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1 ;; -*- Mode: Lisp; Package: Maxima; Syntax: Common-Lisp; Base: 10 -*- ;;;;
2 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
3 ;;; The data in this file contains enhancements. ;;;;;
4 ;;; ;;;;;
5 ;;; Copyright (c) 1984,1987 by William Schelter,University of Texas ;;;;;
6 ;;; All rights reserved ;;;;;
7 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
8 ;;; (c) copyright 1982 massachusetts institute of technology ;;;
9 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
15 ;;; this is the definite integration package.
16 ;; defint does definite integration by trying to find an
17 ;;appropriate method for the integral in question. the first thing that
18 ;;is looked at is the endpoints of the problem.
19 ;;
20 ;; i(grand,var,a,b) will be used for integrate(grand,var,a,b)
22 ;; References are to "Evaluation of Definite Integrals by Symbolic
23 ;; Manipulation", by Paul S. Wang,
24 ;; (http://www.lcs.mit.edu/publications/pubs/pdf/MIT-LCS-TR-092.pdf;
25 ;; a better copy might be: https://maxima.sourceforge.io/misc/Paul_Wang_dissertation.pdf)
26 ;;
27 ;; nointegrate is a macsyma level flag which inhibits indefinite
28 ;;integration.
29 ;; abconv is a macsyma level flag which inhibits the absolute
30 ;;convergence test.
31 ;;
32 ;; $defint is the top level function that takes the user input
33 ;;and does minor changes to make the integrand ready for the package.
34 ;;
35 ;; next comes defint, which is the function that does the
36 ;;integration. it is often called recursively from the bowels of the
37 ;;package. defint does some of the easy cases and dispatches to:
38 ;;
39 ;; dintegrate. this program first sees if the limits of
40 ;;integration are 0,inf or minf,inf. if so it sends the problem to
41 ;;ztoinf or mtoinf, respectively.
42 ;; else, dintegrate tries:
43 ;;
44 ;; intsc1 - does integrals of sin's or cos's or exp(%i var)'s
45 ;; when the interval is 0,2 %pi or 0,%pi.
46 ;; method is conversion to rational function and find
47 ;; residues in the unit circle. [wang, pp 107-109]
48 ;;
49 ;; ratfnt - does rational functions over finite interval by
50 ;; doing polynomial part directly, and converting
51 ;; the rational part to an integral on 0,inf and finding
52 ;; the answer by residues.
53 ;;
54 ;; zto1 - i(x^(k-1)*(1-x)^(l-1),x,0,1) = beta(k,l) or
55 ;; i(log(x)*x^(x-1)*(1-x)^(l-1),x,0,1) = psi...
56 ;; [wang, pp 116,117]
57 ;;
58 ;; dintrad- i(x^m/(a*x^2+b*x+c)^(n+3/2),x,0,inf) [wang, p 74]
59 ;;
60 ;; dintlog- i(log(g(x))*f(x),x,0,inf) = 0 (by symmetry) or
61 ;; tries an integration by parts. (only routine to
62 ;; try integration by parts) [wang, pp 93-95]
63 ;;
64 ;; dintexp- i(f(exp(k*x)),x,a,inf) = i(f(x+1)/(x+1),x,0,inf)
65 ;; or i(f(x)/x,x,0,inf)/k. First case hold for a=0;
66 ;; the second for a=minf. [wang 96-97]
67 ;;
68 ;;dintegrate also tries indefinite integration based on certain
69 ;;predicates (such as abconv) and tries breaking up the integrand
70 ;;over a sum or tries a change of variable.
71 ;;
72 ;; ztoinf is the routine for doing integrals over the range 0,inf.
73 ;; it goes over a series of routines and sees if any will work:
74 ;;
75 ;; scaxn - sc(b*x^n) (sc stands for sin or cos) [wang, pp 81-83]
76 ;;
77 ;; ssp - a*sc^n(r*x)/x^m [wang, pp 86,87]
78 ;;
79 ;; zmtorat- rational function. done by multiplication by plog(-x)
80 ;; and finding the residues over the keyhole contour
81 ;; [wang, pp 59-61]
82 ;;
83 ;; log*rat- r(x)*log^n(x) [wang, pp 89-92]
84 ;;
85 ;; logquad0 log(x)/(a*x^2+b*x+c) uses formula
86 ;; i(log(x)/(x^2+2*x*a*cos(t)+a^2),x,0,inf) =
87 ;; t*log(a)/sin(t). a better formula might be
88 ;; i(log(x)/(x+b)/(x+c),x,0,inf) =
89 ;; (log^2(b)-log^2(c))/(2*(b-c))
90 ;;
91 ;; batapp - x^(p-1)/(b*x^n+a)^m uses formula related to the beta
92 ;; function [wang, p 71]
93 ;; there is also a special case when m=1 and a*b<0
94 ;; see [wang, p 65]
95 ;;
96 ;; sinnu - x^-a*n(x)/d(x) [wang, pp 69-70]
97 ;;
98 ;; ggr - x^r*exp(a*x^n+b)
99 ;;
100 ;; dintexp- see dintegrate
101 ;;
102 ;; ztoinf also tries 1/2*mtoinf if the integrand is an even function
103 ;;
104 ;; mtoinf is the routine for doing integrals on minf,inf.
105 ;; it too tries a series of routines and sees if any succeed.
106 ;;
107 ;; scaxn - when the integrand is an even function, see ztoinf
108 ;;
109 ;; mtosc - exp(%i*m*x)*r(x) by residues on either the upper half
110 ;; plane or the lower half plane, depending on whether
111 ;; m is positive or negative.
112 ;;
113 ;; zmtorat- does rational function by finding residues in upper
114 ;; half plane
115 ;;
116 ;; dintexp- see dintegrate
117 ;;
118 ;; rectzto%pi2 - poly(x)*rat(exp(x)) by finding residues in
119 ;; rectangle [wang, pp98-100]
120 ;;
121 ;; ggrm - x^r*exp((x+a)^n+b)
122 ;;
123 ;; mtoinf also tries 2*ztoinf if the integrand is an even function.
132 ;;;rsn* is in comdenom. does a ratsimp of numerator.
133 ;expvar
135 ;impvar
137 context
138 ;;LIMITP T Causes $ASKSIGN to do special things
139 ;;For DEFINT like eliminate epsilon look for prin-inf
140 ;;take realpart and imagpart.
141 integer-info
142 ;;If LIMITP is non-null ask-integer conses
143 ;;its assumptions onto this list.
144 ))
149 "When @code{true}, definite integration tries to find poles in the integrand
152 ;; Currently, if true, $solvetrigwarn is set to true. No additional
153 ;; debugging information is displayed.
158 nil
159 "When NIL, print a message that the principal value of the integral has
163 nil
164 "When non-NIL, a divergent integral will throw to `divergent.
172 ;; Set to true when OSCIP-VAR returns true in DINTEGRATE.
190 ;; Distribute $defint over equations, lists, and matrices.
