@cindex Laurent expansion
Expressions know how to expand themselves as a Taylor series or (more
-generally) a Laurent series. Similar to most conventional Computer
-Algebra Systems, no distinction is made between those two. There is a
-class of its own for storing such series as well as a class for storing
-the order of the series. A sample program could read:
+generally) a Laurent series. As in most conventional Computer Algebra
+Systems, no distinction is made between those two. There is a class of
+its own for storing such series as well as a class for storing the order
+of the series. As a consequence, if you want to work with series,
+i.e. multiply two series, you need to call the method @code{ex::series}
+again to convert it to a series object with the usual structure
+(expansion plus order term). A sample application from special
+relativity could read:
@example
#include <ginac/ginac.h>
int main()
@{
- symbol x("x");
- numeric point(0);
- ex MyExpr1 = sin(x);
- ex MyExpr2 = 1/(x - pow(x, 2) - pow(x, 3));
- ex MyTailor, MySeries;
+ symbol v("v"), c("c");
+
+ ex gamma = 1/sqrt(1 - pow(v/c,2));
+ 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;
- MyTailor = MyExpr1.series(x, point, 5);
- cout << MyExpr1 << " == " << MyTailor
- << " for small " << x << endl;
- MySeries = MyExpr2.series(x, point, 7);
- cout << MyExpr2 << " == " << MySeries
- << " for small " << x << endl;
// ...
@}
@end example
+Only calling the series method makes the last output simplify to
+@math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
+series raised to the power @math{-2}.
+
@cindex M@'echain's formula
-As an instructive application, let us calculate the numerical value of
-Archimedes' constant
+As another instructive application, let us calculate the numerical
+value of Archimedes' constant
@tex
$\pi$
@end tex
return psi(x)*gamma(x);
}
-static ex gamma_series(const ex & x, const symbol & s, const ex & point, int order)
+static ex gamma_series(const ex & x, const symbol & s, const ex & pt, int order)
{
// method:
// Taylor series where there is no pole falls back to psi function
// gamma(x) == gamma(x+1) / x
// from which follows
// series(gamma(x),x,-m,order) ==
- // series(gamma(x+m+1)/(x*(x+1)...*(x+m)),x,-m,order+1);
- ex xpoint = x.subs(s==point);
- if (!xpoint.info(info_flags::integer) || xpoint.info(info_flags::positive))
+ // series(gamma(x+m+1)/(x*(x+1)*...*(x+m)),x,-m,order+1);
+ const ex x_pt = x.subs(s==pt);
+ if (!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive))
throw do_taylor(); // caught by function::series()
// if we got here we have to care for a simple pole at -m:
- numeric m = -ex_to_numeric(xpoint);
+ numeric m = -ex_to_numeric(x_pt);
ex ser_numer = gamma(x+m+_ex1());
ex ser_denom = _ex1();
for (numeric p; p<=m; ++p)
ser_denom *= x+p;
- return (ser_numer/ser_denom).series(s, point, order+1);
+ return (ser_numer/ser_denom).series(s, pt, order+1);
}
REGISTER_FUNCTION(gamma, gamma_eval, gamma_evalf, gamma_diff, gamma_series);
static ex beta_eval(const ex & x, const ex & y)
{
if (x.info(info_flags::numeric) && y.info(info_flags::numeric)) {
- numeric nx(ex_to_numeric(x));
- numeric ny(ex_to_numeric(y));
// treat all problematic x and y that may not be passed into gamma,
// because they would throw there although beta(x,y) is well-defined
// using the formula beta(x,y) == (-1)^y * beta(1-x-y, y)
+ numeric nx(ex_to_numeric(x));
+ numeric ny(ex_to_numeric(y));
if (nx.is_real() && nx.is_integer() &&
ny.is_real() && ny.is_integer()) {
if (nx.is_negative()) {
return retval;
}
-static ex beta_series(const ex & x, const ex & y, const symbol & s, const ex & point, int order)
+static ex beta_series(const ex & x, const ex & y, const symbol & s, const ex & pt, int order)
{
// method:
- // Taylor series where there is no pole falls back to beta function
- // evaluation.
