Hyperbolic Function - Comparison With Circular Trigonometric Functions

Comparison With Circular Trigonometric Functions

Consider these two subsets of the Cartesian plane

Then A forms the right branch of the unit hyperbola {(x,y): x2 − y2 = 1}, while B is the unit circle. Evidently = {(1, 0)}. The primary difference is that the map t → B is a periodic function while t → A is not.

There is a close analogy of A with B through split-complex numbers in comparison with ordinary complex numbers, and its circle group. In particular, the maps t → A and t → B are the exponential map in each case. They are both instances of one-parameter groups in Lie theory where all groups evolve out of the identity For contrast, in the terminology of topological groups, B forms a compact group while A is non-compact since it is unbounded.

The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborn's rule states that one can convert any trigonometric identity into a hyperbolic identity by expanding it completely in terms of integral powers of sines and cosines, changing sine to sinh and cosine to cosh, and switching the sign of every term which contains a product of 2, 6, 10, 14, ... sinhs. This yields for example the addition theorems

\begin{align} \sinh(x + y) &= \sinh (x) \cosh (y) + \cosh (x) \sinh (y) \\ \cosh(x + y) &= \cosh (x) \cosh (y) + \sinh (x) \sinh (y) \\ \tanh(x + y) &= \frac{\tanh (x) + \tanh (y)}{1 + \tanh (x) \tanh (y)} \end{align}

the "double argument formulas"

\begin{align} \sinh 2x &= 2\sinh x \cosh x \\ \cosh 2x &= \cosh^2 x + \sinh^2 x = 2\cosh^2 x - 1 = 2\sinh^2 x + 1 \\ \tanh 2x &= \frac{2\tanh x}{1 + \tanh^2 x} \end{align}

and the "half-argument formulas"

Note: This is equivalent to its circular counterpart multiplied by −1.
Note: This corresponds to its circular counterpart.

The derivative of sinh x is cosh x and the derivative of cosh x is sinh x; this is similar to trigonometric functions, albeit the sign is different (i.e., the derivative of cos x is −sin x).

The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones that does not involve complex numbers.

The graph of the function a cosh(x/a) is the catenary, the curve formed by a uniform flexible chain hanging freely between two fixed points under uniform gravity.

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