- expairseq.cpp: moved expairseq::to_rational to...
[ginac.git] / ginac / inifcns_trans.cpp
index e26050a6453d93b45a7128267b0a8dbd9ff7436a..81f92a93aca908aa6f698e79dd1a4380d5998d61 100644 (file)
@@ -4,7 +4,7 @@
  *  functions. */
 
 /*
- *  GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2000 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
 #include "constant.h"
 #include "numeric.h"
 #include "power.h"
+#include "relational.h"
+#include "symbol.h"
+#include "pseries.h"
+#include "utils.h"
 
+#ifndef NO_NAMESPACE_GINAC
 namespace GiNaC {
+#endif // ndef NO_NAMESPACE_GINAC
 
 //////////
 // exponential function
 //////////
 
-ex exp_evalf(ex const & x)
+static ex exp_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
         TYPECHECK(x,numeric)
@@ -45,23 +51,23 @@ ex exp_evalf(ex const & x)
     return exp(ex_to_numeric(x)); // -> numeric exp(numeric)
 }
 
-ex exp_eval(ex const & x)
+static ex exp_eval(const ex & x)
 {
     // exp(0) -> 1
     if (x.is_zero()) {
-        return exONE();
+        return _ex1();
     }
     // exp(n*Pi*I/2) -> {+1|+I|-1|-I}
-    ex TwoExOverPiI=(2*x)/(Pi*I);
+    ex TwoExOverPiI=(_ex2()*x)/(Pi*I);
     if (TwoExOverPiI.info(info_flags::integer)) {
-        numeric z=mod(ex_to_numeric(TwoExOverPiI),numeric(4));
-        if (z.is_equal(numZERO()))
-            return exONE();
-        if (z.is_equal(numONE()))
+        numeric z=mod(ex_to_numeric(TwoExOverPiI),_num4());
+        if (z.is_equal(_num0()))
+            return _ex1();
+        if (z.is_equal(_num1()))
             return ex(I);
-        if (z.is_equal(numTWO()))
-            return exMINUSONE();
-        if (z.is_equal(numTHREE()))
+        if (z.is_equal(_num2()))
+            return _ex_1();
+        if (z.is_equal(_num3()))
             return ex(-I);
     }
     // exp(log(x)) -> x
@@ -69,26 +75,29 @@ ex exp_eval(ex const & x)
         return x.op(0);
     
     // exp(float)
-    if (x.info(info_flags::numeric) && !x.info(info_flags::rational))
+    if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
         return exp_evalf(x);
     
     return exp(x).hold();
-}    
+}
 
-ex exp_diff(ex const & x, unsigned diff_param)
+static ex exp_deriv(const ex & x, unsigned deriv_param)
 {
-    ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
 
+    // d/dx exp(x) -> exp(x)
     return exp(x);
 }
 
-REGISTER_FUNCTION(exp, exp_eval, exp_evalf, exp_diff, NULL);
+REGISTER_FUNCTION(exp, eval_func(exp_eval).
+                       evalf_func(exp_evalf).
+                       derivative_func(exp_deriv));
 
 //////////
 // natural logarithm
 //////////
 
-ex log_evalf(ex const & x)
+static ex log_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
         TYPECHECK(x,numeric)
@@ -97,46 +106,76 @@ ex log_evalf(ex const & x)
     return log(ex_to_numeric(x)); // -> numeric log(numeric)
 }
 