- // On a pole at -m use the recurrence relation
- // gamma(x) == gamma(x+1) / x
- // from which follows
- // series(gamma(x),x,-m,order) ==
- // series(gamma(x+m+1)/(x*(x+1)...*(x+m)),x,-m,order+1);
- ex xpoint = x.subs(s==point);
- ex ypoint = y.subs(s==point);
- if ((!xpoint.info(info_flags::integer) || xpoint.info(info_flags::positive)) &&
- (!ypoint.info(info_flags::integer) || ypoint.info(info_flags::positive)))
+ // Taylor series where there is no pole of one of the gamma functions
+ // falls back to beta function evaluation. Otherwise, fall back to
+ // gamma series directly.
+ // FIXME: this could need some testing, maybe it's wrong in some cases?
+ const ex x_pt = x.subs(s==pt);
+ const ex y_pt = y.subs(s==pt);
+ ex x_ser, y_ser, xy_ser;
+ if ((!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive)) &&
+ (!y_pt.info(info_flags::integer) || y_pt.info(info_flags::positive)))
throw do_taylor(); // caught by function::series()
- // if we got here we have to care for a simple pole at -m:
- throw (std::domain_error("beta_series(): Mama, please code me!"));
+ // trap the case where x is on a pole directly:
+ if (x.info(info_flags::integer) && !x.info(info_flags::positive))
+ x_ser = gamma(x+s).series(s,pt,order);
+ else
+ x_ser = gamma(x).series(s,pt,order);
+ // trap the case where y is on a pole directly:
+ if (y.info(info_flags::integer) && !y.info(info_flags::positive))
+ y_ser = gamma(y+s).series(s,pt,order);
+ else
+ y_ser = gamma(y).series(s,pt,order);
+ // trap the case where y is on a pole directly:
+ if ((x+y).info(info_flags::integer) && !(x+y).info(info_flags::positive))
+ xy_ser = gamma(y+x+s).series(s,pt,order);
+ else
+ xy_ser = gamma(y+x).series(s,pt,order);
+ // compose the result:
+ return (x_ser*y_ser/xy_ser).series(s,pt,order);
}
REGISTER_FUNCTION(beta, beta_eval, beta_evalf, beta_diff, beta_series);
return psi(_ex1(), x);
}
-static ex psi1_series(const ex & x, const symbol & s, const ex & point, int order)
+static ex psi1_series(const ex & x, const symbol & s, const ex & pt, int order)
{
// method:
// Taylor series where there is no pole falls back to polygamma function
// from which follows
// series(psi(x),x,-m,order) ==
// series(psi(x+m+1) - 1/x - 1/(x+1) - 1/(x+m)),x,-m,order);
- ex xpoint = x.subs(s==point);
- if (!xpoint.info(info_flags::integer) || xpoint.info(info_flags::positive))
+ const ex x_pt = x.subs(s==pt);
+ if (!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive))
throw do_taylor(); // caught by function::series()
// if we got here we have to care for a simple pole at -m:
- numeric m = -ex_to_numeric(xpoint);
+ numeric m = -ex_to_numeric(x_pt);
ex recur;
for (numeric p; p<=m; ++p)
recur += power(x+p,_ex_1());
- return (psi(x+m+_ex1())-recur).series(s, point, order);
+ return (psi(x+m+_ex1())-recur).series(s, pt, order);
}
const unsigned function_index_psi1 = function::register_new("psi", psi1_eval, psi1_evalf, psi1_diff, psi1_series);
return psi(n+_ex1(), x);
}
-static ex psi2_series(const ex & n, const ex & x, const symbol & s, const ex & point, int order)
+static ex psi2_series(const ex & n, const ex & x, const symbol & s, const ex & pt, int order)
{
// method:
// Taylor series where there is no pole falls back to polygamma function
// series(psi(x),x,-m,order) ==
// series(psi(x+m+1) - (-1)^n * n! * ((x)^(-n-1) + (x+1)^(-n-1) + ...
