Hyperbolic functions (cheatsheet)

Hyperbolic functions (cheatsheet) - Hyperbolic...

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Hyperbolic functions (CheatSheet) 1 Intro For historical reasons hyperbolic functions have little or no room at all in the syllabus of a calculus course, but as a matter of fact they have the same dignity as trigonometric functions. Unfortu- nately this can be completely understood only if you have some knowledge of the complex numbers. Roughly speaking ordinary trigonometric functions are trigonometric functions of purely real num- bers, and hyperbolic functions are trigonometric functions of purely imaginary numbers. For the moment we have to postpone this discussion to the end of Calc3 or Calc4, but still we should be aware of the fact that the impressive similarity between trig formulas and hyperbolic formulas is not a pure coincidence. Most of the formulas that follow correspond precisely to a trig formula or they differ by at most a change of sign. For each formula I will explicitly state if some change of sign occurs or not (the different sign is marked in green ). The main purpose of this paper is not to give you a bunch of formulas to memorize, but to make you aware of the fact that hyperbolic formulas are just like trig formulas up to signs; and correct signs can always be checked with some very quick calculation. 2 Definitions Definition of hyperbolic sine and cosine: sinh x = e x - e - x 2 cosh x = e x + e - x 2 There are two equivalent formulas for sine and cosine (Euler’s formulas) but they require some knowledge of the complex numbers: sin x = e ix - e - ix 2 i cos x = e ix + e - ix 2 where i = - 1 or if you prefer i 2 = - 1. Substituting x with ix in these two formulas and keeping in mind that i 2 = - 1 it’s immediate to deduce that cosh x = cos ( ix ) and sinh x = - i sin ( ix ) (I mention this just for the sake of completeness and because it’s fun!).
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