-ex log_eval(ex const & x)
+static ex log_eval(const ex & x)
 {
     if (x.info(info_flags::numeric)) {
-        // log(1) -> 0
-        if (x.is_equal(exONE()))
-            return exZERO();
-        // log(-1) -> I*Pi
-        if (x.is_equal(exMINUSONE()))
-            return (I*Pi);
-        // log(I) -> Pi*I/2
-        if (x.is_equal(I))
-            return (I*Pi*numeric(1,2));
-        // log(-I) -> -Pi*I/2
-        if (x.is_equal(-I))
-            return (I*Pi*numeric(-1,2));
-        // log(0) -> throw singularity
-        if (x.is_equal(exZERO()))
+        if (x.is_equal(_ex0()))  // log(0) -> infinity
             throw(std::domain_error("log_eval(): log(0)"));
+        if (x.info(info_flags::real) && x.info(info_flags::negative))
+            return (log(-x)+I*Pi);
+        if (x.is_equal(_ex1()))  // log(1) -> 0
+            return _ex0();
+        if (x.is_equal(I))       // log(I) -> Pi*I/2
+            return (Pi*I*_num1_2());
+        if (x.is_equal(-I))      // log(-I) -> -Pi*I/2
+            return (Pi*I*_num_1_2());
         // log(float)
-        if (!x.info(info_flags::rational))
+        if (!x.info(info_flags::crational))
             return log_evalf(x);
     }
+    // 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::numeric)) {
+            numeric nt = ex_to_numeric(t);
+            if (nt.is_real())
+                return t;
+        }
+    }
     
     return log(x).hold();
-}    
+}
 
-ex log_diff(ex const & x, unsigned diff_param)
+static ex log_deriv(const ex & x, unsigned deriv_param)
 {
-    ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
+    
+    // d/dx log(x) -> 1/x
+    return power(x, _ex_1());
+}
 
-    return power(x, -1);
+static ex log_series(const ex &x, const relational &rel, int order)
+{
+    const ex x_pt = x.subs(rel);
+    if (!x_pt.info(info_flags::negative) && !x_pt.is_zero())
+        throw do_taylor();  // caught by function::series()
+    // now we either have to care for the branch cut or the branch point:
+    if (x_pt.is_zero()) {  // branch point: return a plain log(x).
+        epvector seq;
+        seq.push_back(expair(log(x), _ex0()));
+        return pseries(rel, seq);
+    } // on the branch cut:
+    const ex point = rel.rhs();
+    const symbol *s = static_cast<symbol *>(rel.lhs().bp);
+    const symbol foo;
+    // compute the formal series:
+    ex replx = series(log(x),*s==foo,order).subs(foo==point);
+    epvector seq;
+    seq.push_back(expair(-I*csgn(x*I)*Pi,_ex0()));
+    seq.push_back(expair(Order(_ex1()),order));
+    return series(replx - I*Pi + pseries(rel, seq),rel,order);
 }
 
-REGISTER_FUNCTION(log, log_eval, log_evalf, log_diff, NULL);
+REGISTER_FUNCTION(log, eval_func(log_eval).
+                       evalf_func(log_evalf).
+                       derivative_func(log_deriv).
+                       series_func(log_series));
 
 //////////
 // sine (trigonometric function)
 //////////
 
-ex sin_evalf(ex const & x)
+static ex sin_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -145,54 +184,79 @@ ex sin_evalf(ex const & x)
     return sin(ex_to_numeric(x)); // -> numeric sin(numeric)
 }
 
-ex sin_eval(ex const & x)
-{
-    // sin(n*Pi) -> 0
-    ex xOverPi=x/Pi;
-    if (xOverPi.info(info_flags::integer))
-        return exZERO();
-    
-    // sin((2n+1)*Pi/2) -> {+|-}1
-    ex xOverPiMinusHalf=xOverPi-exHALF();
-    if (xOverPiMinusHalf.info(info_flags::even))
-        return exONE();
-    else if (xOverPiMinusHalf.info(info_flags::odd))
-        return exMINUSONE();
+static ex sin_eval(const ex & x)
+{
+    // sin(n/d*Pi) -> { all known non-nested radicals }
+    ex SixtyExOverPi = _ex60()*x/Pi;
+    ex sign = _ex1();
+    if (SixtyExOverPi.info(info_flags::integer)) {
+        numeric z = mod(ex_to_numeric(SixtyExOverPi),_num120());
+        if (z>=_num60()) {
+            // wrap to interval [0, Pi)
+            z -= _num60();
+            sign = _ex_1();
+        }
+        if (z>_num30()) {
+            // wrap to interval [0, Pi/2)
+            z = _num60()-z;
+        }
+        if (z.is_equal(_num0()))  // sin(0)       -> 0
+            return _ex0();
+        if (z.is_equal(_num5()))  // sin(Pi/12)   -> sqrt(6)/4*(1-sqrt(3)/3)
+            return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex_1_3()*power(_ex3(),_ex1_2()));
+        if (z.is_equal(_num6()))  // sin(Pi/10)   -> sqrt(5)/4-1/4
+            return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex_1_4());
+        if (z.is_equal(_num10())) // sin(Pi/6)    -> 1/2
+            return sign*_ex1_2();
+        if (z.is_equal(_num15())) // sin(Pi/4)    -> sqrt(2)/2
+            return sign*_ex1_2()*power(_ex2(),_ex1_2());
+        if (z.is_equal(_num18())) // sin(3/10*Pi) -> sqrt(5)/4+1/4
+            return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex1_4());
+        if (z.is_equal(_num20())) // sin(Pi/3)    -> sqrt(3)/2
+            return sign*_ex1_2()*power(_ex3(),_ex1_2());
+        if (z.is_equal(_num25())) // sin(5/12*Pi) -> sqrt(6)/4*(1+sqrt(3)/3)
+            return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex1_3()*power(_ex3(),_ex1_2()));
+        if (z.is_equal(_num30())) // sin(Pi/2)    -> 1
+            return sign*_ex1();
+    }
     
