static symbol x("x");
-static unsigned check_series(const ex &e, const ex &point, const ex &d)
+static unsigned check_series(const ex &e, const ex &point, const ex &d, int order = 8)
{
- ex es = e.series(x, point, 8);
- ex ep = static_cast<series *>(es.bp)->convert_to_poly();
- if ((ep - d).compare(exZERO()) != 0) {
- clog << "series expansion of " << e << " at " << point
+ ex es = e.series(x, point, order);
+ ex ep = static_cast<series *>(es.bp)->convert_to_poly();
+ if ((ep - d).compare(exZERO()) != 0) {
+ clog << "series expansion of " << e << " at " << point
<< " erroneously returned " << ep << " (instead of " << d
<< ")" << endl;
- (ep-d).printtree(clog);
- return 1;
- }
- return 0;
+ (ep-d).printtree(clog);
+ return 1;
+ }
+ return 0;
}
// Series expansion
static unsigned series1(void)
{
- unsigned result = 0;
- ex e, d;
-
- e = sin(x);
- d = x - pow(x, 3) / 6 + pow(x, 5) / 120 - pow(x, 7) / 5040 + Order(pow(x, 8));
- result += check_series(e, exZERO(), d);
-
- e = cos(x);
- d = 1 - pow(x, 2) / 2 + pow(x, 4) / 24 - pow(x, 6) / 720 + Order(pow(x, 8));
- result += check_series(e, exZERO(), d);
-
- e = exp(x);
- d = 1 + x + pow(x, 2) / 2 + pow(x, 3) / 6 + pow(x, 4) / 24 + pow(x, 5) / 120 + pow(x, 6) / 720 + pow(x, 7) / 5040 + Order(pow(x, 8));
- result += check_series(e, exZERO(), d);
-
- e = pow(1 - x, -1);
- d = 1 + x + pow(x, 2) + pow(x, 3) + pow(x, 4) + pow(x, 5) + pow(x, 6) + pow(x, 7) + Order(pow(x, 8));
- result += check_series(e, exZERO(), d);
-
- e = x + pow(x, -1);
- d = x + pow(x, -1);
- result += check_series(e, exZERO(), d);
-
- e = x + pow(x, -1);
- d = 2 + pow(x-1, 2) - pow(x-1, 3) + pow(x-1, 4) - pow(x-1, 5) + pow(x-1, 6) - pow(x-1, 7) + Order(pow(x-1, 8));
- result += check_series(e, exONE(), d);
-
- e = pow(x + pow(x, 3), -1);
- d = pow(x, -1) - x + pow(x, 3) - pow(x, 5) + Order(pow(x, 7));
- result += check_series(e, exZERO(), d);
-
- e = pow(pow(x, 2) + pow(x, 4), -1);
- d = pow(x, -2) - 1 + pow(x, 2) - pow(x, 4) + Order(pow(x, 6));
- result += check_series(e, exZERO(), d);
-
- e = pow(sin(x), -2);
- d = pow(x, -2) + numeric(1,3) + pow(x, 2) / 15 + pow(x, 4) * 2/189 + Order(pow(x, 5));
- result += check_series(e, exZERO(), d);
-
- e = sin(x) / cos(x);
- d = x + pow(x, 3) / 3 + pow(x, 5) * 2/15 + pow(x, 7) * 17/315 + Order(pow(x, 8));
- result += check_series(e, exZERO(), d);
-
- e = cos(x) / sin(x);
- d = pow(x, -1) - x / 3 - pow(x, 3) / 45 - pow(x, 5) * 2/945 + Order(pow(x, 6));
- result += check_series(e, exZERO(), d);
-
- e = pow(numeric(2), x);
- ex t = log(ex(2)) * x;
- d = 1 + t + pow(t, 2) / 2 + pow(t, 3) / 6 + pow(t, 4) / 24 + pow(t, 5) / 120 + pow(t, 6) / 720 + pow(t, 7) / 5040 + Order(pow(x, 8));
