1 | ;;;; -*- Mode: lisp -*- |
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2 | ;;;; |
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3 | ;;;; Copyright (c) 2011 Raymond Toy |
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4 | ;;;; Permission is hereby granted, free of charge, to any person |
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5 | ;;;; obtaining a copy of this software and associated documentation |
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6 | ;;;; files (the "Software"), to deal in the Software without |
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7 | ;;;; restriction, including without limitation the rights to use, |
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8 | ;;;; copy, modify, merge, publish, distribute, sublicense, and/or sell |
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9 | ;;;; copies of the Software, and to permit persons to whom the |
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10 | ;;;; Software is furnished to do so, subject to the following |
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11 | ;;;; conditions: |
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12 | ;;;; |
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13 | ;;;; The above copyright notice and this permission notice shall be |
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14 | ;;;; included in all copies or substantial portions of the Software. |
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15 | ;;;; |
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16 | ;;;; THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, |
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17 | ;;;; EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES |
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18 | ;;;; OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND |
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19 | ;;;; NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT |
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20 | ;;;; HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, |
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21 | ;;;; WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING |
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22 | ;;;; FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR |
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23 | ;;;; OTHER DEALINGS IN THE SOFTWARE. |
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24 | |
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25 | (in-package #:oct) |
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26 | |
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27 | ;;; References: |
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28 | ;;; |
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29 | ;;; [1] Borwein, Borwein, Crandall, "Effective Laguerre Asymptotics", |
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30 | ;;; http://people.reed.edu/~crandall/papers/Laguerre-f.pdf |
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31 | ;;; |
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32 | ;;; [2] Borwein, Borwein, Chan, "The Evaluation of Bessel Functions |
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33 | ;;; via Exp-Arc Integrals", http://web.cs.dal.ca/~jborwein/bessel.pdf |
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34 | ;;; |
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35 | |
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36 | (defvar *debug-exparc* nil) |
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37 | |
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38 | ;; B[k](p) = 1/2^(k+3/2)*integrate(exp(-p*u)*u^(k-1/2),u,0,1) |
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39 | ;; = 1/2^(k+3/2)/p^(k+1/2)*integrate(t^(k-1/2)*exp(-t),t,0,p) |
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40 | ;; = 1/2^(k+3/2)/p^(k+1/2) * g(k+1/2, p) |
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41 | ;; |
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42 | ;; where G(a,z) is the lower incomplete gamma function. |
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43 | ;; |
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44 | ;; There is the continued fraction expansion for G(a,z) (see |
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45 | ;; cf-incomplete-gamma in qd-gamma.lisp): |
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46 | ;; |
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47 | ;; G(a,z) = z^a*exp(-z)/ CF |
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48 | ;; |
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49 | ;; So |
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50 | ;; |
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51 | ;; B[k](p) = 1/2^(k+3/2)/p^(k+1/2)*p^(k+1/2)*exp(-p)/CF |
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52 | ;; = exp(-p)/2^(k+3/2)/CF |
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53 | ;; |
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54 | ;; |
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55 | ;; Note also that [2] gives a recurrence relationship for B[k](p) in |
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56 | ;; eq (2.6), but there is an error there. The correct relationship is |
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57 | ;; |
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58 | ;; B[k](p) = -exp(-p)/(p*sqrt(2)*2^(k+1)) + (k-1/2)*B[k-1](p)/(2*p) |
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59 | ;; |
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60 | ;; The paper is missing the division by p in the term containing |
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61 | ;; B[k-1](p). This is easily derived from the recurrence relationship |
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62 | ;; for the (lower) incomplete gamma function. |
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63 | ;; |
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64 | ;; Note too that as k increases, the recurrence appears to be unstable |
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65 | ;; and B[k](p) begins to increase even though it is strictly bounded. |
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66 | ;; (This is also easy to see from the integral.) Hence, we do not use |
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67 | ;; the recursion. However, it might be stable for use with |
