Question

sin(iz) = i sinh (z) b) cos(iz) = i cosh(z) c) tan(iz) = i tanh(z) which one is correct

Answer 1

Answer:

I'm trying to understand in an intuitive manner the relationship between the circular and hyperbolic functions in the complex plane, i.e.:

cos(z)=cosh(iz)

sin(z)=−isinh(iz)

where z is a complex number.

From a geometric point of view, what I understand is that cos is the composition of a rotation through

π

2

, followed by cosh, and sin is the composition of a rotation through

π

2

, followed by sinh, followed by a rotation through −

π

2

(where sin, cos, sinh, cosh are defined as complex functions).

Where does this connection come from? Is there some way it can be visualized in terms of complex mappings? (I'm not asking for a proof of the identities, I already know one).