     if (is_ex_exactly_of_type(x, function)) {
-        ex t=x.op(0);
+        ex t = x.op(0);
         // sin(asin(x)) -> x
         if (is_ex_the_function(x, asin))
             return t;
-        // sin(acos(x)) -> (1-x^2)^(1/2)
+        // sin(acos(x)) -> sqrt(1-x^2)
         if (is_ex_the_function(x, acos))
-            return power(exONE()-power(t,exTWO()),exHALF());
+            return power(_ex1()-power(t,_ex2()),_ex1_2());
         // sin(atan(x)) -> x*(1+x^2)^(-1/2)
         if (is_ex_the_function(x, atan))
-            return t*power(exONE()+power(t,exTWO()),exMINUSHALF());
+            return t*power(_ex1()+power(t,_ex2()),_ex_1_2());
     }
     
     // sin(float) -> float
-    if (x.info(info_flags::numeric) && !x.info(info_flags::rational))
+    if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
         return sin_evalf(x);
     
     return sin(x).hold();
 }
 
-ex sin_diff(ex const & x, unsigned diff_param)
+static ex sin_deriv(const ex & x, unsigned deriv_param)
 {
-    ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
+    // d/dx sin(x) -> cos(x)
     return cos(x);
 }
 
-REGISTER_FUNCTION(sin, sin_eval, sin_evalf, sin_diff, NULL);
+REGISTER_FUNCTION(sin, eval_func(sin_eval).
+                       evalf_func(sin_evalf).
+                       derivative_func(sin_deriv));
 
 //////////
 // cosine (trigonometric function)
 //////////
 
-ex cos_evalf(ex const & x)
+static ex cos_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
         TYPECHECK(x,numeric)
@@ -201,54 +265,79 @@ ex cos_evalf(ex const & x)
     return cos(ex_to_numeric(x)); // -> numeric cos(numeric)
 }
 
