Appendix: Hyperbolic Trigonometric Functions

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appendix: hyperbolic trigonometric functions
Definitions (a) cosh(x) = (ex+e-x)/2, (b) sinh(x) = (ex-e-x)/2, (c) tanh(x) = sinh(x)/cosh(x) (= (ex-e-x)/(ex+e-x)).

From the definitions, we can easily deduce

(1) cosh2(x) - sinh2(x) = 1,

(2) sinh(2x) = 2cosh(x)sinh(x), (3) cosh(2x) = cosh2(x) + sinh2(x). From (1) and (3), (3') cosh(2x) = 2cosh2(x) - 1.

Using (1) and (2),(3), we get

(4) sinh(2x) = 2tanh(x)/(1-tanh2(x)), (5) cosh(2x) = (1+tanh2(x))/(1-tanh2(x)), (6) tanh(2x) = 2tanh(x)/(1+tanh2(x)).

Again from the definition, after some calculation,

(7) tanh(x+y) = (tanh(x)+tanh(y))/(1+tanh(x)tanh(y)).
Note that, by (c), tanh(x) = (1-e-2x)/(1+e-2x) = 1 - 2/(e2x+1), so that tanh(0) = 0. Also, for x > 0, e2x > 1, and increases with x, so that tanh(x) increases on [0,∞), and tanh(x) tends to 1 as x tends to ∞. Thus tanh(x) is increasing, and maps [0,∞) to [0,1).

It follows that the inverse function arctanh(x) is an increasing function, mapping [0,1) to [0,∞).

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