- Primitive heuristics to make determinants of sparse matrices work.

[ginac.git] / doc / tutorial / ginac.texi

diff --git a/doc/tutorial/ginac.texi b/doc/tutorial/ginac.texi
index 1d060689837613c748f78bc52102d9423c5d36f2..ae19fbaaaf9915903156a7c507cd4e32ee4a68cc 100644 (file)
--- a/doc/tutorial/ginac.texi
+++ b/doc/tutorial/ginac.texi
@@ -376,26 +376,27 @@ polynomials):
 @end example
 
 You can differentiate functions and expand them as Taylor or Laurent
-series (the third argument of @code{series} is the evaluation point, the
-fourth defines the order):
+series in a very natural syntax (the second argument of @code{series} is
+a relation defining the evaluation point, the third specifies the
+order):
 
 @cindex Zeta function
 @example
 > diff(tan(x),x);
 tan(x)^2+1
-> series(sin(x),x,0,4);
+> series(sin(x),x==0,4);
 x-1/6*x^3+Order(x^4)
-> series(1/tan(x),x,0,4);
+> series(1/tan(x),x==0,4);
 x^(-1)-1/3*x+Order(x^2)
-> series(gamma(x),x,0,3);
-x^(-1)-EulerGamma+(1/12*Pi^2+1/2*EulerGamma^2)*x
-+(-1/3*zeta(3)-1/12*Pi^2*EulerGamma-1/6*EulerGamma^3)*x^2+Order(x^3)
+> series(Gamma(x),x==0,3);
+x^(-1)-gamma+(1/12*Pi^2+1/2*gamma^2)*x+
+(-1/3*zeta(3)-1/12*Pi^2*gamma-1/6*gamma^3)*x^2+Order(x^3)
 > evalf(");
 x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
 -(0.90747907608088628905)*x^2+Order(x^3)
-> series(gamma(2*sin(x)-2),x,Pi/2,6);
--(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*EulerGamma^2-1/240)*(x-1/2*Pi)^2
--EulerGamma-1/12+Order((x-1/2*Pi)^3)
+> series(Gamma(2*sin(x)-2),x==Pi/2,6);
+-(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*gamma^2-1/240)*(x-1/2*Pi)^2
+-gamma-1/12+Order((x-1/2*Pi)^3)
 @end example
 
 Here we have made use of the @command{ginsh}-command @code{"} to pop the
@@ -1013,7 +1014,7 @@ following table.
 
 @cindex @code{Pi}
 @cindex @code{Catalan}
-@cindex @code{EulerGamma}
+@cindex @code{gamma}
 @cindex @code{evalf()}
 Constants behave pretty much like symbols except that they return some
 specific number when the method @code{.evalf()} is called.
@@ -1029,7 +1030,7 @@ The predefined known constants are:
 @item @code{Catalan}
 @tab Catalan's constant
 @tab 0.91596559417721901505460351493238411
-@item @code{EulerGamma}
+@item @code{gamma}
 @tab Euler's (or Euler-Mascheroni) constant
 @tab 0.57721566490153286060651209008240243
 @end multitable
@@ -1146,11 +1147,11 @@ int main()
     symbol x("x"), y("y");
     
     ex foo = x+y/2;
-    cout << "gamma(" << foo << ") -> " << gamma(foo) << endl;
+    cout << "Gamma(" << foo << ") -> " << Gamma(foo) << endl;
     ex bar = foo.subs(y==1);
-    cout << "gamma(" << bar << ") -> " << gamma(bar) << endl;
+    cout << "Gamma(" << bar << ") -> " << Gamma(bar) << endl;
     ex foobar = bar.subs(x==7);
-    cout << "gamma(" << foobar << ") -> " << gamma(foobar) << endl;
+    cout << "Gamma(" << foobar << ") -> " << Gamma(foobar) << endl;
     // ...
 @}
 @end example
@@ -1159,9 +1160,9 @@ This program shows how the function returns itself twice and finally an
 expression that may be really useful:
 
 @example
-gamma(x+(1/2)*y) -> gamma(x+(1/2)*y)
-gamma(x+1/2) -> gamma(x+1/2)
-gamma(15/2) -> (135135/128)*Pi^(1/2)
+Gamma(x+(1/2)*y) -> Gamma(x+(1/2)*y)
+Gamma(x+1/2) -> Gamma(x+1/2)
+Gamma(15/2) -> (135135/128)*Pi^(1/2)
 @end example
 
 @cindex branch cut
@@ -1192,7 +1193,8 @@ They are created by simply using the C++ operators @code{==}, @code{!=},
 the @code{.subs()} method show how objects of class relational are used
 as arguments.  There they provide an intuitive syntax for substitutions.
 They can also used for creating systems of equations that are to be
-solved for unknown variables.
+solved for unknown variables.  More applications of this class will
+appear throughout the next chapters.
 
 
 @node Archiving, Important Algorithms, Relations, Basic Concepts
@@ -1604,13 +1606,13 @@ int main()
     symbol v("v"), c("c");
     
     ex gamma = 1/sqrt(1 - pow(v/c,2));
-    ex mass_nonrel = gamma.series(v, 0, 10);
+    ex mass_nonrel = gamma.series(v==0, 10);
     
     cout << "the relativistic mass increase with v is " << endl
          << mass_nonrel << endl;
     
     cout << "the inverse square of this series is " << endl
-         << pow(mass_nonrel,-2).series(v, 0, 10) << endl;
+         << pow(mass_nonrel,-2).series(v==0, 10) << endl;
     
     // ...
 @}
@@ -1648,7 +1650,7 @@ using namespace GiNaC;
 ex mechain_pi(int degr)
 @{
     symbol x;
-    ex pi_expansion = series_to_poly(atan(x).series(x,0,degr));
+    ex pi_expansion = series_to_poly(atan(x).series(x,degr));
     ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
                    -4*pi_expansion.subs(x==numeric(1,239));
     return pi_approx;
@@ -1666,7 +1668,11 @@ int main()
 @}
 @end example
 
-When you run this program, it will type out:
+Note how we just called @code{.series(x,degr)} instead of
+@code{.series(x==0,degr)}.  This is a simple shortcut for @code{ex}'s
+method @code{series()}: if the first argument is a symbol the expression
+is expanded in that symbol around point @code{0}.  When you run this
+program, it will type out:
 
 @example
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