Arguments of exponential and trig functions

[dyn]

What can be said about the arguments of the exponential functions and trig functions ? Can the argument be a vector or must it be a scalar ? If it can only be a scalar must it be dimensionless ?

 

[haruspex]

What can be said about the arguments of the exponential functions and trig functions ? Can the argument be a vector or must it be a scalar ? If it can only be a scalar must it be dimensionless ?

Any function that can be expressed as a power series can be extended to apply to any algebra - that is, to any ring over a scalar field.
Thus if M is a square matrix then e^M can be given a meaning.
This doesn't work for vectors in general because there is no general multiply function with a vector result. (In 3D, you could try to apply it to the cross product, but it becomes rather uninteresting when raising any given vector to a power returns zero.)
If dealing with physical quantities, it will usually be the case that the argument (and result) will be dimensionless. But you could construct artificial situations. E.g. from F=ma you could write e^F=e^ma. Each side has the strange dimension of exponentiated force, and the units could be exponentiated Newtons. Doesn't strike me as useful.