Question

What is period of function sine hyperbolic sinh(z)?

Answer 1

The period `2pi` for `z=|z|e^(i arg z)`, in its `arg z` is indeed the period for `f(z)=sinh z`.

Explanation:

Let `z=r e^(itheta)=r(cos theta + i sin theta)=z(r, theta)=|z|e^( i arg z).`.

Now, `z=z(r, theta)=z(r, theta+2pi)`

So, `sinh (z(r, theta+2pi)=sinh (z(r, theta)=sinh z`,

Thus sinh z is periodic with period 2pi in arg z = theta#.