-ex cos_eval(ex const & x)
-{
-    // cos(n*Pi) -> {+|-}1
-    ex xOverPi=x/Pi;
-    if (xOverPi.info(info_flags::even))
-        return exONE();
-    else if (xOverPi.info(info_flags::odd))
-        return exMINUSONE();
-    
-    // cos((2n+1)*Pi/2) -> 0
-    ex xOverPiMinusHalf=xOverPi-exHALF();
-    if (xOverPiMinusHalf.info(info_flags::integer))
-        return exZERO();
+static ex cos_eval(const ex & x)
+{
+    // cos(n/d*Pi) -> { all known non-nested radicals }
+    ex SixtyExOverPi = _ex60()*x/Pi;
+    ex sign = _ex1();
+    if (SixtyExOverPi.info(info_flags::integer)) {
+        numeric z = mod(ex_to_numeric(SixtyExOverPi),_num120());
+        if (z>=_num60()) {
+            // wrap to interval [0, Pi)
+            z = _num120()-z;
+        }
+        if (z>=_num30()) {
+            // wrap to interval [0, Pi/2)
+            z = _num60()-z;
+            sign = _ex_1();
+        }
+        if (z.is_equal(_num0()))  // cos(0)       -> 1
+            return sign*_ex1();
+        if (z.is_equal(_num5()))  // cos(Pi/12)   -> sqrt(6)/4*(1+sqrt(3)/3)
+            return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex1_3()*power(_ex3(),_ex1_2()));
+        if (z.is_equal(_num10())) // cos(Pi/6)    -> sqrt(3)/2
+            return sign*_ex1_2()*power(_ex3(),_ex1_2());
+        if (z.is_equal(_num12())) // cos(Pi/5)    -> sqrt(5)/4+1/4
+            return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex1_4());
+        if (z.is_equal(_num15())) // cos(Pi/4)    -> sqrt(2)/2
+            return sign*_ex1_2()*power(_ex2(),_ex1_2());
+        if (z.is_equal(_num20())) // cos(Pi/3)    -> 1/2
+            return sign*_ex1_2();
+        if (z.is_equal(_num24())) // cos(2/5*Pi)  -> sqrt(5)/4-1/4x
+            return sign*(_ex1_4()*power(_ex5(),_ex1_2())+_ex_1_4());
+        if (z.is_equal(_num25())) // cos(5/12*Pi) -> sqrt(6)/4*(1-sqrt(3)/3)
+            return sign*_ex1_4()*power(_ex6(),_ex1_2())*(_ex1()+_ex_1_3()*power(_ex3(),_ex1_2()));
+        if (z.is_equal(_num30())) // cos(Pi/2)    -> 0
+            return sign*_ex0();
+    }
     
     if (is_ex_exactly_of_type(x, function)) {
-        ex t=x.op(0);
+        ex t = x.op(0);
         // cos(acos(x)) -> x
         if (is_ex_the_function(x, acos))
             return t;
         // cos(asin(x)) -> (1-x^2)^(1/2)
         if (is_ex_the_function(x, asin))
-            return power(exONE()-power(t,exTWO()),exHALF());
+            return power(_ex1()-power(t,_ex2()),_ex1_2());
         // cos(atan(x)) -> (1+x^2)^(-1/2)
         if (is_ex_the_function(x, atan))
-            return power(exONE()+power(t,exTWO()),exMINUSHALF());
+            return power(_ex1()+power(t,_ex2()),_ex_1_2());
     }
     
     // cos(float) -> float
-    if (x.info(info_flags::numeric) && !x.info(info_flags::rational))
+    if (x.info(info_flags::numeric) && !x.info(info_flags::crational))
         return cos_evalf(x);
     
     return cos(x).hold();
 }
 
-ex cos_diff(ex const & x, unsigned diff_param)
+static ex cos_deriv(const ex & x, unsigned deriv_param)
 {
-    ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
 
-    return numMINUSONE()*sin(x);
+    // d/dx cos(x) -> -sin(x)
+    return _ex_1()*sin(x);
 }
 
-REGISTER_FUNCTION(cos, cos_eval, cos_evalf, cos_diff, NULL);
+REGISTER_FUNCTION(cos, eval_func(cos_eval).
+                       evalf_func(cos_evalf).
+                       derivative_func(cos_deriv));
 
 //////////
 // tangent (trigonometric function)
 //////////
 
-ex tan_evalf(ex const & x)
+static ex tan_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -257,60 +346,89 @@ ex tan_evalf(ex const & x)
     return tan(ex_to_numeric(x));
 }
 
