home *** CD-ROM | disk | FTP | other *** search
/ NetNews Usenet Archive 1992 #26 / NN_1992_26.iso / spool / sci / math / symbolic / 2926 < prev    next >
Encoding:
Text File  |  1992-11-09  |  6.7 KB  |  168 lines
  1. Newsgroups: sci.math.symbolic
  2. Path: sparky!uunet!utcsri!torn!watserv2.uwaterloo.ca!watdragon.uwaterloo.ca!daisy.uwaterloo.ca!gjfee
  3. From: gjfee@daisy.uwaterloo.ca (Greg Fee)
  4. Subject: definite integration
  5. Message-ID: <BxGIzL.386@watdragon.uwaterloo.ca>
  6. Sender: news@watdragon.uwaterloo.ca (USENET News System)
  7. Organization: University of Waterloo
  8. Date: Mon, 9 Nov 1992 16:26:08 GMT
  9. Lines: 157
  10.  
  11. Re: the definite integral
  12.  
  13. integrate(ln(1 - b*x/(a+c*x^2))/x, x=0..infinity)
  14.  
  15. Here is another derivation of the closed form based on differentiation
  16. with respect to a parameter.
  17.  
  18.     |\^/|      MAPLE V
  19. ._|\|   |/|_.  Copyright (c) 1981-1990 by the University of Waterloo.
  20.  \  MAPLE  /   All rights reserved.  MAPLE is a registered trademark of
  21.  <____ ____>   Waterloo Maple Software.
  22.       |        Type ? for help.
  23. > f := ln(1 - b*x/(a+c*x^2))/x;
  24.                                             b x
  25.                                   ln(1 - --------)
  26.                                                 2
  27.                                          a + c x
  28.                              f := ----------------
  29.                                           x
  30.  
  31. > M := a^(1/2)/c^(1/2);
  32.                                          1/2
  33.                                         a
  34.                                    M := ----
  35.                                          1/2
  36.                                         c
  37.  
  38. # Invert integrand about M to transform [M,infinity] to [0,M] .
  39. > f1 := -subs(x=M^2/x,f)*diff(M^2/x,x);
  40.                                              b a
  41.                                 ln(1 - --------------)
  42.                                            /      2 \
  43.                                            |     a  |
  44.                                        c x |a + ----|
  45.                                            |       2|
  46.                                            \    c x /
  47.                           f1 := ----------------------
  48.                                            x
  49.  
  50. > g := normal(f+f1);
  51.                                               2
  52.                                      - a - c x  + b x
  53.                                 ln(- ----------------)
  54.                                                 2
  55.                                          a + c x
  56.                          g := 2 ----------------------
  57.                                            x
  58.  
  59. # New upper limit of integration is M, but transform it to 1
  60. > g := normal(subs(x=M*x,g)*diff(M*x,x));
  61.                                    1/2      2  1/2      1/2
  62.                               - a c    - a x  c    + b a    x
  63.                          ln(- -------------------------------)
  64.                                        1/2         2
  65.                                       c    a (1 + x )
  66.                   g := 2 -------------------------------------
  67.                                            x
  68.  
  69. > g1 := op(1,x/2*g);
  70.                                  1/2      2  1/2      1/2
  71.                             - a c    - a x  c    + b a    x
  72.                     g1 := - -------------------------------
  73.                                      1/2         2
  74.                                     c    a (1 + x )
  75.  
  76. > g1 := convert(g1,parfrac,x);
  77.                                             b x
  78.                           g1 := 1 - ------------------
  79.                                      1/2  1/2       2
  80.                                     a    c    (1 + x )
  81.  
  82. # Let  b/a^(1/2)/c^(1/2) = 2*sin(k)
  83. > ss := b=2*sin(k)*a^(1/2)*c^(1/2);
  84.                                               1/2  1/2
  85.                           ss := b = 2 sin(k) a    c
  86.  
  87. > h1 := subs(ss,g1);
  88.                                           sin(k) x
  89.                               h1 := 1 - 2 --------
  90.                                                 2
  91.                                            1 + x
  92.  
  93. > h := 2/x*ln(h1);
  94.                                            sin(k) x
  95.                                   ln(1 - 2 --------)
  96.                                                  2
  97.                                             1 + x
  98.                            h := 2 ------------------
  99.                                            x
  100.  
  101. > hk := normal(diff(h,k));
  102.                                           cos(k)
  103.                          hk := - 4 -------------------
  104.                                         2
  105.                                    1 + x  - 2 sin(k) x
  106.  
  107. > hk := -4*cos(k)/(cos(k)^2+sin(k)^2+x^2-2*sin(k)*x);
  108.                                           cos(k)
  109.                  hk := - 4 -----------------------------------
  110.                                  2         2    2
  111.                            cos(k)  + sin(k)  + x  - 2 sin(k) x
  112.  
  113. # integrate with resect to x between 0 and 1
  114. > ih := int(hk,x);
  115.                                            2 x - 2 sin(k)
  116.                       ih := - 4 arctan(1/2 --------------)
  117.                                                cos(k)
  118.  
  119. > nih := normal(ih);
  120.                                            x - sin(k)
  121.                          nih := - 4 arctan(----------)
  122.                                              cos(k)
  123.  
  124. > ihx := eval(subs(x=1,nih)-subs(x=0,nih));
  125.                                   1 - sin(k)             sin(k)
  126.                 ihx := - 4 arctan(----------) - 4 arctan(------)
  127.                                     cos(k)               cos(k)
  128.  
  129. > a1 := eval(subs(sin(k)/cos(k)=tan(k),ihx));
  130.                                         1 - sin(k)
  131.                        a1 := - 4 arctan(----------) - 4 k
  132.                                           cos(k)
  133.  
  134. > a2 := subs(arctan((1-sin(k))/cos(k))=Pi/4-k/2,a1); 
  135.                                 a2 := - Pi - 2 k
  136.  
  137. # Now integrate k back
  138. > a3 := int(a2,k);
  139.                                                2
  140.                                a3 := - Pi k - k
  141.  
  142. # Find the missing constant
  143. > int(eval(subs(k=0,h)),x=0..1);
  144.                                        0
  145.  
  146. > subs(k=0,a3);
  147.                                        0
  148.  
  149. # Missing constant in a3 is 0 so a3 is correct
  150. # Put back original parameters
  151. > sk := solve(ss,{k});
  152.                                                  b
  153.                        sk := {k = arcsin(1/2 ---------)}
  154.                                               1/2  1/2
  155.                                              a    c
  156.  
  157. > answer := subs(sk,a3);
  158.                                        b                       b     2
  159.          answer := - Pi arcsin(1/2 ---------) - arcsin(1/2 ---------)
  160.                                     1/2  1/2                1/2  1/2
  161.                                    a    c                  a    c
  162.  
  163. > answer := factor(answer);
  164.                                     b      /                    b     \
  165.          answer := - arcsin(1/2 ---------) |Pi + arcsin(1/2 ---------)|
  166.                                  1/2  1/2  |                 1/2  1/2 |
  167.                                 a    c     \                a    c    /
  168.