@example
> a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
--3*y^4+x^4+12*x*y^3+2*x^2*y^2+4*x^3*y
+12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
> b = x^2 + 4*x*y - y^2;
--y^2+x^2+4*x*y
+4*x*y-y^2+x^2
> expand(a*b);
-3*y^6+x^6-24*x*y^5+43*x^2*y^4+16*x^3*y^3+17*x^4*y^2+8*x^5*y
-> collect(a*b,x);
-3*y^6+48*x*y^4+2*x^2*y^2+x^4*(-y^2+x^2+4*x*y)+4*x^3*y*(-y^2+x^2+4*x*y)
+8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
+> collect(a+b,x);
+4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
+> collect(a+b,y);
+12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
> normal(a/b);
3*y^2+x^2
@end example
* Mathematical functions:: Mathematical functions.
* Relations:: Equality, Inequality and all that.
* Indexed objects:: Handling indexed quantities.
+* Non-commutative objects:: Algebras with non-commutative products.
@end menu
@item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
@item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
@item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
+@item @code{ncmul} @tab Products of non-commutative objects
@item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
@tex
$\sqrt{2}$
@item @code{tensor} @tab Special tensor like the delta and metric tensors
@item @code{idx} @tab Index of an indexed object
@item @code{varidx} @tab Index with variance
+@item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
@end multitable
@end cartouche
@code{expand()} must be called explicitly.
-@node Indexed objects, Methods and Functions, Relations, Basic Concepts
+@node Indexed objects, Non-commutative objects, Relations, Basic Concepts
@c node-name, next, previous, up
@section Indexed objects
a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
the index lives in) which can both be arbitrary expressions but are usually
a number or a simple symbol. In addition, indices of class @code{varidx} have
-a @dfn{variance} (they can be co- or contravariant).
+a @dfn{variance} (they can be co- or contravariant), and indices of class
+@code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
@item Indexed objects are of class @code{indexed} or a subclass. They
contain a @dfn{base expression} (which is the expression being indexed), and
@end itemize
@strong{Note:} when printing expressions, covariant indices and indices
-without variance are denoted @samp{.i} while contravariant indices are denoted
-@samp{~i}. In the following, we are going to use that notation in the text
-so instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions
-are not visible in the output.
+without variance are denoted @samp{.i} while contravariant indices are
+denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
+value. In the following, we are going to use that notation in the text so
+instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
+not visible in the output.
A simple example shall illustrate the concepts:
which makes a new index with the same value and dimension but the opposite
variance. By using it you only have to define the index once.
+@cindex @code{spinidx} (class)
+The @code{spinidx} class provides dotted and undotted variant indices, as
+used in the Weyl-van-der-Waerden spinor formalism:
+
+@example
+ ...
+ symbol K("K"), C_sym("C"), D_sym("D");
+ spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
+ // contravariant, undotted
+ spinidx C_co(C_sym, 2, true); // covariant index
+ spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
+ spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
+
+ cout << indexed(K, C, D) << endl;
+ // -> K~C~D
+ cout << indexed(K, C_co, D_dot) << endl;
+ // -> K.C~*D
+ cout << indexed(K, D_co_dot, D) << endl;
+ // -> K.*D~D
+ ...
+@end example
+
+A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
+dotted or undotted. The default is undotted but this can be overridden by
+supplying a fourth argument to the @code{spinidx} constructor. The two
+methods
+
+@example
+bool spinidx::is_dotted(void);
+bool spinidx::is_undotted(void);
+@end example
+
+allow you to check whether or not a @code{spinidx} object is dotted (use
+@code{ex_to_spinidx()} to get the object reference from an expression).
+Finally, the two methods
+
+@example
+ex spinidx::toggle_dot(void);
+ex spinidx::toggle_variance_dot(void);
+@end example
+
+create a new index with the same value and dimension but opposite dottedness
+and the same or opposite variance.
+
@subsection Substituting indices
@cindex @code{subs()}
To be recognized as a dummy index pair, the two indices must be of the same
class and dimension and their value must be the same single symbol (an index
like @samp{2*n+1} is never a dummy index). If the indices are of class
-@code{varidx}, they must also be of opposite variance.