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68 | ;; double-float precision; this has not been tested. |
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69 | ;; |
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70 | (defun bk (k p) |
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71 | (/ (exp (- p)) |
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72 | (* (sqrt (float 2 (realpart p))) (ash 1 (+ k 1))) |
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73 | (let ((a (float (+ k 1/2) (realpart p)))) |
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74 | (lentz #'(lambda (n) |
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75 | (+ n a)) |
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76 | #'(lambda (n) |
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77 | (if (evenp n) |
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78 | (* (ash n -1) p) |
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79 | (- (* (+ a (ash n -1)) p)))))))) |
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80 | |
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81 | ;; exp-arc I function, as given in the Laguerre paper |
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82 | ;; |
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83 | ;; I(p, q) = 4*exp(p) * sum(g[k](-2*%i*q)/(2*k)!*B[k](p), k, 0, inf) |
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84 | ;; |
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85 | ;; where g[k](p) = product(p^2+(2*j-1)^2, j, 1, k) and B[k](p) as above. |
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86 | ;; |
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87 | ;; For computation, note that g[k](p) = g[k-1](p) * (p^2 + (2*k-1)^2) |
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88 | ;; and (2*k)! = (2*k-2)! * (2*k-1) * (2*k). Then, let |
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89 | ;; |
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90 | ;; R[k](p) = g[k](p)/(2*k)! |
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91 | ;; |
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92 | ;; Then |
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93 | ;; |
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94 | ;; R[k](p) = g[k](p)/(2*k)! |
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95 | ;; = g[k-1](p)/(2*k-2)! * (p^2 + (2*k-1)^2)/((2*k-1)*(2*k) |
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96 | ;; = R[k-1](p) * (p^2 + (2*k-1)^2)/((2*k-1)*(2*k) |
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97 | ;; |
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98 | ;; In the exp-arc paper, the function is defined (equivalently) as |
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99 | ;; |
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100 | ;; I(p, q) = 2*%i*exp(p)/q * sum(r[2*k+1](-2*%i*q)/(2*k)!*B[k](p), k, 0, inf) |
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101 | ;; |
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102 | ;; where r[2*k+1](p) = p*product(p^2 + (2*j-1)^2, j, 1, k) |
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103 | ;; |
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104 | ;; Let's note some properties of I(p, q). |
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105 | ;; |
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106 | ;; I(-%i*z, v) = 2*%i*exp(-%i*z)/q * sum(r[2*k+1](-2*%i*v)/(2*k)!*B[k](-%i*z)) |
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107 | ;; |
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108 | ;; Note thate B[k](-%i*z) = 1/2^(k+3/2)*integrate(exp(%i*z*u)*u^(k-1/2),u,0,1) |
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109 | ;; = conj(B[k](%i*z). |
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110 | ;; |
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111 | ;; Hence I(-%i*z, v) = conj(I(%i*z, v)) when both z and v are real. |
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112 | (defun exp-arc-i (p q) |
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113 | (let* ((sqrt2 (sqrt (float 2 (realpart p)))) |
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114 | (exp/p/sqrt2 (/ (exp (- p)) p sqrt2)) |
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115 | (v (* #c(0 -2) q)) |
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116 | (v2 (expt v 2)) |
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117 | (eps (epsilon (realpart p)))) |
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118 | (when *debug-exparc* |
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119 | (format t "sqrt2 = ~S~%" sqrt2) |
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120 | (format t "exp/p/sqrt2 = ~S~%" exp/p/sqrt2)) |
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121 | (do* ((k 0 (1+ k)) |
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122 | (bk (/ (incomplete-gamma 1/2 p) |
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123 | 2 sqrt2 (sqrt p)) |
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124 | (- (/ (* bk (- k 1/2)) 2 p) |
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125 | (/ exp/p/sqrt2 (ash 1 (+ k 1))))) |
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126 | ;; ratio[k] = r[2*k+1](v)/(2*k)!. |
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127 | ;; r[1] = v and r[2*k+1](v) = r[2*k-1](v)*(v^2 + (2*k-1)^2) |
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128 | ;; ratio[0] = v |
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129 | ;; and ratio[k] = r[2*k-1](v)*(v^2+(2*k-1)^2) / ((2*k-2)! * (2*k-1) * 2*k) |
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130 | ;; = ratio[k]*(v^2+(2*k-1)^2)/((2*k-1) * 2 * k) |
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131 | (ratio v |
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132 | (* ratio (/ (+ v2 (expt (1- (* 2 k)) 2)) |
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133 | (* 2 k (1- (* 2 k)))))) |
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134 | (term (* ratio bk) |
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135 | (* ratio bk)) |
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136 | (sum term (+ sum term))) |
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137 | ((< (abs term) (* (abs sum) eps)) |
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138 | (* sum #c(0 2) (/ (exp p) q))) |
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139 | (when *debug-exparc* |
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140 | (format t "k = ~D~%" k) |