222 (merror (intl:gettext "defint: variable of integration must be a simple or subscripted variable.~%defint: found ~M") ivar))
251 ;; If antideriv can't do it, returns nil
252 ;; use limit to evaluate every answer returned by antideriv.
256 ;;;Hack the expression up for exponentials.
259 ;; Is this expr a candidate for SININT ?
268 t)))
269 ;;ERF type things go here.
274 t))))))
294 ;; We have some subscripted variable that's not our variable
295 ;; (I think), so it's deg-lessp.
296 ;;
297 ;; FIXME: Is this the best way to handle this? Are there
298 ;; other cases we're mising here?
299 t)))
309 ;; This routine tries to take a limit a couple of ways.
315 ans
318 ivar
319 0
328 ;; Test whether fun2 is inverse of fun1 at val.
334 ;; integration change of variable
342 ())
344 ;; This is a hack! If nv is of the form b*x^n+a, we can
345 ;; solve the equation manually instead of using solve.
346 ;; Why? Because solve asks us for the sign of yx and
347 ;; that's bogus.
349 ;; Solve yx = b*x^n+a, for x. Any root will do. So we
350 ;; have x = ((yx-a)/b)^(1/n).
352 d
358 ))))
375 ;; d: original variable (ivar) as a function of 'yx
376 ;; ind: boolean flag
377 ;; nv: new variable ('yx) as a function of original variable (ivar)
386 ;; converts limits of integration to values for new variable 'yx
389 ;; rewrites exp, the integrand in terms of ivar, the
390 ;; integrand in terms of 'yx, and returns the new
391 ;; integrand.
396 ll1 ul1)
405 ;; wrapper around limit, returns nil if
406 ;; limit not found (nounform returned), or undefined ($und or $ind)
413 ans))))
430 ;; D (not used? -- cwh)
443 ;; Look for integrals which involve log and exp functions.
444 ;; Maxima has a special algorithm to get general results.
455 %tan %sinh %cosh %tanh
456 %log %asin %acos %atan
457 %cot %acot %sec
458 %asec %csc %acsc
460 ;; Call ANTIDERIV with logabs disabled,
461 ;; because the Risch algorithm assumes
462 ;; the integral of 1/x is log(x), not log(abs(x)).
463 ;; Why not just assume logabs = false within RISCHINT itself?
464 ;; Well, there's at least one existing result which requires
465 ;; logabs = true in RISCHINT, so try to make a minimal change here instead.
506 ans)
509 ans)
523 ;;Should be a PROG inside of unwind-protect, but Multics has a compiler
524 ;;bug wrt. and I want to test this code now.
532 ;;;This seems((and (and (eq *ul* '$inf)
533 ;;;fairly losing (setq exp (subin (m+ *ll* ivar) exp))
534 ;;; (setq *ll* 0.))
535 ;;; (ztoinf exp ivar)))
540 ;; ((and (and (equal *ul* 1.)
541 ;; (equal *ll* 0.)) (zto1 exp)))
549 ;;;NOT COMPLETE for sin's and cos's.
571 ;; Call ANTIDERIV with logabs disabled,
572 ;; because the Risch algorithm assumes
573 ;; the integral of 1/x is log(x), not log(abs(x)).
574 ;; Why not just assume logabs = false within RISCHINT itself?
575 ;; Well, there's at least one existing result which requires
576 ;; logabs = true in RISCHINT, so try to make a minimal change here instead.
582 ;;;Recursion stopper
607 ;; adds up integrals of ranges between each pair of poles.
608 ;; checks if whole thing is divergent as limits of integration approach poles.
610 ;;; calling $logcontract causes antiderivative of 1/(1-x^5) to blow up
611 ;; (setq anti-deriv (cond ((involve anti-deriv '(%log))
612 ;; ($logcontract anti-deriv))
613 ;; (t anti-deriv)))
623 ivar))))
626 ;;Return section.
636 ;; I think this takes the pole-list and replaces $MINF with -PRIN-INF
637 ;; and $INF with PRIN-INF. The lower and upper integration limits
638 ;; (ll, ul) can also be modified to be -PRIN-INF and PRIN-INF. These
639 ;; special values are used in TAKE-PRINCIPAL.
657 ;; Assumes EXP is a rational expression with no polynomial part and
658 ;; converts the finite integration to integration over a half-infinite
659 ;; interval. The substitution y = (x-a)/(b-x) is used. Equivalently,
660 ;; x = (b*y+a)/(y+1).
661 ;;
662 ;; (I'm guessing CV means Change Variable here.)
665 ;; FIXME! This is a hack. We apply the transformation with
666 ;; symbolic limits and then substitute the actual limits later.
667 ;; That way method-by-limits (usually?) sees a simpler
668 ;; integrand.
669 ;;
670 ;; See Bugs 938235 and 941457. These fail because $FACTOR is
671 ;; unable to factor the transformed result. This needs more
672 ;; work (in other places).
676 ;; If the limit is a number, use $substitute so we simplify
677 ;; the result. Do we really want to do this?
685 ()))
687 ;; Integrate rational functions over a finite interval by doing the
688 ;; polynomial part directly, and converting the rational part to an
689 ;; integral from 0 to inf. This is evaluated via residues.
692 ;; PQR divides the rational expression and returns the quotient
693 ;; and remainder
700 ;; No polynomial part
703 ans
706 ;; Only polynomial part
709 ;; A non-zero quotient and remainder. Combine the results
710 ;; together.
714 ans))
719 ;; I think this takes a rational expression E, and finds the
720 ;; polynomial part. A cons is returned. The car is the quotient and
721 ;; the cdr is the remainder.
745 ;;;If leadop isn't plus don't do anything.
754 t)
760 t)
772 ())
818 ;; *Z* is a "zero parameter" for this package.
819 ;; Its also used to transform.
820 ;; limit(exp,var,val,dir) -- limit(exp,tvar,0,dir)
827 ;; EPSILON is used in principal value code to denote the familiar
828 ;; mathematical entity.
833 ;;; PRIN-INF Is a special symbol in the principal value code used to
834 ;;; denote an end-point which is proceeding to infinity.
858 ;; We have minf <= ll = ul <= inf
859 )
861 ;; We have minf <= ll < ul = inf
862 )
864 ;; We have minf = ll < ul < inf
865 ;;
866 ;; Now substitute
867 ;;
868 ;; ivar -> -ivar
869 ;; ll -> -ul
870 ;; ul -> inf
871 ;;
872 ;; so that minf < ll < ul = inf
878 ;; We have minf <= ul < ll
879 ;;
880 ;; Now substitute
881 ;;
882 ;; exp -> -exp
883 ;; ll <-> ul
884 ;;
885 ;; so that minf <= ll < ul
888 t))
889 ll ul))
903 ;; Substitute a and b into integral e
904 ;;
905 ;; Looks for discontinuties in integral, and works around them.
906 ;; For example, in
907 ;;
908 ;; integrate(x^(2*n)*exp(-(x)^2),x) ==>
909 ;; -gamma_incomplete((2*n+1)/2,x^2)*x^(2*n+1)*abs(x)^(-2*n-1)/2
910 ;;
911 ;; the integral has a discontinuity at x=0.
