1 | |
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2 | [def __form1 [^\]-1;1\[]] |
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3 | [def __form2 [^\[0;+'''∞'''\[]] |
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4 | [def __form3 [^\[+1;+'''∞'''\[]] |
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5 | [def __form4 [^\]-'''∞''';0\]]] |
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6 | [def __form5 [^x '''≥''' 0]] |
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7 | |
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8 | |
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9 | [section Background Information and White Papers] |
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10 | |
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11 | [section The Inverse Hyperbolic Functions] |
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12 | |
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13 | The exponential funtion is defined, for all object for which this makes sense, |
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14 | as the power series |
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15 | [$../../libs/math/special_functions/graphics/special_functions_blurb1.jpeg], |
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16 | with ['[^n! = 1x2x3x4x5...xn]] (and ['[^0! = 1]] by definition) being the factorial of ['[^n]]. |
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17 | In particular, the exponential function is well defined for real numbers, |
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18 | complex number, quaternions, octonions, and matrices of complex numbers, |
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19 | among others. |
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20 | |
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21 | [: ['[*Graph of exp on R]] ] |
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22 | |
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23 | [: [$../../libs/math/special_functions/graphics/exp_on_R.png] ] |
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24 | |
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25 | [: ['[*Real and Imaginary parts of exp on C]]] |
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26 | [: [$../../libs/math/special_functions/graphics/Im_exp_on_C.png]] |
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27 | |
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28 | The hyperbolic functions are defined as power series which |
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29 | can be computed (for reals, complex, quaternions and octonions) as: |
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30 | |
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31 | Hyperbolic cosine: [$../../libs/math/special_functions/graphics/special_functions_blurb5.jpeg] |
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32 | |
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33 | Hyperbolic sine: [$../../libs/math/special_functions/graphics/special_functions_blurb6.jpeg] |
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34 | |
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35 | Hyperbolic tangent: [$../../libs/math/special_functions/graphics/special_functions_blurb7.jpeg] |
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36 | |
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37 | [: ['[*Trigonometric functions on R (cos: purple; sin: red; tan: blue)]]] |
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38 | [: [$../../libs/math/special_functions/graphics/trigonometric.png]] |
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39 | |
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40 | [: ['[*Hyperbolic functions on r (cosh: purple; sinh: red; tanh: blue)]]] |
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41 | [: [$../../libs/math/special_functions/graphics/hyperbolic.png]] |
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42 | |
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43 | The hyperbolic sine is one to one on the set of real numbers, |
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44 | with range the full set of reals, while the hyperbolic tangent is |
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45 | also one to one on the set of real numbers but with range __form1, and |
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46 | therefore both have inverses. The hyperbolic cosine is one to one from __form2 |
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47 | onto __form3 (and from __form4 onto __form3); the inverse function we use |
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48 | here is defined on __form3 with range __form2. |
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49 | |
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50 | The inverse of the hyperbolic tangent is called the Argument hyperbolic tangent, |
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51 | and can be computed as [$../../libs/math/special_functions/graphics/special_functions_blurb15.jpeg]. |
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52 | |
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53 | The inverse of the hyperbolic sine is called the Argument hyperbolic sine, |
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54 | and can be computed (for __form5) as [$../../libs/math/special_functions/graphics/special_functions_blurb17.jpeg]. |
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55 | |
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56 | The inverse of the hyperbolic cosine is called the Argument hyperbolic cosine, |
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57 | and can be computed as [$../../libs/math/special_functions/graphics/special_functions_blurb18.jpeg]. |
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58 | |
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59 | [endsect] |
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60 | |
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61 | [section Sinus Cardinal and Hyperbolic Sinus Cardinal Functions] |
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62 | |
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63 | The Sinus Cardinal family of functions (indexed by the family of indices [^a > 0]) |
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64 | is defined by |
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65 | [$../../libs/math/special_functions/graphics/special_functions_blurb20.jpeg]; |
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66 | it sees heavy use in signal processing tasks. |
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67 | |
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68 | By analogy, the Hyperbolic Sinus Cardinal family of functions |
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69 | (also indexed by the family of indices [^a > 0]) is defined by |
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70 | [$../../libs/math/special_functions/graphics/special_functions_blurb22.jpeg]. |
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71 | |
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72 | These two families of functions are composed of entire functions. |
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73 | |
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74 | [: ['[*Sinus Cardinal of index pi (purple) and Hyperbolic Sinus Cardinal of index pi (red) on R]]] |
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75 | [: [$../../libs/math/special_functions/graphics/sinc_pi_and_sinhc_pi_on_R.png]] |
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76 | |
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77 | [endsect] |
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78 | |
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79 | [section The Quaternionic Exponential] |
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80 | |
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81 | Please refer to the following PDF's: |
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82 | |
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83 | *[@../../libs/math/quaternion/TQE.pdf The Quaternionic Exponential (and beyond)] |
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84 | *[@../../libs/math/quaternion/TQE_EA.pdf The Quaternionic Exponential (and beyond) ERRATA & ADDENDA] |
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85 | |
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86 | [endsect] |
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87 | |
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88 | [endsect] |
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89 | |
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