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141 | (format t " bk = ~S~%" bk) |
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142 | (format t " ratio = ~S~%" ratio) |
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143 | (format t " term = ~S~%" term) |
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144 | (format t " sum - ~S~%" sum))))) |
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145 | |
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146 | (defun exp-arc-i-2 (p q) |
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147 | (let* ((sqrt2 (sqrt (float 2 (realpart p)))) |
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148 | (exp/p/sqrt2 (/ (exp (- p)) p sqrt2)) |
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149 | (v (* #c(0 -2) q)) |
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150 | (v2 (expt v 2)) |
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151 | (eps (epsilon (realpart p)))) |
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152 | (when *debug-exparc* |
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153 | (format t "sqrt2 = ~S~%" sqrt2) |
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154 | (format t "exp/p/sqrt2 = ~S~%" exp/p/sqrt2)) |
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155 | (do* ((k 0 (1+ k)) |
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156 | (bk (bk 0 p) |
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157 | (bk k p)) |
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158 | (ratio v |
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159 | (* ratio (/ (+ v2 (expt (1- (* 2 k)) 2)) |
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160 | (* 2 k (1- (* 2 k)))))) |
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161 | (term (* ratio bk) |
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162 | (* ratio bk)) |
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163 | (sum term (+ sum term))) |
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164 | ((< (abs term) (* (abs sum) eps)) |
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165 | (* sum #c(0 2) (/ (exp p) q))) |
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166 | (when *debug-exparc* |
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167 | (format t "k = ~D~%" k) |
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168 | (format t " bk = ~S~%" bk) |
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169 | (format t " ratio = ~S~%" ratio) |
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170 | (format t " term = ~S~%" term) |
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171 | (format t " sum - ~S~%" sum))))) |
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172 | |
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173 | |
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174 | ;; This currently only works for v an integer. |
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175 | ;; |
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176 | (defun integer-bessel-j-exp-arc (v z) |
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177 | (let* ((iz (* #c(0 1) z)) |
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178 | (i+ (exp-arc-i-2 iz v)) |
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179 | (i- (exp-arc-i-2 (- iz ) v))) |
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180 | (/ (+ (* (cis (* v (float-pi i+) -1/2)) |
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181 | i+) |
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182 | (* (cis (* v (float-pi i+) 1/2)) |
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183 | i-)) |
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184 | (float-pi i+) |
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185 | 2))) |
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186 | |
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187 | ;; alpha[n](z) = integrate(exp(-z*s)*s^n, s, 0, 1/2) |
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188 | ;; beta[n](z) = integrate(exp(-z*s)*s^n, s, -1/2, 1/2) |
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189 | ;; |
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190 | ;; The recurrence in [2] is |
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191 | ;; |
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192 | ;; alpha[n](z) = - exp(-z/2)/2^n/z + n/z*alpha[n-1](z) |
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193 | ;; beta[n]z) = ((-1)^n*exp(z/2)-exp(-z/2))/2^n/z + n/z*beta[n-1](z) |
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194 | ;; |
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195 | ;; We also note that |
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196 | ;; |
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197 | ;; alpha[n](z) = G(n+1,z/2)/z^(n+1) |
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198 | ;; beta[n](z) = G(n+1,z/2)/z^(n+1) - G(n+1,-z/2)/z^(n+1) |
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199 | |
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200 | (defun alpha (n z) |
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201 | (let ((n (float n (realpart z)))) |
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202 | (/ (cf-incomplete-gamma (1+ n) (/ z 2)) |
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203 | (expt z (1+ n))))) |
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204 | |
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205 | (defun beta (n z) |
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206 | (let ((n (float n (realpart z)))) |
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207 | (/ (- (cf-incomplete-gamma (1+ n) (/ z 2)) |
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208 | (cf-incomplete-gamma (1+ n) (/ z -2))) |
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209 | (expt z (1+ n))))) |
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210 | |
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211 | ;; a[0](k,v) := (k+sqrt(k^2+1))^(-v); |
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212 | ;; a[1](k,v) := -v*a[0](k,v)/sqrt(k^2+1); |
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213 | ;; a[n](k,v) := 1/(k^2+1)/(n-1)/n*((v^2-(n-2)^2)*a[n-2](k,v)-k*(n-1)*(2*n-3)*a[n-1](k,v)); |
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214 | |
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215 | ;; Convert this to iteration instead of using this quick-and-dirty |
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216 | ;; memoization? |
