Fixed bug in series expansion of log.
[ginac.git] / ginac / inifcns_trans.cpp
index 239b1ea2ea1d8a3130130a9e32ee77d28458cf3b..872308b93f54e234a99ebf493a4416b4d6b7f1a8 100644 (file)
@@ -4,7 +4,7 @@
  *  functions. */
 
 /*
- *  GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
  *
  *  This program is free software; you can redistribute it and/or modify
  *  it under the terms of the GNU General Public License as published by
@@ -55,6 +55,7 @@ static ex exp_eval(const ex & x)
        if (x.is_zero()) {
                return _ex1;
        }
+
        // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
        const ex TwoExOverPiI=(_ex2*x)/(Pi*I);
        if (TwoExOverPiI.info(info_flags::integer)) {
@@ -68,11 +69,12 @@ static ex exp_eval(const ex & x)
                if (z.is_equal(_num3))
                        return ex(-I);
        }
+
        // exp(log(x)) -> x
        if (is_ex_the_function(x, log))
                return x.op(0);
        
-       // exp(float)
+       // exp(float) -> float
        if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
                return exp(ex_to<numeric>(x));
        
@@ -117,13 +119,18 @@ static ex log_eval(const ex & x)
                        return (Pi*I*_num1_2);
                if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
                        return (Pi*I*_num_1_2);
-               // log(float)
+
+               // log(float) -> float
                if (!x.info(info_flags::crational))
                        return log(ex_to<numeric>(x));
        }
+
        // log(exp(t)) -> t (if -Pi < t.imag() <= Pi):
        if (is_ex_the_function(x, exp)) {
                const ex &t = x.op(0);
+               if (is_a<symbol>(t) && t.info(info_flags::real)) {
+                       return t;
+               }
                if (t.info(info_flags::numeric)) {
                        const numeric &nt = ex_to<numeric>(t);
                        if (nt.is_real())
@@ -194,6 +201,22 @@ static ex log_series(const ex &arg,
                        // in this case n more (or less) terms are needed
                        // (sadly, to generate them, we have to start from the beginning)
                        const ex newarg = ex_to<pseries>((arg/coeff).series(rel, order+n, options)).shift_exponents(-n).convert_to_poly(true);
+                       if (n == 0 && coeff == 1) {
+                               epvector epv;
+                               ex acc = (new pseries(rel, epv))->setflag(status_flags::dynallocated);
+                               epv.reserve(2);
+                               epv.push_back(expair(-1, _ex0));
+                               epv.push_back(expair(Order(_ex1), order));
+                               ex rest = pseries(rel, epv).add_series(argser);
+                               for (int i = order-1; i>0; --i) {
+                                       epvector cterm;
+                                       cterm.reserve(1);
+                                       cterm.push_back(expair(i%2 ? _ex1/i : _ex_1/i, _ex0));
+                                       acc = pseries(rel, cterm).add_series(ex_to<pseries>(acc));
+                                       acc = (ex_to<pseries>(rest)).mul_series(ex_to<pseries>(acc));
+                               }
+                               return acc;
+                       }
                        return pseries(rel, seq).add_series(ex_to<pseries>(log(newarg).series(rel, order, options)));
                } else  // it was a monomial
                        return pseries(rel, seq);
@@ -268,15 +291,18 @@ static ex sin_eval(const ex & x)
                if (z.is_equal(_num30)) // sin(Pi/2)    -> 1
                        return sign;
        }
-       
+
        if (is_exactly_a<function>(x)) {
                const ex &t = x.op(0);
+
                // sin(asin(x)) -> x
                if (is_ex_the_function(x, asin))
                        return t;
+
                // sin(acos(x)) -> sqrt(1-x^2)
                if (is_ex_the_function(x, acos))
                        return sqrt(_ex1-power(t,_ex2));
+
                // sin(atan(x)) -> x/sqrt(1+x^2)
                if (is_ex_the_function(x, atan))
                        return t*power(_ex1+power(t,_ex2),_ex_1_2);
@@ -285,6 +311,10 @@ static ex sin_eval(const ex & x)
        // sin(float) -> float
        if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
                return sin(ex_to<numeric>(x));
+
+       // sin() is odd
+       if (x.info(info_flags::negative))
+               return -sin(-x);
        
        return sin(x).hold();
 }
@@ -349,15 +379,18 @@ static ex cos_eval(const ex & x)
                if (z.is_equal(_num30)) // cos(Pi/2)    -> 0
                        return _ex0;
        }
-       
+
        if (is_exactly_a<function>(x)) {
                const ex &t = x.op(0);
+
                // cos(acos(x)) -> x
                if (is_ex_the_function(x, acos))
                        return t;
+
                // cos(asin(x)) -> sqrt(1-x^2)
                if (is_ex_the_function(x, asin))
                        return sqrt(_ex1-power(t,_ex2));
+
                // cos(atan(x)) -> 1/sqrt(1+x^2)
                if (is_ex_the_function(x, atan))
                        return power(_ex1+power(t,_ex2),_ex_1_2);
@@ -367,6 +400,10 @@ static ex cos_eval(const ex & x)
        if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
                return cos(ex_to<numeric>(x));
        
