Starting from this formula, the exponential function as well as all the trigonometric and hyperbolic functions can be expressed in terms of the gamma function.
32.
These also generate rotations through the exponential map, but instead of trigonometric functions, hyperbolic functions are needed, which generates a rotor as follows:
33.
The analogy between circular and hyperbolic functions was described by Augustus De Morgan in his " Trigonometry and Double Algebra " ( 1849 ).
34.
As functions of a complex variable, inverse hyperbolic functions are multivalued functions that are arcs ( usually half lines or line segments ) have been removed.
35.
Using the inverse hyperbolic function, the rapidity corresponding to velocity is artanh ( v / c ) } } where c is the velocity of light.
36.
:: Those are missing Bessel functions, Gamma functions, elliptic functions, parabolic cylinder functions, hyperbolic functions, hypergeometric functions, and Legendre functions.
37.
The topics you mentioned, such as calculus of hyperbolic functions, are only taught at H3 Maths in JC, which is reserved for the best students.
38.
The hyperbolic functions arise from the " difference " between the squares of the time and spatial coordinates in the spacetime interval, rather than a sum.
39.
Thus, hyperbolic functions are periodic with respect to the imaginary component, with period 2 \ pi i ( \ pi i for hyperbolic tangent and cotangent ).
40.
These transcendental hyperbolic functions can be converted into circular functions sine and cosine by introducing ( & minus; 1 ) k into the series, resulting in alternating series.
