The exponential series converges for all complex numbers to define a holomorphic function exp. The derivative of exp is exp, and the identity exp(z + w) = expz×expw is still valid. Euler’s identity expz = cosz + isinz is true for all z∈ℂ, though it need not be the case that cosz and sinz are the real and imaginary parts of expz. The range of exp is ℂ\\{0}, and exp is periodic with period 2π‎i. Compare complex logarithm.