912 ;;
919 ivar a b)))))
933 total))))))
935 ;; look for terms with a negative exponent
936 ;;
937 ;; recursively traverses exp in order to find discontinuities such as
938 ;; erfc(1/x-x) at x=0
950 ;; returns list of places where exp might be discontinuous in ivar.
951 ;; list begins with *ll* and ends with *ul*, and include any values between
952 ;; *ll* and *ul*.
953 ;; return '$no or '$unknown if no discontinuities found.
970 ;; (multiplicity (cdar dummy)) ;; not used
975 ;; Carefully substitute the integration limits A and B into the
976 ;; expression E.
986 ;; Try easy substitutions. Return NIL if we can't.
989 ;; Infinite limits aren't easy
990 nil)
994 %gamma_incomplete %expintegral_ei)))
996 ;; It's easy if we have a polynomial. I (rtoy) think
997 ;; it's also easy if the denominator is free of the
998 ;; integration variable and also if the expression
999 ;; doesn't involve inverse functions.
1000 ;;
1001 ;; gamma_incomplete and expintegral_ie
1002 ;; included because of discontinuity in
1003 ;; gamma_incomplete(0, exp(%i*x)) and
1004 ;; expintegral_ei(exp(%i*x))
1005 ;;
1006 ;; XXX: Are there other cases we've forgotten about?
1007 ;;
1008 ;; So just try to substitute the limits into the
1009 ;; expression. If no errors are produced, we're done.
1014 ;; no-err-sub has returned T. An error was catched.
1015 nil)
1029 ;; check for divergent integral
1041 ;;;This function works only on things with ATAN's in them now.
1043 ;; POLES-IN-INTERVAL doesn't know about the poles of tan(x). Call
1044 ;; trigsimp to convert tan into sin/cos, which POLES-IN-INTERVAL
1045 ;; knows how to handle.
1046 ;;
1047 ;; XXX Should we fix POLES-IN-INTERVAL instead?
1048 ;;
1049 ;; XXX Is calling trigsimp too much? Should we just only try to
1050 ;; substitute sin/cos for tan?
1051 ;;
1052 ;; XXX Should the result try to convert sin/cos back into tan? (A
1053 ;; call to trigreduce would do it, among other things.)
1056 ;;POLES -> ((mlist) ((mequal) ((%atan) foo) replacement) ......)
1057 ;;We can then use $SUBSTITUTE
1062 ul-ans)
1064 nil)))
1078 ivar ll ul)))
1095 llist)))))))))))))))
1110 ;; There are no calls where fn1 is ever equal to
1111 ;; 'mtorat. Hence this case is never true. What is
1112 ;; this testing for?
1119 nil)
1138 ;; e is of form poly(x)*exp(m*%i*x)
1139 ;; s is degree of denominator
1140 ;; adds e to *bptu* or *bptd* according to sign of m
1157 ;; Check term is of form poly(x)*exp(m*%i*x)
1158 ;; n is degree of denominator.
1168 ;; Set updn to T if the coefficient of IVAR in the
1169 ;; polynomial is known to be positive. Otherwise set to NIL.
1170 ;; (What does updn really mean?)
1181 term)
1216 ;; exponentialize sin and cos
1222 '$%i
1244 exp)
1260 ;; computes integral from minf to inf for expressions of the form
1261 ;; exp(%i*m*x)*r(x) by residues on either the upper half
1262 ;; plane or the lower half plane, depending on whether
1263 ;; m is positive or negative. [wang p. 77]
1264 ;;
1265 ;; exponentializes sin and cos before applying residue method.
1266 ;; can handle some expressions with poles on real line, such as
1267 ;; sin(x)*cos(x)/x.
1277 ivar
1278 nil)
1281 ;;;Above tests for applicability of this method.
1290 ;; call to $expand can create rational terms
1291 ;; with no exp(m*%i*x)
1295 ;;;Function RIB sets up the values of BPTU and BPTD
1302 ;;;If there is BPTU then CSEMIUP must succeed.
1303 ;;;Likewise for BPTD.
1307 ;; The original integrand was already
1308 ;; determined to have no poles by initial-analysis.
1309 ;; If individual terms of the expansion have poles, the poles
1310 ;; must cancel each other out, so we can ignore them.
1313 ;; if integral of ratterms is divergent, ratans is nil,
1314 ;; and mtosc returns nil
1331 t)
1333 t)
1341 t)
1343 t)
1348 s nc dc
1350 nn-var dn-var)
1371 sd new-sd))
1372 result))
1375 sn nil)
1377 sd nil))
1380 ;;;
1381 ;;;New section.
1399 s
1402 ivar)))
1412 findout
1416 on
1439 s)
1443 ;; It's very important here to bind VAR
1444 ;; because the PLOG simplifier checks
1445 ;; for VAR. Without this, the
1446 ;; simplifier converts plog(-x) to
1447 ;; log(x)+%i*%pi, which messes up the
1448 ;; keyhole routine.
1450 d
1451 ivar)))
1454 ;; Let's not signal an error here. Return nil so that we
1455 ;; eventually return a noun form if no other algorithm gives
1456 ;; a result.
1459 nil)))
1461 ;;(setq *dflag* nil)
1490 nn-var dn-var)
1517 sd new-sd))
1518 result))
1554 ;; Test to see if exp is of the form p(x)*f(exp(x)). If so, set pp to
1555 ;; be p(x) and set pe to f(exp(x)).
1568 s
1571 ivar)))
1578 ;; We have an integral of the form p(x)*F(exp(x)), where
1579 ;; p(x) is a polynomial.
1581 ;; No polynomial
1585 ;; These limits must exist for the integral to converge.
1588 ;; This only handles the case when the F(z) is a
1589 ;; rational function.
1592 ;; If we get here, F(z) is not a rational function.
1593 ;; We transform it using the substitution x=log(y)
1594 ;; which gives us an integral of the form
1595 ;; p(log(y))*F(y)/y, which maxima should be able to
1596 ;; handle.
1599 ;; Give up. We don't know how to handle this.
1601 en
1615 exp)))
1617 ;;; given (b*x+a)^n+c returns (a b n c)
1625 nil)
1627 ;;;get high degree of poly
1629 ;;;diff down to linear.
1631 ;;;all the way to constant.
1634 ;;;get rid of factorial from differentiation.
1637 ;;;Sees if can be expressed as (a*x+b)^n + part freeof x.
1647 s)
1672 n
1673 d)))))
1676 on
1731 e nil)
1745 e))))))
1751 ;; Compute the integral(n/d,x,0,inf) by computing the negative of the
1752 ;; sum of residues of log(-x)*n/d over the poles of n/d inside the
1753 ;; keyhole contour. This contour is basically an disk with a slit
1754 ;; along the positive real axis. n/d must be a rational function.
1759 ;; Ok if not on the positive real axis.
