Couth and uncouth function pairs

“You can’t always get what you want. But sometimes you get what you need.” — The Rolling Stones Circular functions and hyperbolic functions aren’t invertible, but we invert them anyway. These functions map many points in the domain to each point in the range, and we invert them by mapping a point in the range back to some point in the domain. Often this works as expected, but sometimes it doesn’t. In the previous post I said You can relate each trig function “foo” with its hyperbolic counterpart “fooh” by applying one of the functions to iz then multiplying by a constant c that depends on foo: c = i for sin and tan, c = 1 for cos and sec, and c = −i for csc and cot. In symbols, c foo(z) = fooh(iz). Let’s suppose foo and fooh are invertible, ignoring any complications, and solve foo(z) = w for z. We get i foo−1(w) = fooh−1(cw) or using “arc” nomenclature for inverse functions i arcfoo(w) = arcfooh(cw). When the naive calculation above holds, except possibly at a finite number of…