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217 | (let ((hash (make-hash-table :test 'equal))) |
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218 | (defun an-clrhash () |
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219 | (clrhash hash)) |
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220 | (defun an-dump-hash () |
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221 | (maphash #'(lambda (k v) |
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222 | (format t "~S -> ~S~%" k v)) |
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223 | hash)) |
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224 | (defun an (n k v) |
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225 | (or (gethash (list n k v) hash) |
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226 | (let ((result |
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227 | (cond ((= n 0) |
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228 | (expt (+ k (sqrt (float (1+ (* k k)) (realpart v)))) (- v))) |
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229 | ((= n 1) |
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230 | (- (/ (* v (an 0 k v)) |
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231 | (sqrt (float (1+ (* k k)) (realpart v)))))) |
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232 | (t |
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233 | (/ (- (* (- (* v v) (expt (- n 2) 2)) (an (- n 2) k v)) |
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234 | (* k (- n 1) (+ n n -3) (an (- n 1) k v))) |
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235 | (+ 1 (* k k)) |
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236 | (- n 1) |
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237 | n))))) |
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238 | (setf (gethash (list n k v) hash) result) |
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239 | result)))) |
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240 | |
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241 | ;; SUM-AN computes the series |
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242 | ;; |
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243 | ;; sum(exp(-k*z)*a[n](k,v), k, 1, N) |
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244 | ;; |
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245 | (defun sum-an (big-n n v z) |
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246 | (let ((sum 0)) |
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247 | (loop for k from 1 upto big-n |
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248 | do |
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249 | (incf sum (* (exp (- (* k z))) |
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250 | (an n k v)))) |
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251 | sum)) |
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252 | |
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253 | ;; SUM-AB computes the series |
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254 | ;; |
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255 | ;; sum(alpha[n](z)*a[n](0,v) + beta[n](z)*sum_an(N, n, v, z), n, 0, inf) |
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256 | (defun sum-ab (big-n v z) |
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257 | (let ((eps (epsilon (realpart z)))) |
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258 | (an-clrhash) |
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259 | (do* ((n 0 (+ 1 n)) |
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260 | (term (+ (* (alpha n z) (an n 0 v)) |
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261 | (* (beta n z) (sum-an big-n n v z))) |
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262 | (+ (* (alpha n z) (an n 0 v)) |
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263 | (* (beta n z) (sum-an big-n n v z)))) |
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264 | (sum term (+ sum term))) |
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265 | ((<= (abs term) (* eps (abs sum))) |
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266 | sum) |
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267 | (when nil |
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268 | (format t "n = ~D~%" n) |
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269 | (format t " term = ~S~%" term) |
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270 | (format t " sum = ~S~%" sum))))) |
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271 | |
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272 | ;; Convert to iteration instead of this quick-and-dirty memoization? |
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273 | (let ((hash (make-hash-table :test 'equal))) |
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274 | (defun %big-a-clrhash () |
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275 | (clrhash hash)) |
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276 | (defun %big-a-dump-hash () |
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277 | (maphash #'(lambda (k v) |
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278 | (format t "~S -> ~S~%" k v)) |
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279 | hash)) |
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280 | (defun %big-a (n v) |
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281 | (or (gethash (list n v) hash) |
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282 | (let ((result |
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283 | (cond ((zerop n) |
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284 | (expt 2 (- v))) |
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285 | (t |
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286 | (* (%big-a (- n 1) v) |
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287 | (/ (* (+ v n n -2) (+ v n n -1)) |
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288 | (* 4 n (+ n v)))))))) |
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289 | (setf (gethash (list n v) hash) result) |
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290 | result)))) |
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291 | |
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292 | ;; Computes A[n](v) = |
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293 | ;; (-1)^n*v*2^(-v)*pochhammer(v+n+1,n-1)/(2^(2*n)*n!) If v is a |