1764 t)
1775 ;; Look at an expression e of the form sin(r*x)^k, where k is an
1776 ;; integer. Return the list (1 r k). (Not sure if the first element
1777 ;; of the list is always 1 because I'm not sure what partition is
1778 ;; trying to do here.)
1792 ;; Look at an expression e of the form sin(r*x) and return r.
1803 ;; integrate(a*sc(r*x)^k/x^n,x,0,inf).
1806 ;; Get the argument of the involved trig function.
1809 ;; I don't think this needs to be special.
1810 #+nil
1812 ;; Replace (1-cos(arg)^2) with sin(arg)^2.
1814 ;(m+t 1. (m- (m^t `((%cos) ,ivar) 2.)))
1815 ;; The code from above generates expressions with
1816 ;; a missing simp flag. Furthermore, the
1817 ;; substitution has to be done for the complete
1818 ;; argument of the trig function. (DK 02/2010)
1821 exp))
1826 ;; n is the power of the denominator.
1828 ;; The simple case.
1834 ;; Do this for a sum of such terms.
1836 c))))))))))
1838 ;; We have an integral of the form sin(r*x)^k/x^n. C is the list (1 r k).
1839 ;;
1840 ;; The substitution y=r*x converts this integral to
1841 ;;
1842 ;; r^(n-1)*integral(sin(y)^k/y^n,y,0,inf)
1843 ;;
1844 ;; (If r is negative, we need to negate the result.)
1845 ;;
1846 ;; The recursion Wang gives on p. 87 has a typo. The second term
1847 ;; should be subtracted from the first. This formula is given in G&R,
1848 ;; 3.82, formula 12.
1849 ;;
1850 ;; integrate(sin(x)^r/x^s,x) =
1851 ;; r*(r-1)/(s-1)/(s-2)*integrate(sin(x)^(r-2)/x^(s-2),x)
1852 ;; - r^2/(s-1)/(s-2)*integrate(sin(x)^r/x^(s-2),x)
1853 ;;
1854 ;; (Limits are assumed to be 0 to inf.)
1855 ;;
1856 ;; This recursion ends up with integrals with s = 1 or 2 and
1857 ;;
1858 ;; integrate(sin(x)^p/x,x,0,inf) = integrate(sin(x)^(p-1),x,0,%pi/2)
1859 ;;
1860 ;; with p > 0 and odd. This latter integral is known to maxima, and
1861 ;; it's value is beta(p/2,1/2)/2.
1862 ;;
1863 ;; integrate(sin(x)^2/x^2,x,0,inf) = %pi/2*binomial(q-3/2,q-1)
1864 ;;
1865 ;; where q >= 2.
1866 ;;
1868 ;; Compute sign(r)*r^(n-1)*integrate(sin(y)^k/y^n,y,0,inf)
1870 c
1876 recursion)
1877 ;; Recursion failed. Return the integrand
1878 ;; The following code generates expressions with a missing simp flag
1879 ;; for the sin function. Use better simplifying code. (DK 02/2010)
1880 ; (let ((integrand (div (pow `((%sin) ,(m* r ivar))
1881 ; k)
1882 ; (pow ivar n))))
1884 k)
1889 ;; integrate(sin(x)^n/x^2,x,0,inf) = pi/2*binomial(n-3/2,n-1).
1890 ;; Express in terms of Gamma functions, though.
1896 ;; integrate(sin(x)^n/x,x,0,inf) = beta((n+1)/2,1/2)/2, for n odd and
1897 ;; n > 0.
1900 half%pi)
1903 ;; This implements the recursion for computing
1904 ;; integrate(sin(y)^l/y^k,y,0,inf). (Note the change in notation from
1905 ;; the above!)
1911 ;; Integral diverges if l-(k-1) < 0.
1914 ;; If l + k is not even, return NIL. (Is this the right
1915 ;; thing to do?)
1916 nil)
1918 ;; We have integrate(sin(y)^l/y^2). Use sevn to evaluate.
1921 ;; We have integrate(sin(y)^l/y,y)
1927 ;; j = k-2, i = (k-1)*(k-2)
1928 ;;
1929 ;;
1930 ;; The main recursion:
1931 ;;
1932 ;; i(sin(y)^l/y^k)
1933 ;; = l*(l-1)/(k-1)/(k-2)*i(sin(y)^(l-2)/y^k)
1934 ;; - l^2/(k-1)/(k-1)*i(sin(y)^l/y^(k-2))
1942 ;; Returns the fractional part of a?
1946 ;; Why do we return 0 if a is a number? Perhaps we really
1947 ;; mean integer?
1950 ;; If we're here, this basically assumes a is a rational.
1951 ;; Compute the remainder and return the result.
1959 ;; Doesn't appear to be used anywhere in Maxima. Not sure what this
1960 ;; was intended to do.
1961 #+nil
1972 ;;; THE FOLLOWING FUNCTION IS FOR TRIG FUNCTIONS OF THE FOLLOWING TYPE:
1973 ;;; LOWER LIMIT=0 B A MULTIPLE OF %PI SCA FUNCTION OF SIN (X) COS (X)
1974 ;;; B<=%PI2
1979 ;; means there was no error
1982 ; returns cons of (integer_part . fractional_part) of a
1984 ;; I think we really want to compute how many full periods are in a
1985 ;; and the remainder.
1990 ; returns cons of (integer_part . fractional_part) of a
1992 ;; I think we really want to compute how many full periods are in a
1993 ;; and the remainder.
2000 ;; Return the integer part of r.
2002 ;; r - fpart(r)
2006 ;;;Try making exp(%i*ivar) --> yy, if result is rational then do integral
2007 ;;;around unit circle. Make corrections for limits of integration if possible.
2027 rat-form
2028 ivar)))
2033 ;;; Do integrals of sin and cos. this routine makes sure lower limit
2034 ;;; is zero.
2036 ;; integrate(e,var,a,b)
2052 ivar e)
2053 cc))
2063 ;; Exit if b or a is not a constant or if the integrand
2064 ;; doesn't appear to have a period of 2 pi.
2067 ;; Multiples of 2*%pi in limits.
2074 ;; The integral is not over a full period (2*%pi) or multiple
2075 ;; of a full period.
2077 ;; Wang p. 111: The integral integrate(f(x),x,a,b) can be
2078 ;; written as:
2079 ;;
2080 ;; n * integrate(f,x,0,2*%pi) + integrate(f,x,0,c)
2081 ;; - integrate(f,x,0,d)
2082 ;;
2083 ;; for some integer n and d >= 0, c < 2*%pi because there exist
2084 ;; integers p and q such that a = 2 * p *%pi + d and b = 2 * q
2085 ;; * %pi + c. Then n = q - p.
2087 ;; Compute q and c for the upper limit b.
2094 ;; Compute p and d for the lower limit a.
2096 ;; avoid an extra trip around the circle - helps skip principal values
2101 ;; Compute -integrate(f,x,0,d)
2106 ;; Compute n = q - p (stored in nzp2)
2108 out
2109 ;; Compute n*integrate(f,x,0,2*%pi)
2111 ;; n = 0
2114 ;; n is not zero, so compute
2115 ;; integrate(f,x,0,2*%pi)
2117 ;; Unable to compute integrate(f,x,0,2*%pi)
2119 ;; Compute integrate(f,x,0,c)
2123 ;; All three pieces succeeded.