-ex tan_eval(ex const & x)
-{
-    // tan(n*Pi/3) -> {0|3^(1/2)|-(3^(1/2))}
-    ex ThreeExOverPi=numTHREE()*x/Pi;
-    if (ThreeExOverPi.info(info_flags::integer)) {
-        numeric z=mod(ex_to_numeric(ThreeExOverPi),numeric(3));
-        if (z.is_equal(numZERO()))
-            return exZERO();
-        if (z.is_equal(numONE()))
-            return power(exTHREE(),exHALF());
-        if (z.is_equal(numTWO()))
-            return -power(exTHREE(),exHALF());
+static ex tan_eval(const ex & x)
+{
+    // tan(n/d*Pi) -> { all known non-nested radicals }
+    ex SixtyExOverPi = _ex60()*x/Pi;
+    ex sign = _ex1();
+    if (SixtyExOverPi.info(info_flags::integer)) {
+        numeric z = mod(ex_to_numeric(SixtyExOverPi),_num60());
+        if (z>=_num60()) {
+            // wrap to interval [0, Pi)
+            z -= _num60();
+        }
+        if (z>=_num30()) {
+            // wrap to interval [0, Pi/2)
+            z = _num60()-z;
+            sign = _ex_1();
+        }
+        if (z.is_equal(_num0()))  // tan(0)       -> 0
+            return _ex0();
+        if (z.is_equal(_num5()))  // tan(Pi/12)   -> 2-sqrt(3)
+            return sign*(_ex2()-power(_ex3(),_ex1_2()));
+        if (z.is_equal(_num10())) // tan(Pi/6)    -> sqrt(3)/3
+            return sign*_ex1_3()*power(_ex3(),_ex1_2());
+        if (z.is_equal(_num15())) // tan(Pi/4)    -> 1
+            return sign*_ex1();
+        if (z.is_equal(_num20())) // tan(Pi/3)    -> sqrt(3)
+            return sign*power(_ex3(),_ex1_2());
+        if (z.is_equal(_num25())) // tan(5/12*Pi) -> 2+sqrt(3)
+            return sign*(power(_ex3(),_ex1_2())+_ex2());
+        if (z.is_equal(_num30())) // tan(Pi/2)    -> infinity
+            throw (std::domain_error("tan_eval(): simple pole"));
     }
     
-    // tan((2n+1)*Pi/2) -> throw
-    ex ExOverPiMinusHalf=x/Pi-exHALF();
-    if (ExOverPiMinusHalf.info(info_flags::integer))
-        throw (std::domain_error("tan_eval(): infinity"));
-    
     if (is_ex_exactly_of_type(x, function)) {
-        ex t=x.op(0);
+        ex t = x.op(0);
         // tan(atan(x)) -> x
         if (is_ex_the_function(x, atan))
             return t;
         // tan(asin(x)) -> x*(1+x^2)^(-1/2)
         if (is_ex_the_function(x, asin))
-            return t*power(exONE()-power(t,exTWO()),exMINUSHALF());
+            return t*power(_ex1()-power(t,_ex2()),_ex_1_2());
         // tan(acos(x)) -> (1-x^2)^(1/2)/x
         if (is_ex_the_function(x, acos))
-            return power(t,exMINUSONE())*power(exONE()-power(t,exTWO()),exHALF());
+            return power(t,_ex_1())*power(_ex1()-power(t,_ex2()),_ex1_2());
     }
     
     // tan(float) -> float
-    if (x.info(info_flags::numeric) && !x.info(info_flags::rational)) {
+    if (x.info(info_flags::numeric) && !x.info(info_flags::crational)) {
         return tan_evalf(x);
     }
     
     return tan(x).hold();
 }
 
-ex tan_diff(ex const & x, unsigned diff_param)
+static ex tan_deriv(const ex & x, unsigned deriv_param)
 {
-    ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
-    return (1+power(tan(x),exTWO()));
+    // d/dx tan(x) -> 1+tan(x)^2;
+    return (_ex1()+power(tan(x),_ex2()));
 }
 
-REGISTER_FUNCTION(tan, tan_eval, tan_evalf, tan_diff, NULL);
+static ex tan_series(const ex &x, const relational &rel, int order)
+{
+    // method:
+    // Taylor series where there is no pole falls back to tan_deriv.
+    // On a pole simply expand sin(x)/cos(x).
+    const ex x_pt = x.subs(rel);
+    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);
+}
+
+REGISTER_FUNCTION(tan, eval_func(tan_eval).
+                       evalf_func(tan_evalf).
+                       derivative_func(tan_deriv).
+                       series_func(tan_series));
 