+       // cos() is even
+       if (x.info(info_flags::negative))
+               return cos(-x);
+       
        return cos(x).hold();
 }
 
@@ -426,15 +463,18 @@ static ex tan_eval(const ex & x)
                if (z.is_equal(_num30)) // tan(Pi/2)    -> infinity
                        throw (pole_error("tan_eval(): simple pole",1));
        }
-       
+
        if (is_exactly_a<function>(x)) {
                const ex &t = x.op(0);
+
                // tan(atan(x)) -> x
                if (is_ex_the_function(x, atan))
                        return t;
+
                // tan(asin(x)) -> x/sqrt(1+x^2)
                if (is_ex_the_function(x, asin))
                        return t*power(_ex1-power(t,_ex2),_ex_1_2);
+
                // tan(acos(x)) -> sqrt(1-x^2)/x
                if (is_ex_the_function(x, acos))
                        return power(t,_ex_1)*sqrt(_ex1-power(t,_ex2));
@@ -445,6 +485,10 @@ static ex tan_eval(const ex & x)
                return tan(ex_to<numeric>(x));
        }
        
+       // tan() is odd
+       if (x.info(info_flags::negative))
+               return -tan(-x);
+       
        return tan(x).hold();
 }
 
@@ -469,7 +513,7 @@ static ex tan_series(const ex &x,
        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(rel, order+2, options);
+       return (sin(x)/cos(x)).series(rel, order, options);
 }
 
 REGISTER_FUNCTION(tan, eval_func(tan_eval).
@@ -493,24 +537,34 @@ static ex asin_evalf(const ex & x)
 static ex asin_eval(const ex & x)
 {
        if (x.info(info_flags::numeric)) {
+
                // asin(0) -> 0
                if (x.is_zero())
                        return x;
+
                // asin(1/2) -> Pi/6
                if (x.is_equal(_ex1_2))
                        return numeric(1,6)*Pi;
+
                // asin(1) -> Pi/2
                if (x.is_equal(_ex1))
                        return _num1_2*Pi;
+
                // asin(-1/2) -> -Pi/6
                if (x.is_equal(_ex_1_2))
                        return numeric(-1,6)*Pi;
+
                // asin(-1) -> -Pi/2
                if (x.is_equal(_ex_1))
                        return _num_1_2*Pi;
+
                // asin(float) -> float
                if (!x.info(info_flags::crational))
                        return asin(ex_to<numeric>(x));
+
+               // asin() is odd
+               if (x.info(info_flags::negative))
+                       return -asin(-x);
        }
        
        return asin(x).hold();
@@ -544,24 +598,34 @@ static ex acos_evalf(const ex & x)
 static ex acos_eval(const ex & x)
 {
        if (x.info(info_flags::numeric)) {
+
                // acos(1) -> 0
                if (x.is_equal(_ex1))
                        return _ex0;
+
                // acos(1/2) -> Pi/3
                if (x.is_equal(_ex1_2))
                        return _ex1_3*Pi;
+
                // acos(0) -> Pi/2
                if (x.is_zero())
                        return _ex1_2*Pi;
+
                // acos(-1/2) -> 2/3*Pi
                if (x.is_equal(_ex_1_2))
                        return numeric(2,3)*Pi;
+
                // acos(-1) -> Pi
                if (x.is_equal(_ex_1))
                        return Pi;
+
                // acos(float) -> float
                if (!x.info(info_flags::crational))
                        return acos(ex_to<numeric>(x));
+
+               // acos(-x) -> Pi-acos(x)
+               if (x.info(info_flags::negative))
+                       return Pi-acos(-x);
        }
        
        return acos(x).hold();
@@ -595,20 +659,29 @@ static ex atan_evalf(const ex & x)
 static ex atan_eval(const ex & x)
 {
        if (x.info(info_flags::numeric)) {
+
                // atan(0) -> 0
                if (x.is_zero())
                        return _ex0;
+
                // atan(1) -> Pi/4
                if (x.is_equal(_ex1))
                        return _ex1_4*Pi;
+
                // atan(-1) -> -Pi/4
                if (x.is_equal(_ex_1))
                        return _ex_1_4*Pi;
+
                if (x.is_equal(I) || x.is_equal(-I))
                        throw (pole_error("atan_eval(): logarithmic pole",0));
+
                // atan(float) -> float
                if (!x.info(info_flags::crational))
                        return atan(ex_to<numeric>(x));
+
+               // atan() is odd
+               if (x.info(info_flags::negative))
+                       return -atan(-x);
        }
        