2126 ;; Less than 1 full period, so intsc can integrate it.
2127 ;; Apply the substitution to make the lower limit 0.
2128 ;; This is last resort because substitution often causes intsc to fail.
2130 limit-diff ivar))
2131 ;; nothing worked
2134 ;; integrate(sc, var, 0, b), where sc is f(sin(x), cos(x)).
2135 ;; calls intsc with a wrapper to just return nil if integral is divergent,
2136 ;; rather than generating an error.
2141 nil
2142 ans)))
2144 ;; integrate(sc, ivar, 0, b), where sc is f(sin(x), cos(x)). I (rtoy)
2145 ;; think this expects b to be less than 2*%pi.
2148 0
2155 ;; Partition the integrand SC into the factors that do not
2156 ;; contain VAR (the car part) and the parts that do (the cdr
2157 ;; part).
2162 ;; integrate(sc, ivar, 0, b), where sc is f(sin(x), cos(x)).
2164 ;; Determine if sc is a product of sin's and cos's.
2168 ;; We have a product of sin's and cos's. We handle some
2169 ;; special cases here.
2171 ;; Wang p. 110, formula (1):
2172 ;; integrate(sin(x)^m*cos(x)^n, x, 0, %pi/2) =
2173 ;; gamma((m+1)/2)*gamma((n+1)/2)/2/gamma((n+m+2)/2)
2176 ;; Wang p. 110, near the bottom, says
2177 ;;
2178 ;; int(f(sin(x),cos(x)), x, 0, %pi) =
2179 ;; int(f(sin(x),cos(x)) + f(sin(x),-cos(x)),x,0,%pi/2)
2200 ;; Wang, p. 111 says
2201 ;;
2202 ;; int(f(sin(x),cos(x)),x,0,3*%pi/2) =
2203 ;; int(f(sin(x),cos(x)),x,0,%pi)
2204 ;; + int(f(-sin(x),-cos(x)),x,0,%pi/2)
2208 ;; We don't have a product of sin's and cos's.
2213 dn*)
2218 ;;;Is careful about substitution of limits where the denominator may be zero
2219 ;;;because of various assumptions made.
2240 nil
2241 ans)))
2247 nil
2248 ans)))
2250 ;; Determine whether E is of the form sin(x)^m*cos(x)^n and return the
2251 ;; list (m n).
2255 m n)
2274 ;; Says whether (m^t value exponent) has at least one real branch.
2275 ;; Only works for values of 1 and -1 now. Returns $yes $no
2276 ;; $unknown.
2287 ;; Compute beta((m+1)/2,(n+1)/2)/2.
2293 ;;Seems like Guys who call this don't agree on what it should return.
2306 ;; Check e for an expression of the form x^kk*(b*x^n+a)^l. If it
2307 ;; matches, Return the two values kk and (list l a n b).
2312 ;; We have f(x)^y. Look to see if f(x) has the desired
2313 ;; form. Then f(x)^y has the desired form too, with
2314 ;; suitably modified values.
2315 ;;
2316 ;; XXX: Should we ask for the sign of f(x) if y is not an
2317 ;; integer? This transformation we're going to do requires
2318 ;; that f(x)^y be real.
2320 e
2325 ;; Got a match. Adjust kk and cc appropriately.
2327 cc
2342 ;;(DEFUN BATAP (E)
2343 ;; (PROG (K C L)
2344 ;; (COND ((NOT (BATA0 E)) (RETURN NIL))
2345 ;; ((AND (EQUAL -1. (CADDDR C))
2346 ;; (EQ ($askSIGN (SETQ K (m+ 1. K)))
2347 ;; '$pos)
2348 ;; (EQ ($askSIGN (SETQ L (m+ 1. (CAR C))))
2349 ;; '$pos)
2350 ;; (ALIKE1 (CADR C)
2351 ;; (m^ UL (CADDR C)))
2352 ;; (SETQ E (CADR C))
2353 ;; (EQ ($askSIGN (SETQ C (CADDR C))) '$pos))
2354 ;; (RETURN (M// (m* (m^ UL (m+t K (m* C (m+t -1. L))))
2355 ;; `(($BETA) ,(SETQ NN* (M// K C))
2356 ;; ,(SETQ DN* L)))
2357 ;; C))))))
2360 ;; Integrals of the form i(log(x)^m*x^k*(a+b*x^n)^l,x,0,ul). There
2361 ;; are two cases to consider: One case has ul>0, b<0, a=-b*ul^n, k>-1,
2362 ;; l>-1, n>0, m a nonnegative integer. The second case has ul=inf, l < 0.
2363 ;;
2364 ;; These integrals are essentially partial derivatives of the Beta
2365 ;; function (i.e. the Eulerian integral of the first kind). Note
2366 ;; that, currently, with the default setting intanalysis:true, this
2367 ;; function might not even be called for some of these integrals.
2368 ;; However, this can be palliated by setting intanalysis:false.
2387 ;; We have i(x^kk/(d+cc*x^r)^s,x,0,inf) =
2388 ;; beta(aa,bb)/(cc^aa*d^bb*r). Compute this, and then
2389 ;; differentiate it m times to get the log term
2390 ;; incorporated.
2395 0
2396 m)))
2400 r))))
2409 ;; The result looks like B(k/n,l) ( ... ).
2410 ;; Perhaps, we should just $factor, instead of
2411 ;; pulling out beta like this.
2413 beta
2414 ($fullratsimp
2422 ivar m)
2424 beta))))))))))))
2427 ;;; If e is of the form given below, make the obvious change
2428 ;;; of variables (substituting ul*x^(1/n) for x) in order to reduce
2429 ;;; e to the usual form of the integrand in the Eulerian
2430 ;;; integral of the first kind.
2431 ;;; N. B: The old version of ZTO1 completely ignored this
2432 ;;; substitution; the log(x)s were just thrown in, which,
2433 ;;; of course would give wrong results.
2436 ;; Parse e
2440 ;; e=x^k*(a+b*x^n)^l
2442 c
2454 ;; Wang p. 71 gives the following formula for a beta function:
2455 ;;
2456 ;; integrate(x^(k-1)/(c*x^r+d)^s,x,0,inf)
2457 ;; = beta(a,b)/(c^a*d^b*r)
2458 ;;
2459 ;; where a = k/r > 0, b = s - a > 0, s > k > 0, r > 0, c*d > 0.
2460 ;;
2461 ;; This function matches this and returns k-1, d, r, c, a, b. And
2462 ;; also checks that all the conditions hold. If not, NIL is returned.
2463 ;;
2469 c
2486 ;; Handles beta integrals.
2494 nil)
2497 c
2498 ;; e = x^k*(d+c*x^al)^l.
2503 new-k)
2511 ;; Compute exp(d)*gamma((c+1)/b)/b/a^((c+1)/b). In essence, this is
2512 ;; the value of integrate(x^c*exp(d-a*x^b),x,0,inf).
