No Branch Cut Needed for cos(sqrt(z))

[NT123]

Homework Statement

Branch cut for cos(sqrt(z)).

Homework Equations

The Attempt at a Solution

Apparently there is no need for a branch cut for this function, but I am not sure why - I heard it has something to do with cos being an even function. Any clarification would be appreciated.

 

[Hurkyl]

I'm sure you've done an exercise like writing down all of the values of a multivalued function like sqrt(z) in terms of ordinary functions. (e.g. in terms of the principal branch Sqrt(z))

That seems like an obvious place to start to me.

 

[NT123]

I'm sure you've done an exercise like writing down all of the values of a multivalued function like sqrt(z) in terms of ordinary functions. (e.g. in terms of the principal branch Sqrt(z))

That seems like an obvious place to start to me.

You mean write z as r*exp(i(t+2*k*pi)), so sqrt(z) = (r^1/2)*exp(it/2) or (r^1/2)*exp(i(t/2+pi)? = -(r^1/2)*exp(it/2) ?