        return atan(x).hold();
@@ -678,19 +751,79 @@ REGISTER_FUNCTION(atan, eval_func(atan_eval).
 static ex atan2_evalf(const ex &y, const ex &x)
 {
        if (is_exactly_a<numeric>(y) && is_exactly_a<numeric>(x))
-               return atan2(ex_to<numeric>(y), ex_to<numeric>(x));
+               return atan(ex_to<numeric>(y), ex_to<numeric>(x));
        
        return atan2(y, x).hold();
 }
 
 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_evalf(y,x);
+       if (y.info(info_flags::numeric) && x.info(info_flags::numeric)) {
+
+               if (y.is_zero()) {
+
+                       // atan(0, 0) -> 0
+                       if (x.is_zero())
+                               return _ex0;
+
+                       // atan(0, x), x real and positive -> 0
+                       if (x.info(info_flags::positive))
+                               return _ex0;
+
+                       // atan(0, x), x real and negative -> -Pi
+                       if (x.info(info_flags::negative))
+                               return _ex_1*Pi;
+               }
+
+               if (x.is_zero()) {
+
+                       // atan(y, 0), y real and positive -> Pi/2
+                       if (y.info(info_flags::positive))
+                               return _ex1_2*Pi;
+
+                       // atan(y, 0), y real and negative -> -Pi/2
+                       if (y.info(info_flags::negative))
+                               return _ex_1_2*Pi;
+               }
+
+               if (y.is_equal(x)) {
+
+                       // atan(y, y), y real and positive -> Pi/4
+                       if (y.info(info_flags::positive))
+                               return _ex1_4*Pi;
+
+                       // atan(y, y), y real and negative -> -3/4*Pi
+                       if (y.info(info_flags::negative))
+                               return numeric(-3, 4)*Pi;
+               }
+
+               if (y.is_equal(-x)) {
+
+                       // atan(y, -y), y real and positive -> 3*Pi/4
+                       if (y.info(info_flags::positive))
+                               return numeric(3, 4)*Pi;
+
+                       // atan(y, -y), y real and negative -> -Pi/4
+                       if (y.info(info_flags::negative))
+                               return _ex_1_4*Pi;
+               }
+
+               // atan(float, float) -> float
+               if (!y.info(info_flags::crational) && !x.info(info_flags::crational))
+                       return atan(ex_to<numeric>(y), ex_to<numeric>(x));
+
+               // atan(real, real) -> atan(y/x) +/- Pi
+               if (y.info(info_flags::real) && x.info(info_flags::real)) {
+                       if (x.info(info_flags::positive))
+                               return atan(y/x);
+                       else if(y.info(info_flags::positive))
+                               return atan(y/x)+Pi;
+                       else
+                               return atan(y/x)-Pi;
+               }
        }
-       
-       return atan2(y,x).hold();
+
+       return atan2(y, x).hold();
 }    
 
 static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
@@ -724,10 +857,18 @@ static ex sinh_evalf(const ex & x)
 static ex sinh_eval(const ex & x)
 {
        if (x.info(info_flags::numeric)) {
-               if (x.is_zero())  // sinh(0) -> 0
+
+               // sinh(0) -> 0
+               if (x.is_zero())
                        return _ex0;        
-               if (!x.info(info_flags::crational))  // sinh(float) -> float
+
+               // sinh(float) -> float
+               if (!x.info(info_flags::crational))
                        return sinh(ex_to<numeric>(x));
+
+               // sinh() is odd
+               if (x.info(info_flags::negative))
+                       return -sinh(-x);
        }
        
        if ((x/Pi).info(info_flags::numeric) &&
@@ -736,12 +877,15 @@ static ex sinh_eval(const ex & x)
        
        if (is_exactly_a<function>(x)) {
                const ex &t = x.op(0);
+
                // sinh(asinh(x)) -> x
                if (is_ex_the_function(x, asinh))
                        return t;
+
                // sinh(acosh(x)) -> sqrt(x-1) * sqrt(x+1)
                if (is_ex_the_function(x, acosh))
                        return sqrt(t-_ex1)*sqrt(t+_ex1);
+
                // sinh(atanh(x)) -> x/sqrt(1-x^2)
                if (is_ex_the_function(x, atanh))
                        return t*power(_ex1-power(t,_ex2),_ex_1_2);
@@ -778,10 +922,18 @@ static ex cosh_evalf(const ex & x)
 static ex cosh_eval(const ex & x)
 {
        if (x.info(info_flags::numeric)) {
-               if (x.is_zero())  // cosh(0) -> 1
+
+               // cosh(0) -> 1
+               if (x.is_zero())
                        return _ex1;
-               if (!x.info(info_flags::crational))  // cosh(float) -> float
+
+               // cosh(float) -> float
+               if (!x.info(info_flags::crational))
                        return cosh(ex_to<numeric>(x));
+
+               // cosh() is even
+               if (x.info(info_flags::negative))
+                       return cosh(-x);
        }
        