2523 ;; Evaluates the contour integral of GRAND around the unit circle
2524 ;; using residues.
2531 ;; Is pt stricly inside the unit circle?
2537 ;; Is pt on the unit circle? (Use
2538 ;; the cached value computed
2539 ;; above.)
2560 ;; Wang 81-83. Unfortunately, the pdf version has page 82 as all
2561 ;; black, so here is, as best as I can tell, what Wang is doing.
2562 ;; Fortunately, p. 81 has the necessary hints.
2563 ;;
2564 ;; First consider integrate(exp(%i*k*x^n),x) around the closed contour
2565 ;; consisting of the real axis from 0 to R, the arc from the angle 0
2566 ;; to %pi/(2*n) and the ray from the arc back to the origin.
2567 ;;
2568 ;; There are no poles in this region, so the integral must be zero.
2569 ;; But consider the integral on the three parts. The real axis is the
2570 ;; integral we want. The return ray is
2571 ;;
2572 ;; exp(%i*%pi/2/n) * integrate(exp(%i*k*(t*exp(%i*%pi/2/n))^n),t,R,0)
2573 ;; = exp(%i*%pi/2/n) * integrate(exp(%i*k*t^n*exp(%i*%pi/2)),t,R,0)
2574 ;; = -exp(%i*%pi/2/n) * integrate(exp(-k*t^n),t,0,R)
2575 ;;
2576 ;; As R -> infinity, this last integral is gamma(1/n)/k^(1/n)/n.
2577 ;;
2578 ;; We assume the integral on the circular arc approaches 0 as R ->
2579 ;; infinity. (Need to prove this.)
2580 ;;
2581 ;; Thus, we have
2582 ;;
2583 ;; integrate(exp(%i*k*t^n),t,0,inf)
2584 ;; = exp(%i*%pi/2/n) * gamma(1/n)/k^(1/n)/n.
2585 ;;
2586 ;; Equating real and imaginary parts gives us the desired results:
2587 ;;
2588 ;; integrate(cos(k*t^n),t,0,inf) = G * cos(%pi/2/n)
2589 ;; integrate(sin(k*t^n),t,0,inf) = G * sin(%pi/2/n)
2590 ;;
2591 ;; where G = gamma(1/n)/k^(1/n)/n.
2592 ;;
2600 ;; Ok, we have cos(b*x^n) or sin(b*x^n), and we set e = (n
2601 ;; b)
2603 ;; n = 1. Give up. (Why not divergent?)
2604 nil)
2609 ;; s is the sign of b. Give up if it's zero.
2610 nil)
2612 ;; Give up if n-1 <= 0. (Why give up? Isn't the
2613 ;; integral divergent?)
2614 nil)
2616 ;; We can apply our formula now. g = gamma(1/n)/n/b^(1/n)
2623 e)))))))
2626 ;; this is the second part of the definite integral package
2631 ())
2633 ivar t)
2655 nil)
2657 t)))))
2667 n
2670 plm))))
2682 plm
2683 factors
2684 pl
2685 rl
2686 pl1
2687 rl1))))
2695 ans)
2697 ;; AFAICT, this call to PLOG doesn't need
2698 ;; to bind VAR.
2700 ans))))
2707 ;; AFAICT, this call to PLOG doesn't need
2708 ;; to bind VAR. An example where this is
2709 ;; used is
2710 ;; integrate(log(x)^2/(1+x^2),x,0,1) =
2711 ;; %pi^3/16.
2714 n)
2715 d
2716 plm)))))
2720 0
2723 ;; Handle integral(n(x)/d(x)*log(x)^m,x,0,inf). n and d are
2724 ;; polynomials.
2742 ans))))
2766 n
2770 ;;Makes a new poly such that np(x)-np(x+2*%i*%pi)=p(x).
2771 ;;Constructs general POLY of degree one higher than P with
2772 ;;arbitrary coeff. and then solves for coeffs by equating like powers
2773 ;;of the varibale of integration.
2774 ;;Can probably be made simpler now.
2782 ;;;Now, poly(x)-poly(x+2*%i*%pi)=p(x), P is the original poly.
2791 -1
2800 ;; Integrate(p(x)*f(exp(x))/g(exp(x)),x,minf,inf) by applying the
2801 ;; transformation y = exp(x) to get
2802 ;; integrate(p(log(y))*f(y)/g(y)/y,y,0,inf). This should be handled
2803 ;; by dintlog.
2810 ;; This implements Wang's algorithm in Chapter 5.2, pp. 98-100.
2811 ;;
2812 ;; This is a very brief description of the algorithm. Basically, we
2813 ;; have integrate(R(exp(x))*p(x),x,minf,inf), where R(x) is a rational
2814 ;; function and p(x) is a polynomial.
2815 ;;
2816 ;; We find a polynomial q(x) such that q(x) - q(x+2*%i*%pi) = p(x).
2817 ;; Then consider a contour integral of R(exp(z))*q(z) over a
2818 ;; rectangular contour. Opposite corners of the rectangle are (-R,
2819 ;; 2*%i*%pi) and (R, 0).
2820 ;;
2821 ;; Wang shows that this contour integral, in the limit, is the
2822 ;; integral of R(exp(x))*q(x)-R(exp(x))*q(x+2*%i*%pi), which is
2823 ;; exactly the integral we're looking for.
2824 ;;
2825 ;; Thus, to find the value of the contour integral, we just need the
2826 ;; residues of R(exp(z))*q(z). The only tricky part is that we want
2827 ;; the log function to have an imaginary part between 0 and 2*%pi
2828 ;; instead of -%pi to %pi.
2830 ;; We have R(exp(x))*p(x) represented as p(x)*pe(exp(x))/d(exp(x)).
2836 ;; At this point denom-exponential has converted d(exp(x)) to the
2837 ;; polynomial d(z), where z = exp(x).
2840 pe))
2842 ;; Find the poles of the denominator. denom-exponential is the
2843 ;; denominator of R(x).
2844 ;;
2845 ;; It seems as if polelist returns a list of several items.
2846 ;; The first element is a list consisting of the pole and (z -
2847 ;; pole). We don't care about this, so we take the rest of the
2848 ;; result.
2851 ;; The imaginary part is nonzero,
2852 ;; or the realpart is negative.
2856 ;; The realpart is not zero.
2858 ;; Not sure what this does.
2860 ;; No roots at all, so return
2864 ;; We have simple roots or roots in REGION1
2867 ;; The cadr of pl is the list of the simple poles of
2868 ;; denom-exponential. Take the log of them to find the
2869 ;; poles of the original expression. Then compute the
2870 ;; residues at each of these poles and sum them up and put
2871 ;; the result in B. (If no simple poles set B to 0.)
2877 ;; I think this handles the case of poles outside the
2878 ;; regions. The sum of these residues are placed in C.
2882 temp)))
2887 ;; We have the repeated poles of deonom-exponential, so we
2888 ;; need to convert them to the actual pole values for
2889 ;; R(exp(x)), by taking the log of the value of poles.
