The complex sine function is, as in the real case, defined as the solution of the
differential equation (ODE) sin''(z) = -sin(z) to the initial conditions sin(0) = 0, sin'(0) = 1. The
real and the complex sine function therefore agree for real arguments x. The function g(z) = sin(i*z)
satisfies the differential equation g''(z) = +g(z) with g(0) = 1, g'(0) = i. By definition of the
hyperbolic sine as solution of this ODE we conclude sin(i*z) = i*sinh(z).