the complex inverse trigonometric functions

On the real line we restrict sin and cos to a chosen interval to invert them, since they repeat. In the complex plane the same need arises, but because the trig functions are built from the exponential, their inverses can be written explicitly in terms of the logarithm and the square root, and so they inherit exactly the branch structure of those multi-valued functions.

Start from sin z = (e^(i z) - e^(-i z)) / (2 i) and set sin w = z. Letting u = e^(i w) turns the equation into the quadratic u^2 - 2 i z u - 1 = 0, whose solution u = i z + sqrt(1 - z^2) leads to arcsin z = -i log(i z + sqrt(1 - z^2)). Similarly arccos z = -i log(z + sqrt(z^2 - 1)) and arctan z = (1 / (2 i)) log((i - z) / (i + z)) = (i/2) log((i + z)/(i - z)). The inverse hyperbolic functions are the same kind of expression: arcsinh z = log(z + sqrt(z^2 + 1)), arccosh z = log(z + sqrt(z^2 - 1)), arctanh z = (1/2) log((1 + z)/(1 - z)).

Because each formula contains a logarithm (and usually a square root), each is multi-valued, and you make it single-valued by choosing branches for the log and the root. The branch points come from where the inside of the square root or the log vanishes: arcsin and arccos have branch points at z = 1 and z = -1, while arctan has branch points at z = i and z = -i (where i + z or i - z hits zero). A principal value is obtained from the principal log and principal root, after which arcsin, arccos, and arctan are honest holomorphic functions on suitably cut planes. The takeaway is that nothing genuinely new is needed: complex inverse trig and inverse hyperbolic functions are just logarithms and square roots wearing trigonometric names, so their branch cuts are bookkeeping you already understand.