2894 ;; Compute the residues at all of these poles and sum
2895 ;; them up.
2898 poles))
2913 ;; Check to see if each term in exp that is of the form exp(k*x) has
2914 ;; an integer value for k.
2930 ;; Test to see if exp is of the form f(exp(x)), and if so, replace
2931 ;; exp(x) with 'z*. That is, basically return f(z*).
2934 exp)
2938 ivar))
2941 ;; Does e really need to be special here? Seems to be ok without
2942 ;; it; testsuite works.
2943 #+nil
2946 e)
2977 ;; I think this is looking at f(exp(x)) and tries to find some
2978 ;; rational function R and some number k such that f(exp(x)) =
2979 ;; R(exp(k*x)).
2996 ;; Integration by parts.
2997 ;;
2998 ;; integrate(u(x)*diff(v(x),x),x,a,b)
2999 ;; |b
3000 ;; = u(x)*v(x)| - integrate(v(x)*diff(u(x),x))
3001 ;; |a
3002 ;;
3004 ;;;SINCE ONLY CALLED FROM DINTLOG TO get RID OF LOGS - IF LOG REMAINS, QUIT
3008 nil)
3015 nil)
3022 ;; integrate(f(exp(k*x)),x,a,b), where f(z) is rational.
3023 ;;
3024 ;; See Wang p. 96-97.
3025 ;;
3026 ;; If the limits are minf to inf, we use the substitution y=exp(k*x)
3027 ;; to get integrate(f(y)/y,y,0,inf)/k. If the limits are 0 to inf,
3028 ;; use the substitution s+1=exp(k*x) to get
3029 ;; integrate(f(s+1)/(s+1),s,0,inf).
3035 ;; If we can integrate it directly, do so and take the
3036 ;; appropriate limits.
3037 )
3039 ;; ans is the list (f(x) exp(k*x)).
3042 ;; Use the substitution s + 1 = exp(k*x). The
3043 ;; integral becomes integrate(f(s+1)/(s+1),s,0,inf)
3046 ;; Use the substitution y=exp(k*x) because the
3047 ;; limits are minf to inf.
3049 ;; Apply the substitution and integrate it.
3052 ;; integrate(log(g(x))*f(x),x,0,inf)
3060 exp
3061 ivar))
3063 ;; Make the substitution y=1/x. If the integrand has
3064 ;; exactly the same form, the answer has to be 0.
3070 ;; It's easy if we have the antiderivative.
3071 ;; but intsubs sometimes gives results containing %limit
3073 ;; Ok, the easy cases didn't work. We now try integration by
3074 ;; parts. Set ANS to f(x).
3077 ;; Bad. f(x) contains a log term, so we give up.
3082 ivar ll ul)))
3083 ;; The arg of the log function is the same as the
3084 ;; integration variable. We can do something a little
3085 ;; simpler than integration by parts. We have something
3086 ;; like f(x)*log(x). Consider f(x)*x^z. If we
3087 ;; differentiate this wrt to z, the integrand becomes
3088 ;; f(x)*log(x)*x^z. When we evaluate this at z = 0, we
3089 ;; get the desired integrand.
3090 ;;
3091 ;; So we need f(0) to be 0 at 0. If we can integrate
3092 ;; f(x)*x^z, then we differentiate the result and
3093 ;; evaluate it at z = 0.
3096 ;; Try integration by parts.
3099 ;; Compute diff(e,ivar,n) at the point pt.
3103 ;;; GGR and friends
3105 ;; MAYBPC returns (COEF EXPO CONST)
3106 ;;
3107 ;; This basically picks off b*x^n+a and returns the list
3108 ;; (b n a).
3115 ;; At this point, e is of the form (a n b)
3120 ;; If we're here, b is complex, and n > 0. zn = imagpart(b).
3121 ;;
3122 ;; Set ivar to the same sign as zn.
3127 ;; zd = exp(ivar*%i*%pi*(1+nd)/(2*n). (ZD is special!)
3130 ;; Return zn, n, a, zd.
3136 ;; We're here if realpart(b) <= 0, and n >= 0. Then return -b, n, a.
3139 ;; Integrate x^m*exp(b*x^n+a), with realpart(m) > -1.
3140 ;;
3141 ;; See Wang, pp. 84-85.
3142 ;;
3143 ;; I believe the formula Wang gives is incorrect. The derivation is
3144 ;; correct except for the last step.
3145 ;;
3146 ;; Let J = integrate(x^m*exp(%i*k*x^n),x,0,inf), with real k.
3147 ;;
3148 ;; Consider the case for k < 0. Take a sector of a circle bounded by
3149 ;; the real line and the angle -%pi/(2*n), and by the radii, r and R.
3150 ;; Since there are no poles inside this contour, the integral
3151 ;;
3152 ;; integrate(z^m*exp(%i*k*z^n),z) = 0
3153 ;;
3154 ;; Then J = exp(-%pi*%i*(m+1)/(2*n))*integrate(R^m*exp(k*R^n),R,0,inf)
3155 ;;
3156 ;; because the integral along the boundary is zero except for the part
3157 ;; on the real axis. (Proof?)
3158 ;;
3159 ;; Wang seems to say this last integral is gamma(s/n/(-k)^s) where s =
3160 ;; (m+1)/n. But that seems wrong. If we use the substitution R =
3161 ;; (y/(-k))^(1/n), we end up with the result:
3162 ;;
3163 ;; integrate(y^((m+1)/n-1)*exp(-y),y,0,inf)/(n*k^((m+1)/n).
3164 ;;
3165 ;; or gamma((m+1)/n)/k^((m+1)/n)/n.
3166 ;;
3167 ;; Note that this also handles the case of
3168 ;;
3169 ;; integrate(x^m*exp(-k*x^n),x,0,inf);
3170 ;;
3171 ;; where k is positive real number. A simple change of variables,
3172 ;; y=k*x^n, gives
3173 ;;
3174 ;; integrate(y^((m+1)/n-1)*exp(-y),y,0,inf)/(n*k^((m+1)/n))
3175 ;;
3176 ;; which is the same form above.
3195 ;; e = (m b n a). That is, the integral is of the form
3196 ;; x^m*exp(b*x^n+a). I think we want to compute
3197 ;; gamma((m+1)/n)/b^((m+1)/n)/n.
3198 ;;
3199 ;; FIXME: If n > m + 1, the integral converges. We need
3200 ;; to check for this.
3202 e
3205 ;; The derivation only holds if b is purely real or
3206 ;; purely imaginary. Give up if it's not.
3208 ;; Check for convergence. If b is complex, we need n -
3209 ;; m > 1. If b is real, we need b < 0.
3218 ;; If we're here, b must be negative.
3221 ;; Complex b. Take the imaginary part
3223 n a))
3225 ;; FIXME: Why do we set %emode here? Shouldn't we just
3226 ;; bind it? And why do we want it bound to T anyway?
3227 ;; Shouldn't the user control that? The same goes for
3228 ;; dosimp.