// ... + (x+m)^(-n-1))),x,-m,order);
- ex xpoint = x.subs(s==point);
- if (!xpoint.info(info_flags::integer) || xpoint.info(info_flags::positive))
+ const ex x_pt = x.subs(s==pt);
+ if (!x_pt.info(info_flags::integer) || x_pt.info(info_flags::positive))
throw do_taylor(); // caught by function::series()
// if we got here we have to care for a pole of order n+1 at -m:
- numeric m = -ex_to_numeric(xpoint);
+ numeric m = -ex_to_numeric(x_pt);
ex recur;
for (numeric p; p<=m; ++p)
recur += power(x+p,-n+_ex_1());
recur *= factorial(n)*power(_ex_1(),n);
- return (psi(n, x+m+_ex1())-recur).series(s, point, order);
+ return (psi(n, x+m+_ex1())-recur).series(s, pt, order);
}
const unsigned function_index_psi2 = function::register_new("psi", psi2_eval, psi2_evalf, psi2_diff, psi2_series);
// exponential function
//////////
-static ex exp_evalf(ex const & x)
+static ex exp_evalf(const ex & x)
{
BEGIN_TYPECHECK
TYPECHECK(x,numeric)
return exp(ex_to_numeric(x)); // -> numeric exp(numeric)
}
-static ex exp_eval(ex const & x)
+static ex exp_eval(const ex & x)
{
// exp(0) -> 1
if (x.is_zero()) {
return exp(x).hold();
}
-static ex exp_diff(ex const & x, unsigned diff_param)
+static ex exp_diff(const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param==0);
// natural logarithm
//////////
-static ex log_evalf(ex const & x)
+static ex log_evalf(const ex & x)
{
BEGIN_TYPECHECK
TYPECHECK(x,numeric)
return log(ex_to_numeric(x)); // -> numeric log(numeric)
}
-static ex log_eval(ex const & x)
+static ex log_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
if (x.is_equal(_ex1())) // log(1) -> 0
if (!x.info(info_flags::crational))
return log_evalf(x);
}
- // log(exp(t)) -> t (for real-valued t):
+ // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
if (is_ex_the_function(x, exp)) {
ex t = x.op(0);
- if (t.info(info_flags::real))
- return t;
+ if (t.info(info_flags::numeric)) {
+ numeric nt = ex_to_numeric(t);
+ if (nt.is_real())
+ return t;
+ }
}
return log(x).hold();
}
-static ex log_diff(ex const & x, unsigned diff_param)
+static ex log_diff(const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param==0);
// sine (trigonometric function)
//////////
-static ex sin_evalf(ex const & x)
+static ex sin_evalf(const ex & x)
{
BEGIN_TYPECHECK
TYPECHECK(x,numeric)
return sin(ex_to_numeric(x)); // -> numeric sin(numeric)
}
-static ex sin_eval(ex const & x)
+static ex sin_eval(const ex & x)
{
// sin(n/d*Pi) -> { all known non-nested radicals }
ex SixtyExOverPi = _ex60()*x/Pi;
return sin(x).hold();
}
-static ex sin_diff(ex const & x, unsigned diff_param)
+static ex sin_diff(const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param==0);
// cosine (trigonometric function)
//////////
-static ex cos_evalf(ex const & x)
+static ex cos_evalf(const ex & x)
{
BEGIN_TYPECHECK
TYPECHECK(x,numeric)
return cos(ex_to_numeric(x)); // -> numeric cos(numeric)
}
-static ex cos_eval(ex const & x)
+static ex cos_eval(const ex & x)
{
// cos(n/d*Pi) -> { all known non-nested radicals }
ex SixtyExOverPi = _ex60()*x/Pi;
return cos(x).hold();
}
-static ex cos_diff(ex const & x, unsigned diff_param)
+static ex cos_diff(const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param==0);
// tangent (trigonometric function)
//////////
-static ex tan_evalf(ex const & x)
+static ex tan_evalf(const ex & x)
{
BEGIN_TYPECHECK
TYPECHECK(x,numeric)
return tan(ex_to_numeric(x));
}
-static ex tan_eval(ex const & x)
+static ex tan_eval(const ex & x)
{
// tan(n/d*Pi) -> { all known non-nested radicals }
ex SixtyExOverPi = _ex60()*x/Pi;
return tan(x).hold();
}
-static ex tan_diff(ex const & x, unsigned diff_param)
+static ex tan_diff(const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param==0);
return (_ex1()+power(tan(x),_ex2()));
}
-static ex tan_series(ex const & x, symbol const & s, ex const & point, int order)
+static ex tan_series(const ex & x, const symbol & s, const ex & pt, int order)
{
// method:
// Taylor series where there is no pole falls back to tan_diff.
// On a pole simply expand sin(x)/cos(x).