- result += check_series(e, exZERO(), d.expand());
-
- e = pow(Pi, x);
- t = log(Pi) * x;
- d = 1 + t + pow(t, 2) / 2 + pow(t, 3) / 6 + pow(t, 4) / 24 + pow(t, 5) / 120 + pow(t, 6) / 720 + pow(t, 7) / 5040 + Order(pow(x, 8));
- result += check_series(e, exZERO(), d.expand());
-
- return result;
+ unsigned result = 0;
+ ex e, d;
+
+ e = sin(x);
+ d = x - pow(x, 3) / 6 + pow(x, 5) / 120 - pow(x, 7) / 5040 + Order(pow(x, 8));
+ result += check_series(e, exZERO(), d);
+
+ e = cos(x);
+ d = 1 - pow(x, 2) / 2 + pow(x, 4) / 24 - pow(x, 6) / 720 + Order(pow(x, 8));
+ result += check_series(e, exZERO(), d);
+
+ e = exp(x);
+ d = 1 + x + pow(x, 2) / 2 + pow(x, 3) / 6 + pow(x, 4) / 24 + pow(x, 5) / 120 + pow(x, 6) / 720 + pow(x, 7) / 5040 + Order(pow(x, 8));
+ result += check_series(e, exZERO(), d);
+
+ e = pow(1 - x, -1);
+ d = 1 + x + pow(x, 2) + pow(x, 3) + pow(x, 4) + pow(x, 5) + pow(x, 6) + pow(x, 7) + Order(pow(x, 8));
+ result += check_series(e, exZERO(), d);
+
+ e = x + pow(x, -1);
+ d = x + pow(x, -1);
+ result += check_series(e, exZERO(), d);
+
+ e = x + pow(x, -1);
+ d = 2 + pow(x-1, 2) - pow(x-1, 3) + pow(x-1, 4) - pow(x-1, 5) + pow(x-1, 6) - pow(x-1, 7) + Order(pow(x-1, 8));
+ result += check_series(e, exONE(), d);
+
+ e = pow(x + pow(x, 3), -1);
+ d = pow(x, -1) - x + pow(x, 3) - pow(x, 5) + Order(pow(x, 7));
+ result += check_series(e, exZERO(), d);
+
+ e = pow(pow(x, 2) + pow(x, 4), -1);
+ d = pow(x, -2) - 1 + pow(x, 2) - pow(x, 4) + Order(pow(x, 6));
+ result += check_series(e, exZERO(), d);
+
+ e = pow(sin(x), -2);
+ d = pow(x, -2) + numeric(1,3) + pow(x, 2) / 15 + pow(x, 4) * 2/189 + Order(pow(x, 5));
+ result += check_series(e, exZERO(), d);
+
+ e = sin(x) / cos(x);
+ d = x + pow(x, 3) / 3 + pow(x, 5) * 2/15 + pow(x, 7) * 17/315 + Order(pow(x, 8));
+ result += check_series(e, exZERO(), d);
+
+ e = cos(x) / sin(x);
+ d = pow(x, -1) - x / 3 - pow(x, 3) / 45 - pow(x, 5) * 2/945 + Order(pow(x, 6));
+ result += check_series(e, exZERO(), d);
+
+ e = pow(numeric(2), x);
+ ex t = log(ex(2)) * x;
+ d = 1 + t + pow(t, 2) / 2 + pow(t, 3) / 6 + pow(t, 4) / 24 + pow(t, 5) / 120 + pow(t, 6) / 720 + pow(t, 7) / 5040 + Order(pow(x, 8));
+ result += check_series(e, exZERO(), d.expand());
+
+ e = pow(Pi, x);
+ t = log(Pi) * x;
+ d = 1 + t + pow(t, 2) / 2 + pow(t, 3) / 6 + pow(t, 4) / 24 + pow(t, 5) / 120 + pow(t, 6) / 720 + pow(t, 7) / 5040 + Order(pow(x, 8));
+ result += check_series(e, exZERO(), d.expand());
+
+ return result;
}
// Series addition
static unsigned series2(void)
{
- unsigned result = 0;
- ex e, d;
-
- e = pow(sin(x), -1).series(x, exZERO(), 8) + pow(sin(-x), -1).series(x, exZERO(), 12);
