Re: [Axiom-developer] Re: About Schaums.

[Waldek Hebisch]

I wrote:
> Martin Rubey wrote:
> > root <address@hidden> writes:
> > >   Axiom has a closed form for 2 integrals where Schaums has series.
> > 
> > But at least one of them seems to be wrong. Since it seems that my message 
> > was
> > overlooked, I repeat it here:
> > 
> > address@hidden writes:
> > 
> > >   14:668 SCHAUMS AND AXIOM DIFFER (Axiom has closed form)
> > 
> > But I'm not so sure that it is correct, at least not for a=1 and x in 0..1.
> > 
> > draw(D(integrate(asech(x)/x,x),x)-asech(x)/x, x=0..1)
> > 
> 
> Integral of 'asech(x)/x' is definitely wrong.  I belive that the
> error is in 'lfextendedint':
> 
>     lfextendedint(f, x, g) ==
>       empty?(l := varselect(kernels f, x)) => [x::F * f, 0]
>       symbolIfCan(k := kmax(l))
>         case SE =>
>           map(multivariate(#1, k), extendedint(univariate(f, k),
>                                                univariate(g, k)))
>       ...
> 
> AFAICS we need stronger condition than 'case SE' (and extra code
> to handle remaining cases).
>  

For FriCAS I applied the following fix:

Index: src/algebra/intef.spad.pamphlet
===================================================================
--- src/algebra/intef.spad.pamphlet     (wersja 274)
+++ src/algebra/intef.spad.pamphlet     (kopia robocza)
@@ -232,8 +232,12 @@
       empty?(l := varselect(kernels f, x)) => [x::F * f, 0]
       symbolIfCan(k := kmax(l))
         case SE =>
+          g1 :=
+               empty?(l1 := varselect(kernels g, x)) => 0::F
+               kmax(l1) = k => g
+               0::F
           map(multivariate(#1, k), extendedint(univariate(f, k),
-                                               univariate(g, k)))
+                                               univariate(g1, k)))
       is?(k, "exp"::SE) => expextint(f, x, k, g)
       prim?(k, x)       => primextint(f, x, k, g)
       has?(operator k, ALGOP) => alglfextint(f, k, l, x, g)
Index: src/input/bugs2008.input.pamphlet
===================================================================
--- src/input/bugs2008.input.pamphlet   (wersja 274)
+++ src/input/bugs2008.input.pamphlet   (kopia robocza)
@@ -71,6 +71,15 @@
 testEquals("gcd([67108859*67108859*x^2-1, 67108859*67108859*x^2+2*67108859*x 
+1])", "67108859*x+1")
 testEquals("gcd([0, 0, x^2-1, x^2+2*x+1])$HeuGcd(SparseUnivariatePolynomial 
Integer)", "(x+1)::(SparseUnivariatePolynomial Integer)")
 
+testcase "Integral of asech(x)/x"
+i1 := integrate(asech(x)/x, x)
+isym := "integral"::Symbol
+testTrue "is?(operator(kernels(i1).1), isym)"
+i2 := integrate(log((sqrt(1-x^2)+1)/x)/x, x)
+testTrue "is?(operator(kernels(i2).1), isym)"
+
+testEquals("integrate(log(sqrt(1-x^2)/x)/(x^2), x)", "(-log(sqrt(1-x^2)/x) 
-x*log(x+1)/2 +  x*log(x-1)/2 + 1)/x")
+
 )set output algebra on
 statistics()
 @
Index: ChangeLog.wh
===================================================================
--- ChangeLog.wh        (wersja 278)
+++ ChangeLog.wh        (kopia robocza)
@@ -1,3 +1,9 @@
+2008-05-08  Waldek Hebisch  <address@hidden>
+
+       * src/algebra/intef.spad.pamphlet: lfextendedint(): 
+       when integrating rational functions use only logarithms
+       of rational functions.
+
 2008-05-07  Waldek Hebisch  <address@hidden>
 
        * src/etc/Makefile.in: stamp-databases: Avoid error


-- 
                              Waldek Hebisch
address@hidden