+@code{varidx} they must also be of opposite variance; if they are of class
+@code{spinidx} they must be both dotted or both undotted.
The method @code{.get_free_indices()} returns a vector containing the free
indices of an expression. It also checks that the free indices of the terms
@}
@end example
+@cindex @code{spinor_metric()}
+@subsubsection Spinor metric tensor
+
+The function @code{spinor_metric()} creates an antisymmetric tensor with
+two indices that is used to raise/lower indices of 2-component spinors.
+It is output as @samp{eps}:
+
+@example
+@{
+ symbol psi("psi");
+
+ spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
+ ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
+
+ e = spinor_metric(A, B) * indexed(psi, B_co);
+ cout << e.simplify_indexed() << endl;
+ // -> psi~A
+
+ e = spinor_metric(A, B) * indexed(psi, A_co);
+ cout << e.simplify_indexed() << endl;
+ // -> -psi~B
+
+ e = spinor_metric(A_co, B_co) * indexed(psi, B);
+ cout << e.simplify_indexed() << endl;
+ // -> -psi.A
+
+ e = spinor_metric(A_co, B_co) * indexed(psi, A);
+ cout << e.simplify_indexed() << endl;
+ // -> psi.B
+
+ e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
+ cout << e.simplify_indexed() << endl;
+ // -> 2
+
+ e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
+ cout << e.simplify_indexed() << endl;
+ // -> -delta.A~C
+@}
+@end example
+
+The matrix representation of the spinor metric is @code{[[ [[ 0, 1 ]], [[ -1, 0 ]]}.
+
+@cindex @code{epsilon_tensor()}
+@cindex @code{lorentz_eps()}
@subsubsection Epsilon tensor
The epsilon tensor is totally antisymmetric, its number of indices is equal
of the metric tensor.
-@node Methods and Functions, Information About Expressions, Indexed objects, Top
+@node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
+@c node-name, next, previous, up
+@section Non-commutative objects
+
+GiNaC is equipped to handle certain non-commutative algebras. Three classes of
+non-commutative objects are built-in which are mostly of use in high energy
+physics:
+
+@itemize
+@item Clifford (Dirac) algebra (class @code{clifford})
+@item su(3) Lie algebra (class @code{color})
+@item Matrices (unindexed) (class @code{matrix})
+@end itemize
+
+The @code{clifford} and @code{color} classes are subclasses of
+@code{indexed} because the elements of these algebras ususally carry
+indices.
+
+Unlike most computer algebra systems, GiNaC does not primarily provide an
+operator (often denoted @samp{&*}) for representing inert products of
+arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
+classes of objects involved, and non-commutative products are formed with
+the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
+figuring out by itself which objects commute and will group the factors
+by their class. Consider this example:
+
+@example
+ ...
+ varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
+ idx a(symbol("a"), 8), b(symbol("b"), 8);
+ ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
+ cout << e << endl;
+ // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
+ ...
+@end example
+
+As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
+groups the non-commutative factors (the gammas and the su(3) generators)
+together while preserving the order of factors within each class (because
+Clifford objects commute with color objects). The resulting expression is a
+@emph{commutative} product with two factors that are themselves non-commutative
+products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
+parentheses are placed around the non-commutative products in the output.
+
+@cindex @code{ncmul} (class)
+Non-commutative products are internally represented by objects of the class
+@code{ncmul}, as opposed to commutative products which are handled by the
+@code{mul} class. You will normally not have to worry about this distinction,
+though.
+
+The advantage of this approach is that you never have to worry about using
+(or forgetting to use) a special operator when constructing non-commutative
+expressions. Also, non-commutative products in GiNaC are more intelligent
+than in other computer algebra systems; they can, for example, automatically
+canonicalize themselves according to rules specified in the implementation
+of the non-commutative classes. The drawback is that to work with other than
+the built-in algebras you have to implement new classes yourself. Symbols
+always commute and it's not possible to construct non-commutative products
+using symbols to represent the algebra elements or generators. User-defined
+functions can, however, be specified as being non-commutative.