        if ((x/Pi).info(info_flags::numeric) &&
@@ -790,12 +942,15 @@ static ex cosh_eval(const ex & x)
        
        if (is_exactly_a<function>(x)) {
                const ex &t = x.op(0);
+
                // cosh(acosh(x)) -> x
                if (is_ex_the_function(x, acosh))
                        return t;
+
                // cosh(asinh(x)) -> sqrt(1+x^2)
                if (is_ex_the_function(x, asinh))
                        return sqrt(_ex1+power(t,_ex2));
+
                // cosh(atanh(x)) -> 1/sqrt(1-x^2)
                if (is_ex_the_function(x, atanh))
                        return power(_ex1-power(t,_ex2),_ex_1_2);
@@ -832,10 +987,18 @@ static ex tanh_evalf(const ex & x)
 static ex tanh_eval(const ex & x)
 {
        if (x.info(info_flags::numeric)) {
-               if (x.is_zero())  // tanh(0) -> 0
+
+               // tanh(0) -> 0
+               if (x.is_zero())
                        return _ex0;
-               if (!x.info(info_flags::crational))  // tanh(float) -> float
+
+               // tanh(float) -> float
+               if (!x.info(info_flags::crational))
                        return tanh(ex_to<numeric>(x));
+
+               // tanh() is odd
+               if (x.info(info_flags::negative))
+                       return -tanh(-x);
        }
        
        if ((x/Pi).info(info_flags::numeric) &&
@@ -844,12 +1007,15 @@ static ex tanh_eval(const ex & x)
        
        if (is_exactly_a<function>(x)) {
                const ex &t = x.op(0);
+
                // tanh(atanh(x)) -> x
                if (is_ex_the_function(x, atanh))
                        return t;
+
                // tanh(asinh(x)) -> x/sqrt(1+x^2)
                if (is_ex_the_function(x, asinh))
                        return t*power(_ex1+power(t,_ex2),_ex_1_2);
+
                // tanh(acosh(x)) -> sqrt(x-1)*sqrt(x+1)/x
                if (is_ex_the_function(x, acosh))
                        return sqrt(t-_ex1)*sqrt(t+_ex1)*power(t,_ex_1);
@@ -879,7 +1045,7 @@ static ex tanh_series(const ex &x,
        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(rel, order+2, options);
+       return (sinh(x)/cosh(x)).series(rel, order, options);
 }
 
 REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
@@ -903,12 +1069,18 @@ static ex asinh_evalf(const ex & x)
 static ex asinh_eval(const ex & x)
 {
        if (x.info(info_flags::numeric)) {
+
                // asinh(0) -> 0
                if (x.is_zero())
                        return _ex0;
+
                // asinh(float) -> float
                if (!x.info(info_flags::crational))
                        return asinh(ex_to<numeric>(x));
+
+               // asinh() is odd
+               if (x.info(info_flags::negative))
+                       return -asinh(-x);
        }
        
        return asinh(x).hold();
@@ -941,18 +1113,26 @@ static ex acosh_evalf(const ex & x)
 static ex acosh_eval(const ex & x)
 {
        if (x.info(info_flags::numeric)) {
+
                // acosh(0) -> Pi*I/2
                if (x.is_zero())
                        return Pi*I*numeric(1,2);
+
                // acosh(1) -> 0
                if (x.is_equal(_ex1))
                        return _ex0;
+
                // acosh(-1) -> Pi*I
                if (x.is_equal(_ex_1))
                        return Pi*I;
+
                // acosh(float) -> float
                if (!x.info(info_flags::crational))
                        return acosh(ex_to<numeric>(x));
+
+               // acosh(-x) -> Pi*I-acosh(x)
+               if (x.info(info_flags::negative))
+                       return Pi*I-acosh(-x);
        }
        
        return acosh(x).hold();
@@ -985,15 +1165,22 @@ static ex atanh_evalf(const ex & x)
 static ex atanh_eval(const ex & x)
 {
        if (x.info(info_flags::numeric)) {
+
                // atanh(0) -> 0
                if (x.is_zero())
                        return _ex0;
+
                // atanh({+|-}1) -> throw
                if (x.is_equal(_ex1) || x.is_equal(_ex_1))
                        throw (pole_error("atanh_eval(): logarithmic pole",0));
+
                // atanh(float) -> float
                if (!x.info(info_flags::crational))
                        return atanh(ex_to<numeric>(x));
+
+               // atanh() is odd
+               if (x.info(info_flags::negative))
+                       return -atanh(-x);
        }
        
        return atanh(x).hold();