- ex xpoint = x.subs(s==point);
- if (!(2*xpoint/Pi).info(info_flags::odd))
+ const ex x_pt = x.subs(s==pt);
+ if (!(2*x_pt/Pi).info(info_flags::odd))
throw do_taylor(); // caught by function::series()
// if we got here we have to care for a simple pole
- return (sin(x)/cos(x)).series(s, point, order+2);
+ return (sin(x)/cos(x)).series(s, pt, order+2);
}
REGISTER_FUNCTION(tan, tan_eval, tan_evalf, tan_diff, tan_series);
// inverse sine (arc sine)
//////////
-static ex asin_evalf(ex const & x)
+static ex asin_evalf(const ex & x)
{
BEGIN_TYPECHECK
TYPECHECK(x,numeric)
return asin(ex_to_numeric(x)); // -> numeric asin(numeric)
}
-static ex asin_eval(ex const & x)
+static ex asin_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
// asin(0) -> 0
return asin(x).hold();
}
-static ex asin_diff(ex const & x, unsigned diff_param)
+static ex asin_diff(const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param==0);
// inverse cosine (arc cosine)
//////////
-static ex acos_evalf(ex const & x)
+static ex acos_evalf(const ex & x)
{
BEGIN_TYPECHECK
TYPECHECK(x,numeric)
return acos(ex_to_numeric(x)); // -> numeric acos(numeric)
}
-static ex acos_eval(ex const & x)
+static ex acos_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
// acos(1) -> 0
return acos(x).hold();
}
-static ex acos_diff(ex const & x, unsigned diff_param)
+static ex acos_diff(const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param==0);
// inverse tangent (arc tangent)
//////////
-static ex atan_evalf(ex const & x)
+static ex atan_evalf(const ex & x)
{
BEGIN_TYPECHECK
TYPECHECK(x,numeric)
return atan(ex_to_numeric(x)); // -> numeric atan(numeric)
}
-static ex atan_eval(ex const & x)
+static ex atan_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
// atan(0) -> 0
return atan(x).hold();
}
-static ex atan_diff(ex const & x, unsigned diff_param)
+static ex atan_diff(const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param==0);
// inverse tangent (atan2(y,x))
//////////
-static ex atan2_evalf(ex const & y, ex const & x)
+static ex atan2_evalf(const ex & y, const ex & x)
{
BEGIN_TYPECHECK
TYPECHECK(y,numeric)
return atan(ex_to_numeric(y),ex_to_numeric(x)); // -> numeric atan(numeric)
}
-static ex atan2_eval(ex const & y, ex const & x)
+static ex atan2_eval(const ex & y, const ex & x)
{
if (y.info(info_flags::numeric) && !y.info(info_flags::crational) &&
x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
return atan2(y,x).hold();
}
-static ex atan2_diff(ex const & y, ex const & x, unsigned diff_param)
+static ex atan2_diff(const ex & y, const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param<2);
// hyperbolic sine (trigonometric function)
//////////
-static ex sinh_evalf(ex const & x)
+static ex sinh_evalf(const ex & x)
{
BEGIN_TYPECHECK
TYPECHECK(x,numeric)
return sinh(ex_to_numeric(x)); // -> numeric sinh(numeric)
}
-static ex sinh_eval(ex const & x)
+static ex sinh_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
if (x.is_zero()) // sinh(0) -> 0
return sinh(x).hold();
}
-static ex sinh_diff(ex const & x, unsigned diff_param)
+static ex sinh_diff(const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param==0);
// hyperbolic cosine (trigonometric function)
//////////
-static ex cosh_evalf(ex const & x)
+static ex cosh_evalf(const ex & x)
{
BEGIN_TYPECHECK
TYPECHECK(x,numeric)
return cosh(ex_to_numeric(x)); // -> numeric cosh(numeric)
}
-static ex cosh_eval(ex const & x)
+static ex cosh_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
if (x.is_zero()) // cosh(0) -> 1
return cosh(x).hold();
}
-static ex cosh_diff(ex const & x, unsigned diff_param)
+static ex cosh_diff(const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param==0);
// hyperbolic tangent (trigonometric function)
//////////
-static ex tanh_evalf(ex const & x)
+static ex tanh_evalf(const ex & x)
{
BEGIN_TYPECHECK
TYPECHECK(x,numeric)
return tanh(ex_to_numeric(x)); // -> numeric tanh(numeric)
}
-static ex tanh_eval(ex const & x)
+static ex tanh_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
if (x.is_zero()) // tanh(0) -> 0
return tanh(x).hold();
}
-static ex tanh_diff(ex const & x, unsigned diff_param)
+static ex tanh_diff(const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param==0);
return _ex1()-power(tanh(x),_ex2());
}
-static ex tanh_series(ex const & x, symbol const & s, ex const & point, int order)
+static ex tanh_series(const ex & x, const symbol & s, const ex & pt, int order)
{
// method:
// Taylor series where there is no pole falls back to tanh_diff.