3229 ;;(setq $%emode t)
3235 ;; Match x^m*exp(b*x^n+a). If it does, return (list m b n a).
3241 ;; We're looking at something like exp(f(ivar)). See if it's
3242 ;; of the form b*x^n+a, and return (list 0 b n a). (The 0 is
3243 ;; so we can graft something onto it if needed.)
3248 ;; E should be the product of exactly 2 terms
3250 ;; Check to see if one of the terms is of the form
3251 ;; ivar^p. If so, make sure the realpart of p > -1. If
3252 ;; so, check the other term has the right form via
3253 ;; another call to ggr1.
3264 ;; Both terms have the right form and nn* contains the ivar of
3265 ;; the exponential term. Put dn* as the car of nn*. The
3266 ;; result is something like (m b n a) when we have the
3267 ;; expression x^m*exp(b*x^n+a).
3271 ;; Match b*x^n+a. If a match is found, return the list (a n b).
3272 ;; Otherwise, return NIL
3285 ;; Match b*x^n. Return the list (n b) if found or NIL if not.
3295 ;; nn* should be the value of var. This is only called by bx**n with
3296 ;; the second arg of var.
3307 ;;; given (b*x^n+a)^m returns (m a n b)
3314 nil)
3321 ;;;Catches Unfactored forms.
3324 ivar)
3337 ;;;Is E = VAR raised to some power? If so return power or 0.
3343 ;;;Is E = VAR1 raised to some power? If so return power.
3364 ans)))))
3392 ;; Note: This doesn't seem be called from anywhere.
3417 ivar)
3445 ;;;Looks at the IMAGINARY part of a log and puts it in the interval 0 2*%pi.
3448 ;; We take the $rectform above to make sure that the log is
3449 ;; expanded out for the situations where simplifying plog itself
3450 ;; doesn't do it. This should probably be considered a bug in the
3451 ;; plog simplifier and should be fixed there.
3464 ;;; Temporary fix for a lacking in taylor, which loses with %i in denom.
3465 ;;; Besides doesn't seem like a bad thing to do in general.
3469 ;; Multiply the denominator by it's conjugate to get rid of
3470 ;; %i.
3475 ;; If the new denominator still contains %i, just give up.
3477 exp
3478 new-exp)))
3481 ;;; LL and UL must be real otherwise this routine return $UNKNOWN.
3482 ;;; Returns $no $unknown or a list of poles in the interval (ll ul)
3483 ;;; for exp w.r.t. ivar.
3484 ;;; Form of list ((pole . multiplicity) (pole1 . multiplicity) ....)
3501 ;; this clause handles cases where we can't find the exact roots,
3502 ;; but we know that they occur outside the interval of integration.
3503 ;; example: integrate ((1+exp(t))/sqrt(t+exp(t)), t, 0, 1);
3519 ;; (multiplicity (cdar dummy)) (not used? -- cwh)
3529 pole-list))))))))))))
3532 ;;;Returns $YES if there is no pole and $NO if there is one.
3545 ;;;Takes care of forms that the ratio test fails on.
3561 (%einvolve-var
3564 ivar))
3598 ;; real values for ll and ul; place can be imaginary.
3605 '$no
3610 ;; returns true or nil
3612 ;; real values for ll and ul; place can be imaginary.
3628 ;; If *failures is set, we may have missed some roots.
3629 ;; We still return the roots that we have found.
3634 rootlist)
3641 rootlist)))))))))
3644 ;;; Is x > y. X or Y can be $MINF or $INF, zeroA or zeroB.
3684 nil)
3687 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
3688 ;;;
3689 ;;; Integrate Definite Integrals involving log and exp functions. The algorithm
3690 ;;; are taken from the paper "Evaluation of CLasses of Definite Integrals ..."
3691 ;;; by K.O.Geddes et. al.
3692 ;;;
3693 ;;; 1. CASE: Integrals generated by the Gamma function.
3694 ;;;
3695 ;;; inf
3696 ;;; /
3697 ;;; [ w m s - m - 1
3698 ;;; I t log (t) expt(- t ) dt = s signum(s)
3699 ;;; ]
3700 ;;; /
3701 ;;; 0
3702 ;;; !
3703 ;;; m !
3704 ;;; d !
3705 ;;; (--- (gamma(z))! )
3706 ;;; m !
3707 ;;; dz ! w + 1
3708 ;;; !z = -----
3709 ;;; s
3710 ;;;
3711 ;;; The integral converges for:
3712 ;;; s # 0, m = 0, 1, 2, ... and realpart((w+1)/s) > 0.
3713 ;;;
3714 ;;; 2. CASE: Integrals generated by the Incomplete Gamma function.
3715 ;;;
3716 ;;; inf !
3717 ;;; / m !
3718 ;;; [ w m s d s !
3719 ;;; I t log (t) exp(- t ) dt = (--- (gamma_incomplete(a, x ))! )
3720 ;;; ] m !
3721 ;;; / da ! w + 1
3722 ;;; x !z = -----
3723 ;;; s
3724 ;;; - m - 1
3725 ;;; s signum(s)
3726 ;;;
3727 ;;; The integral converges for:
3728 ;;; s # 0, m = 0, 1, 2, ... and realpart((w+1)/s) > 0.
3729 ;;; The shown solution is valid for s>0. For s<0 gamma_incomplete has to be
3730 ;;; replaced by gamma(a) - gamma_incomplete(a,x^s).
3731 ;;;
3732 ;;; 3. CASE: Integrals generated by the beta function.
3733 ;;;
3734 ;;; 1
3735 ;;; /
3736 ;;; [ m s r n
3737 ;;; I log (1 - t) (1 - t) t log (t) dt =
3738 ;;; ]
3739 ;;; /
3740 ;;; 0
3741 ;;; !
3742 ;;; ! !
3743 ;;; n m ! !
3744 ;;; d d ! !
3745 ;;; --- (--- (beta(z, w))! )!
3746 ;;; n m ! !
3747 ;;; dz dw ! !
3748 ;;; !w = s + 1 !
3749 ;;; !z = r + 1
3750 ;;;
3751 ;;; The integral converges for:
3752 ;;; n, m = 0, 1, 2, ..., s > -1 and r > -1.
3753 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
3757 ;;; Recognize c*z^w*log(z)^m*exp(-t^s)
3769 ;;; Recognize c*z^r*log(z)^n*(1-z)^s*log(1-z)^m
3782 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
3789 ;; var1 is used as a parameter for differentiation. Add var1>0 to the
3790 ;; database, to get the desired simplification of the differentiation of
3791 ;; the gamma_incomplete function.
3799 ;; The integrand matches the cases 1 and 2.
3818 ;; Case 1: Generated by the Gamma function.
3825 var1
3830 ; derivate the involved hypergeometric
3831 ; functions.
3836 ;; Case 2: Generated by the Incomplete Gamma function.
3852 ;; Case 3: Generated by the Beta function.
3878 var2 m)
3880 var1 n)
3886 ;; Simplify result and set $gamma_expand to global value
3889 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;