+
+@cindex @code{return_type()}
+@cindex @code{return_type_tinfo()}
+Information about the commutativity of an object or expression can be
+obtained with the two member functions
+
+@example
+unsigned ex::return_type(void) const;
+unsigned ex::return_type_tinfo(void) const;
+@end example
+
+The @code{return_type()} function returns one of three values (defined in
+the header file @file{flags.h}), corresponding to three categories of
+expressions in GiNaC:
+
+@itemize
+@item @code{return_types::commutative}: Commutes with everything. Most GiNaC
+ classes are of this kind.
+@item @code{return_types::noncommutative}: Non-commutative, belonging to a
+ certain class of non-commutative objects which can be determined with the
+ @code{return_type_tinfo()} method. Expressions of this category commute
+ with everything except @code{noncommutative} expressions of the same
+ class.
+@item @code{return_types::noncommutative_composite}: Non-commutative, composed
+ of non-commutative objects of different classes. Expressions of this
+ category don't commute with any other @code{noncommutative} or
+ @code{noncommutative_composite} expressions.
+@end itemize
+
+The value returned by the @code{return_type_tinfo()} method is valid only
+when the return type of the expression is @code{noncommutative}. It is a
+value that is unique to the class of the object and usually one of the
+constants in @file{tinfos.h}, or derived therefrom.
+
+Here are a couple of examples:
+
+@cartouche
+@multitable @columnfractions 0.33 0.33 0.34
+@item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
+@item @code{42} @tab @code{commutative} @tab -
+@item @code{2*x-y} @tab @code{commutative} @tab -
+@item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
+@item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
+@item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
+@item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
+@end multitable
+@end cartouche
+
+Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
+@code{TINFO_clifford} for objects with a representation label of zero.
+Other representation labels yield a different @code{return_type_tinfo()},
+but it's the same for any two objects with the same label. This is also true
+for color objects.
+
+
+@cindex @code{clifford} (class)
+@subsection Clifford algebra
+
+@cindex @code{dirac_gamma()}
+Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
+doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
+@samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
+is the Minkowski metric tensor. Dirac gammas are constructed by the function
+
+@example
+ex dirac_gamma(const ex & mu, unsigned char rl = 0);
+@end example
+
+which takes two arguments: the index and a @dfn{representation label} in the
+range 0 to 255 which is used to distinguish elements of different Clifford
+algebras (this is also called a @dfn{spin line index}). Gammas with different
+labels commute with each other. The dimension of the index can be 4 or (in
+the framework of dimensional regularization) any symbolic value. Spinor
+indices on Dirac gammas are not supported in GiNaC.
+
+@cindex @code{dirac_ONE()}
+The unity element of a Clifford algebra is constructed by
+
+@example
+ex dirac_ONE(unsigned char rl = 0);
+@end example
+
+@cindex @code{dirac_gamma5()}
+and there's a special element @samp{gamma5} that commutes with all other
+gammas and in 4 dimensions equals @samp{gamma~0 gamma~1 gamma~2 gamma~3},
+provided by
+
+@example
+ex dirac_gamma5(unsigned char rl = 0);
+@end example
+
+@cindex @code{dirac_gamma6()}
+@cindex @code{dirac_gamma7()}
+The two additional functions
+
+@example
+ex dirac_gamma6(unsigned char rl = 0);
+ex dirac_gamma7(unsigned char rl = 0);
+@end example
+
+return @code{dirac_ONE(rl) + dirac_gamma5(rl)} and @code{dirac_ONE(rl) - dirac_gamma5(rl)},
+respectively.
+
+@cindex @code{dirac_slash()}
+Finally, the function
+
+@example
+ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
+@end example
+
+creates a term of the form @samp{e.mu gamma~mu} with a new and unique index
+whose dimension is given by the @code{dim} argument.
+
+The @code{simplify_indexed()} function performs contractions in gamma strings
+if possible, for example
+
+@example
+@{
+ ...
+ symbol a("a"), b("b"), D("D");
+ varidx mu(symbol("mu"), D);
+ ex e = dirac_gamma(mu) * dirac_slash(a, D)
+ * dirac_gamma(mu.toggle_variance());
+ cout << e << endl;
+ // -> (gamma~mu*gamma~symbol10*gamma.mu)*a.symbol10
+ e = e.simplify_indexed();
+ cout << e << endl;
+ // -> -gamma~symbol10*a.symbol10*D+2*gamma~symbol10*a.symbol10
+ cout << e.subs(D == 4) << endl;
+ // -> -2*gamma~symbol10*a.symbol10
+ // [ == -2 * dirac_slash(a, D) ]
+ ...
