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90 lines
3.6 KiB
Plaintext
90 lines
3.6 KiB
Plaintext
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[def __form1 [^\]-1;1\[]]
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[def __form2 [^\[0;+'''∞'''\[]]
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[def __form3 [^\[+1;+'''∞'''\[]]
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[def __form4 [^\]-'''∞''';0\]]]
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[def __form5 [^x '''≥''' 0]]
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[section Background Information and White Papers]
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[section The Inverse Hyperbolic Functions]
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The exponential funtion is defined, for all object for which this makes sense,
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as the power series
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[$../../libs/math/special_functions/graphics/special_functions_blurb1.jpeg],
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with ['[^n! = 1x2x3x4x5...xn]] (and ['[^0! = 1]] by definition) being the factorial of ['[^n]].
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In particular, the exponential function is well defined for real numbers,
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complex number, quaternions, octonions, and matrices of complex numbers,
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among others.
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[: ['[*Graph of exp on R]] ]
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[: [$../../libs/math/special_functions/graphics/exp_on_R.png] ]
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[: ['[*Real and Imaginary parts of exp on C]]]
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[: [$../../libs/math/special_functions/graphics/Im_exp_on_C.png]]
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The hyperbolic functions are defined as power series which
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can be computed (for reals, complex, quaternions and octonions) as:
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Hyperbolic cosine: [$../../libs/math/special_functions/graphics/special_functions_blurb5.jpeg]
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Hyperbolic sine: [$../../libs/math/special_functions/graphics/special_functions_blurb6.jpeg]
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Hyperbolic tangent: [$../../libs/math/special_functions/graphics/special_functions_blurb7.jpeg]
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[: ['[*Trigonometric functions on R (cos: purple; sin: red; tan: blue)]]]
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[: [$../../libs/math/special_functions/graphics/trigonometric.png]]
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[: ['[*Hyperbolic functions on r (cosh: purple; sinh: red; tanh: blue)]]]
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[: [$../../libs/math/special_functions/graphics/hyperbolic.png]]
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The hyperbolic sine is one to one on the set of real numbers,
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with range the full set of reals, while the hyperbolic tangent is
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also one to one on the set of real numbers but with range __form1, and
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therefore both have inverses. The hyperbolic cosine is one to one from __form2
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onto __form3 (and from __form4 onto __form3); the inverse function we use
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here is defined on __form3 with range __form2.
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The inverse of the hyperbolic tangent is called the Argument hyperbolic tangent,
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and can be computed as [$../../libs/math/special_functions/graphics/special_functions_blurb15.jpeg].
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The inverse of the hyperbolic sine is called the Argument hyperbolic sine,
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and can be computed (for __form5) as [$../../libs/math/special_functions/graphics/special_functions_blurb17.jpeg].
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The inverse of the hyperbolic cosine is called the Argument hyperbolic cosine,
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and can be computed as [$../../libs/math/special_functions/graphics/special_functions_blurb18.jpeg].
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[endsect]
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[section Sinus Cardinal and Hyperbolic Sinus Cardinal Functions]
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The Sinus Cardinal family of functions (indexed by the family of indices [^a > 0])
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is defined by
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[$../../libs/math/special_functions/graphics/special_functions_blurb20.jpeg];
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it sees heavy use in signal processing tasks.
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By analogy, the Hyperbolic Sinus Cardinal family of functions
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(also indexed by the family of indices [^a > 0]) is defined by
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[$../../libs/math/special_functions/graphics/special_functions_blurb22.jpeg].
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These two families of functions are composed of entire functions.
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[: ['[*Sinus Cardinal of index pi (purple) and Hyperbolic Sinus Cardinal of index pi (red) on R]]]
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[: [$../../libs/math/special_functions/graphics/sinc_pi_and_sinhc_pi_on_R.png]]
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[endsect]
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[section The Quaternionic Exponential]
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Please refer to the following PDF's:
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*[@../../libs/math/quaternion/TQE.pdf The Quaternionic Exponential (and beyond)]
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*[@../../libs/math/quaternion/TQE_EA.pdf The Quaternionic Exponential (and beyond) ERRATA & ADDENDA]
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[endsect]
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[endsect]
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