My bad. I should have encouraged more curiosity by pointing out that these functions actually have additive as well as multiplicative identities https://msp.o

My bad. I should have encouraged more curiosity by pointing out that these functions actually have additive as well as multiplicative identities https://msp.org/pjm/1955/5-2/pjm-v5-n2-p02-p.pdf https://msp.org/pjm/1955/5-2/pjm-v5-n2-p02-p.pdf (Might even be relevant to the pendulum, see the comment below on the Arithmetic-Geometric Mean ;)