+@}
+@end example
+
+@cindex @code{dirac_trace()}
+To calculate the trace of an expression containing strings of Dirac gammas
+you use the function
+
+@example
+ex dirac_trace(const ex & e, unsigned char rl = 0);
+@end example
+
+This function takes the trace of all gammas with the specified representation
+label; gammas with other labels are left standing. The @code{dirac_trace()}
+function is a linear functional that is equal to the usual trace only in
+@math{D = 4} dimensions. In particular, the functional is not cyclic in
+@math{D != 4} dimensions when acting on expressions containing @samp{gamma5},
+so it's not a proper trace. This @samp{gamma5} scheme is described in greater
+detail in @cite{The Role of gamma5 in Dimensional Regularization}.
+
+The value of the trace itself is also usually different in 4 and in
+@math{D != 4} dimensions:
+
+@example
+@{
+ // 4 dimensions
+ varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
+ ex e = dirac_gamma(mu) * dirac_gamma(nu) *
+ dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
+ cout << dirac_trace(e).simplify_indexed() << endl;
+ // -> -8*eta~rho~nu
+@}
+...
+@{
+ // D dimensions
+ symbol D("D");
+ varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
+ ex e = dirac_gamma(mu) * dirac_gamma(nu) *
+ dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
+ cout << dirac_trace(e).simplify_indexed() << endl;
+ // -> 8*eta~rho~nu-4*eta~rho~nu*D
+@}
+@end example
+
+Here is an example for using @code{dirac_trace()} to compute a value that
+appears in the calculation of the one-loop vacuum polarization amplitude in
+QED:
+
+@example
+@{
+ symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
+ varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
+
+ scalar_products sp;
+ sp.add(l, l, pow(l, 2));
+ sp.add(l, q, ldotq);
+
+ ex e = dirac_gamma(mu) *
+ (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
+ dirac_gamma(mu.toggle_variance()) *
+ (dirac_slash(l, D) + m * dirac_ONE());
+ e = dirac_trace(e).simplify_indexed(sp);
+ e = e.collect(lst(l, ldotq, m), true);
+ cout << e << endl;
+ // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
+@}
+@end example
+
+
+@cindex @code{color} (class)
+@subsection Color algebra
+
+@cindex @code{color_T()}
+For computations in quantum chromodynamics, GiNaC implements the base elements
+and structure constants of the su(3) Lie algebra (color algebra). The base
+elements @math{T_a} are constructed by the function
+
+@example
+ex color_T(const ex & a, unsigned char rl = 0);
+@end example
+
+which takes two arguments: the index and a @dfn{representation label} in the
+range 0 to 255 which is used to distinguish elements of different color
+algebras. Objects with different labels commute with each other. The
+dimension of the index must be exactly 8 and it should be of class @code{idx},
+not @code{varidx}.
+
+@cindex @code{color_ONE()}
+The unity element of a color algebra is constructed by
+
+@example
+ex color_ONE(unsigned char rl = 0);
+@end example
+
+@cindex @code{color_d()}
+@cindex @code{color_f()}
+and the functions
+
+@example
+ex color_d(const ex & a, const ex & b, const ex & c);
+ex color_f(const ex & a, const ex & b, const ex & c);
+@end example
+
+create the symmetric and antisymmetric structure constants @math{d_abc} and
+@math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
+and @math{[T_a, T_b] = i f_abc T_c}.
+
+@cindex @code{color_h()}
+There's an additional function
+
+@example
+ex color_h(const ex & a, const ex & b, const ex & c);
+@end example
+
+which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
+
+The function @code{simplify_indexed()} performs some simplifications on
+expressions containing color objects:
+
+@example
+@{
+ ...
+ idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
+ k(symbol("k"), 8), l(symbol("l"), 8);
+
+ e = color_d(a, b, l) * color_f(a, b, k);
+ cout << e.simplify_indexed() << endl;
+ // -> 0
+
+ e = color_d(a, b, l) * color_d(a, b, k);
+ cout << e.simplify_indexed() << endl;
+ // -> 5/3*delta.k.l
+
+ e = color_f(l, a, b) * color_f(a, b, k);
+ cout << e.simplify_indexed() << endl;
+ // -> 3*delta.k.l
+
+ e = color_h(a, b, c) * color_h(a, b, c);
+ cout << e.simplify_indexed() << endl;
+ // -> -32/3
+
+ e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
+ cout << e.simplify_indexed() << endl;
+ // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
+ ...
