and c are rational, Sin[a x]+Sin[b x]+Sin[c x] is periodic with period 2 Pi /GCD[a, b, c], and there are a limited number of different spacings between zeros. But in a case like Sin[x]+Sin[Sqrt[2] x]+Sin[Sqrt[3] x] there is a continuous distribution of spacings between zeros, as shown on a logarithmic scale below. (For 0 < x < 10^6 there are a total of 448,494 zeros, with maximum spacing 4.6 and minimum spacing 0.013.)

From Stephen Wolfram: A New Kind of Science [citation]