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294 | ;; negative integer -m, use A[n](-m) = (-1)^(m+1)*A[n-m](m) for n >= |
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295 | ;; m. |
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296 | (defun big-a (n v) |
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297 | (let ((m (ftruncate v))) |
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298 | (cond ((and (= m v) (minusp m)) |
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299 | (if (< n m) |
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300 | (%big-a n v) |
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301 | (let ((result (%big-a (+ n m) v))) |
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302 | (if (oddp (truncate m)) |
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303 | result |
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304 | (- result))))) |
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305 | (t |
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306 | (%big-a n v))))) |
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307 | |
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308 | ;; I[n](t, z, v) = exp(-t*z)/t^(2*n+v-1) * |
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309 | ;; integrate(exp(-t*z*s)*(1+s)^(-2*n-v), s, 0, inf) |
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310 | ;; |
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311 | ;; Use the substitution u=1+s to get a new integral |
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312 | ;; |
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313 | ;; integrate(exp(-t*z*s)*(1+s)^(-2*n-v), s, 0, inf) |
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314 | ;; = exp(t*z) * integrate(u^(-v-2*n)*exp(-t*u*z), u, 1, inf) |
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315 | ;; = exp(t*z)*t^(v+2*n-1)*z^(v+2*n-1)*incomplete_gamma_tail(1-v-2*n,t*z) |
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316 | ;; |
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317 | ;; The continued fraction for incomplete_gamma_tail(a,z) is |
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318 | ;; |
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319 | ;; z^a*exp(-z)/CF |
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320 | ;; |
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321 | ;; So incomplete_gamma_tail(1-v-2*n, t*z) is |
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322 | ;; |
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323 | ;; (t*z)^(1-v-2*n)/CF |
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324 | ;; |
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325 | ;; which finally gives |
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326 | ;; |
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327 | ;; integrate(exp(-t*z*s)*(1+s)^(-2*n-v), s, 0, inf) |
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328 | ;; = (t*z)^(1-v-2*n)/CF |
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329 | ;; |
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330 | ;; and I[n](t, z, v) = exp(-t*z)/CF |
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331 | (defun big-i (n theta z v) |
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332 | (/ (exp (- (* theta z))) |
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333 | (let* ((a (- 1 v n n)) |
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334 | (z-a (- (* theta z) a))) |
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335 | (lentz #'(lambda (n) |
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336 | (+ n n 1 z-a)) |
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337 | #'(lambda (n) |
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338 | (* n (- a n))))))) |
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339 | |
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340 | (defun sum-big-ia (big-n v z) |
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341 | ) |
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342 | |
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343 | (defun bessel-j (v z) |
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344 | (let ((vv (ftruncate v))) |
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345 | (cond ((= vv v) |
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346 | ;; v is an integer |
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347 | (integer-bessel-j-exp-arc v z)) |
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348 | (t |
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349 | (let ((big-n 100) |
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350 | (vpi (* v (float-pi (realpart z))))) |
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351 | (+ (integer-bessel-j-exp-arc v z) |
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352 | (* z |
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353 | (/ (sin vpi) vpi) |
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354 | (+ (/ -1 z) |
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355 | (sum-ab big-n v z))))))))) |
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356 | |
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357 | (defun paris-series (v z n) |
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358 | (labels ((pochhammer (a k) |
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359 | (/ (gamma (+ a k)) |
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360 | (gamma a))) |
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361 | (a (v k) |
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362 | (* (/ (pochhammer (+ 1/2 v) k) |
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363 | (gamma (float (1+ k) z))) |
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364 | (pochhammer (- 1/2 v) k)))) |
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365 | (* (loop for k from 0 below n |
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366 | sum (* (/ (a v k) |
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367 | (expt (* 2 z) k)) |
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368 | (/ (cf-incomplete-gamma (+ k v 1/2) (* 2 z)) |
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369 | (gamma (+ k v 1/2))))) |
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370 | (/ (exp z) |
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371 | (sqrt (* 2 (float-pi z) z)))))) |
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372 | |
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