+@end example
+
+@cindex @code{color_trace()}
+To calculate the trace of an expression containing color objects you use the
+function
+
+@example
+ex color_trace(const ex & e, unsigned char rl = 0);
+@end example
+
+This function takes the trace of all color @samp{T} objects with the
+specified representation label; @samp{T}s with other labels are left
+standing. For example:
+
+@example
+ ...
+ e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
+ cout << e << endl;
+ // -> -I*f.a.c.b+d.a.c.b
+@}
+@end example
+
+
+@node Methods and Functions, Information About Expressions, Non-commutative objects, Top
@c node-name, next, previous, up
@chapter Methods and Functions
@cindex polynomial
@cindex @code{ex_to_@dots{}}
@cindex @code{Converting ex to other classes}
@cindex @code{info()}
+@cindex @code{return_type()}
+@cindex @code{return_type_tinfo()}
Sometimes it's useful to check whether a given expression is a plain number,
a sum, a polynomial with integer coefficients, or of some other specific type.
-GiNaC provides two functions for this (the first one is actually a macro):
+GiNaC provides a couple of functions for this (the first one is actually a macro):
@example
bool is_ex_of_type(const ex & e, TYPENAME t);
bool ex::info(unsigned flag);
+unsigned ex::return_type(void) const;
+unsigned ex::return_type_tinfo(void) const;
@end example
When the test made by @code{is_ex_of_type()} returns true, it is safe to
@end multitable
@end cartouche
+To determine whether an expression is commutative or non-commutative and if
+so, with which other expressions it would commute, you use the methods
+@code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
+for an explanation of these.
+
@subsection Accessing subexpressions
@cindex @code{nops()}
@code{collect()} accomplishes this task:
@example
-ex ex::collect(const ex & s);
+ex ex::collect(const ex & s, bool distributed = false);
@end example
+The first argument to @code{collect()} can also be a list of objects in which
+case the result is either a recursively collected polynomial, or a polynomial
+in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
+by the @code{distributed} flag.
+
Note that the original polynomial needs to be in expanded form in order
-to be able to find the coefficients properly.
+for @code{collect()} to be able to find the coefficients properly.
@subsection Degree and coefficients
@cindex @code{degree()}
void ex::print(const print_context & c, unsigned level = 0);
@end example
+@cindex @code{print_context} (class)
The type of @code{print_context} object passed in determines the format
of the output. The possible types are defined in @file{ginac/print.h}.
All constructors of @code{print_context} and derived classes take an
cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
@end example
-The @code{print_context} type @code{print_tree} provdes a dump of the
+The @code{print_context} type @code{print_tree} provides a dump of the
internal structure of an expression for debugging purposes:
@example
This kind of output is also available in @command{ginsh} as the @code{print()}
function.
+Another useful output format is for LaTeX parsing in mathematical mode.
+It is rather similar to the default @code{print_context} but provides
+some braces needed by LaTeX for delimiting boxes and also converts some
+common objects to conventional LaTeX names. It is possible to give symbols
+a special name for LaTeX output by supplying it as a second argument to
+the @code{symbol} constructor.
+
+For example, the code snippet
+
+@example
+ // ...
+ symbol x("x");
+ ex foo = lgamma(x).series(x==0,3);
+ foo.print(print_latex(std::cout));
+@end example
+
+will print out:
+
+@example
+ @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
+@end example
+
If you need any fancy special output format, e.g. for interfacing GiNaC
with other algebra systems or for producing code for different
programming languages, you can always traverse the expression tree yourself:
@}
if (new_str.length() == 0)
- return _ex0();
+ return 0;
else
return mystring(new_str).hold();
@}
J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
Academic Press, London
+@item
+@cite{The Role of gamma5 in Dimensional Regularization}, D. Kreimer, hep-ph/9401354
+
@end itemize
git://www.ginac.de / ginac.git / blobdiff