- d = Order(pow(x, 6));
- result += check_series(e, exZERO(), d);
-
- return result;
+ unsigned result = 0;
+ ex e, d;
+
+ e = pow(sin(x), -1).series(x, exZERO(), 8) + pow(sin(-x), -1).series(x, exZERO(), 12);
+ d = Order(pow(x, 6));
+ result += check_series(e, exZERO(), d);
+
+ return result;
}
// Series multiplication
static unsigned series3(void)
{
- unsigned result = 0;
- ex e, d;
-
- e = sin(x).series(x, exZERO(), 8) * pow(sin(x), -1).series(x, exZERO(), 12);
- d = 1 + Order(pow(x, 7));
- result += check_series(e, exZERO(), d);
+ unsigned result = 0;
+ ex e, d;
+
+ e = sin(x).series(x, exZERO(), 8) * pow(sin(x), -1).series(x, exZERO(), 12);
+ d = 1 + Order(pow(x, 7));
+ result += check_series(e, exZERO(), d);
+
+ return result;
+}
- return result;
+// Series of special functions
+static unsigned series4(void)
+{
+ unsigned result = 0;
+ ex e, d;
+
+ e = gamma(2*x);
+ d = pow(x+1,-1)*numeric(1,4) +
+ pow(x+1,0)*(numeric(3,4) -
+ numeric(1,2)*EulerGamma) +
+ pow(x+1,1)*(numeric(7,4) -
+ numeric(3,2)*EulerGamma +
+ numeric(1,2)*pow(EulerGamma,2) +
+ numeric(1,12)*pow(Pi,2)) +
+ pow(x+1,2)*(numeric(15,4) -
+ numeric(7,2)*EulerGamma -
+ numeric(1,3)*pow(EulerGamma,3) +
+ numeric(1,4)*pow(Pi,2) +
+ numeric(3,2)*pow(EulerGamma,2) -
+ numeric(1,6)*pow(Pi,2)*EulerGamma -
+ numeric(2,3)*zeta(3)) +
+ pow(x+1,3)*(numeric(31,4) - pow(EulerGamma,3) -
+ numeric(15,2)*EulerGamma +
+ numeric(1,6)*pow(EulerGamma,4) +
+ numeric(7,2)*pow(EulerGamma,2) +
+ numeric(7,12)*pow(Pi,2) -
+ numeric(1,2)*pow(Pi,2)*EulerGamma -
+ numeric(2)*zeta(3) +
+ numeric(1,6)*pow(EulerGamma,2)*pow(Pi,2) +
+ numeric(1,40)*pow(Pi,4) +
+ numeric(4,3)*zeta(3)*EulerGamma) +
+ Order(pow(x+1,4));
+ result += check_series(e, -1, d, 4);
+
+ e = tan(x*Pi/2);
+ d = pow(x-1,-1)/Pi*(-2) +
+ pow(x-1,1)*Pi/6 +
+ pow(x-1,3)*pow(Pi,3)/360 +
+ pow(x-1,5)*pow(Pi,5)/15120 +
+ pow(x-1,7)*pow(Pi,7)/604800 +
+ Order(pow(x-1,8));
+ result += check_series(e,1,d,8);
+
+ return result;
}
unsigned series_expansion(void)
{
- unsigned result = 0;
-
- cout << "checking series expansion..." << flush;
- clog << "---------series expansion:" << endl;
-
- result += series1();
- result += series2();
- result += series3();
-
- if (!result) {
- cout << " passed ";
- clog << "(no output)" << endl;
- } else {
- cout << " failed ";
- }
- return result;
+ unsigned result = 0;
+
+ cout << "checking series expansion..." << flush;
+ clog << "---------series expansion:" << endl;
+
+ result += series1();
+ result += series2();
+ result += series3();
+ result += series4();
+
+ if (!result) {
+ cout << " passed ";
+ clog << "(no output)" << endl;
+ } else {
+ cout << " failed ";
+ }
+ return result;
}