Hyperbolic Angle - Imaginary Circular Angle

Imaginary Circular Angle

The hyperbolic angle is often presented as if it were an imaginary number. In fact, if x is a real number and i2 = −1, then

so that the hyperbolic functions cosh and sinh can be presented through the circular functions. But these identities do not arise from a circle or rotation, rather they can be understood in terms of infinite series. In particular, the one expressing the exponential function ( ) consists of even and odd terms, the former comprise the cosh function, the latter the sinh function . The infinite series for cosine is derived from cosh by turning it into an alternating series, and the series for sine comes from making sinh into an alternating series. The above identities use the number i to remove the alternating factor (−1)n from terms of the series to restore the full halves of the exponential series. Nevertheless, in the theory of holomorphic functions, the hyperbolic sine and cosine functions are incorporated into the complex sine and cosine functions.