the complex hyperbolic functions

The hyperbolic functions cosh and sinh are the exponential's other pair of children: cosh t = (e^t + e^(-t)) / 2 and sinh t = (e^t - e^(-t)) / 2. On the real line they describe hanging chains and growth; in the complex plane the same formulas extend them, and they turn out to be the trig functions in disguise.

Define cosh z = (e^z + e^(-z)) / 2 and sinh z = (e^z - e^(-z)) / 2 for complex z. Because they are built from e^z they are entire, the derivative of sinh z is cosh z and the derivative of cosh z is sinh z, and the identity cosh^2 z - sinh^2 z = 1 holds. Comparing their formulas with the trig formulas reveals the clean dictionary: cos z = cosh(i z), sin z = -i sinh(i z), and equivalently cosh z = cos(i z), sinh z = -i sin(i z). In other words rotating the input by a right angle (multiplying by i) swaps the trigonometric world with the hyperbolic world. This is why complex sine and cosine grow exponentially off the real axis: they are hyperbolic functions seen sideways.

The dictionary is a labour-saving device. Any hyperbolic identity is a trig identity with i inserted, and vice versa, so you never need to memorize both lists. It also explains the cosh y and sinh y factors that appeared when we split sin(x + i y) and cos(x + i y): the imaginary part of the input is feeding the hyperbolic functions directly. Hyperbolic functions also carry the 2 pi i periodicity of e^z (along the imaginary axis), inherited from the exponential.