// On a pole simply expand sinh(x)/cosh(x).
- ex xpoint = x.subs(s==point);
- if (!(2*I*xpoint/Pi).info(info_flags::odd))
+ const ex x_pt = x.subs(s==pt);
+ if (!(2*I*x_pt/Pi).info(info_flags::odd))
throw do_taylor(); // caught by function::series()
// if we got here we have to care for a simple pole
- return (sinh(x)/cosh(x)).series(s, point, order+2);
+ return (sinh(x)/cosh(x)).series(s, pt, order+2);
}
REGISTER_FUNCTION(tanh, tanh_eval, tanh_evalf, tanh_diff, tanh_series);
// inverse hyperbolic sine (trigonometric function)
//////////
-static ex asinh_evalf(ex const & x)
+static ex asinh_evalf(const ex & x)
{
BEGIN_TYPECHECK
TYPECHECK(x,numeric)
return asinh(ex_to_numeric(x)); // -> numeric asinh(numeric)
}
-static ex asinh_eval(ex const & x)
+static ex asinh_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
// asinh(0) -> 0
return asinh(x).hold();
}
-static ex asinh_diff(ex const & x, unsigned diff_param)
+static ex asinh_diff(const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param==0);
// inverse hyperbolic cosine (trigonometric function)
//////////
-static ex acosh_evalf(ex const & x)
+static ex acosh_evalf(const ex & x)
{
BEGIN_TYPECHECK
TYPECHECK(x,numeric)
return acosh(ex_to_numeric(x)); // -> numeric acosh(numeric)
}
-static ex acosh_eval(ex const & x)
+static ex acosh_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
// acosh(0) -> Pi*I/2
return acosh(x).hold();
}
-static ex acosh_diff(ex const & x, unsigned diff_param)
+static ex acosh_diff(const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param==0);
// inverse hyperbolic tangent (trigonometric function)
//////////
-static ex atanh_evalf(ex const & x)
+static ex atanh_evalf(const ex & x)
{
BEGIN_TYPECHECK
TYPECHECK(x,numeric)
return atanh(ex_to_numeric(x)); // -> numeric atanh(numeric)
}
-static ex atanh_eval(ex const & x)
+static ex atanh_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
// atanh(0) -> 0
return atanh(x).hold();
}
-static ex atanh_diff(ex const & x, unsigned diff_param)
+static ex atanh_diff(const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param==0);
// Riemann's Zeta-function
//////////
-static ex zeta1_evalf(ex const & x)
+static ex zeta1_evalf(const ex & x)
{
BEGIN_TYPECHECK
TYPECHECK(x,numeric)
return zeta(ex_to_numeric(x));
}
-static ex zeta1_eval(ex const & x)
+static ex zeta1_eval(const ex & x)
{
if (x.info(info_flags::numeric)) {
numeric y = ex_to_numeric(x);
return zeta(x).hold();
}
-static ex zeta1_diff(ex const & x, unsigned diff_param)
+static ex zeta1_diff(const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param==0);
// Derivatives of Riemann's Zeta-function zeta(0,x)==zeta(x)
//////////
-static ex zeta2_eval(ex const & n, ex const & x)
+static ex zeta2_eval(const ex & n, const ex & x)
{
if (n.info(info_flags::numeric)) {
// zeta(0,x) -> zeta(x)
return zeta(n, x).hold();
}
-static ex zeta2_diff(ex const & n, ex const & x, unsigned diff_param)
+static ex zeta2_diff(const ex & n, const ex & x, unsigned diff_param)
{
GINAC_ASSERT(diff_param<2);
// Series terminates
return series::series(s, point, seq);
}
- coeff = power(fac, -1) * deriv.subs(s == point);
+ coeff = fac.inverse() * deriv.subs(s == point);
if (!coeff.is_zero())
seq.push_back(expair(coeff, numeric(n)));
}