 //////////
 // inverse sine (arc sine)
 //////////
 
-ex asin_evalf(ex const & x)
+static ex asin_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -319,46 +437,49 @@ ex asin_evalf(ex const & x)
     return asin(ex_to_numeric(x)); // -> numeric asin(numeric)
 }
 
-ex asin_eval(ex const & 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(exHALF()))
+        if (x.is_equal(_ex1_2()))
             return numeric(1,6)*Pi;
         // asin(1) -> Pi/2
-        if (x.is_equal(exONE()))
-            return numeric(1,2)*Pi;
+        if (x.is_equal(_ex1()))
+            return _num1_2()*Pi;
         // asin(-1/2) -> -Pi/6
-        if (x.is_equal(exMINUSHALF()))
+        if (x.is_equal(_ex_1_2()))
             return numeric(-1,6)*Pi;
         // asin(-1) -> -Pi/2
-        if (x.is_equal(exMINUSONE()))
-            return numeric(-1,2)*Pi;
+        if (x.is_equal(_ex_1()))
+            return _num_1_2()*Pi;
         // asin(float) -> float
-        if (!x.info(info_flags::rational))
+        if (!x.info(info_flags::crational))
             return asin_evalf(x);
     }
     
     return asin(x).hold();
 }
 
-ex asin_diff(ex const & x, unsigned diff_param)
+static ex asin_deriv(const ex & x, unsigned deriv_param)
 {
-    ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
-    return power(1-power(x,exTWO()),exMINUSHALF());
+    // d/dx asin(x) -> 1/sqrt(1-x^2)
+    return power(1-power(x,_ex2()),_ex_1_2());
 }
 
-REGISTER_FUNCTION(asin, asin_eval, asin_evalf, asin_diff, NULL);
+REGISTER_FUNCTION(asin, eval_func(asin_eval).
+                        evalf_func(asin_evalf).
+                        derivative_func(asin_deriv));
 
 //////////
 // inverse cosine (arc cosine)
 //////////
 
-ex acos_evalf(ex const & x)
+static ex acos_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -367,46 +488,49 @@ ex acos_evalf(ex const & x)
     return acos(ex_to_numeric(x)); // -> numeric acos(numeric)
 }
 
-ex acos_eval(ex const & x)
+static ex acos_eval(const ex & x)
 {
     if (x.info(info_flags::numeric)) {
         // acos(1) -> 0
-        if (x.is_equal(exONE()))
-            return exZERO();
+        if (x.is_equal(_ex1()))
+            return _ex0();
         // acos(1/2) -> Pi/3
-        if (x.is_equal(exHALF()))
-            return numeric(1,3)*Pi;
+        if (x.is_equal(_ex1_2()))
+            return _ex1_3()*Pi;
         // acos(0) -> Pi/2
         if (x.is_zero())
-            return numeric(1,2)*Pi;
+            return _ex1_2()*Pi;
         // acos(-1/2) -> 2/3*Pi
-        if (x.is_equal(exMINUSHALF()))
+        if (x.is_equal(_ex_1_2()))
             return numeric(2,3)*Pi;
         // acos(-1) -> Pi
-        if (x.is_equal(exMINUSONE()))
+        if (x.is_equal(_ex_1()))
             return Pi;
         // acos(float) -> float
-        if (!x.info(info_flags::rational))
+        if (!x.info(info_flags::crational))
             return acos_evalf(x);
     }
     
     return acos(x).hold();
 }
 
-ex acos_diff(ex const & x, unsigned diff_param)
+static ex acos_deriv(const ex & x, unsigned deriv_param)
 {
-    ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
-    return numMINUSONE()*power(1-power(x,exTWO()),exMINUSHALF());
+    // d/dx acos(x) -> -1/sqrt(1-x^2)
+    return _ex_1()*power(1-power(x,_ex2()),_ex_1_2());
 }
 
-REGISTER_FUNCTION(acos, acos_eval, acos_evalf, acos_diff, NULL);
+REGISTER_FUNCTION(acos, eval_func(acos_eval).
+                        evalf_func(acos_evalf).
+                        derivative_func(acos_deriv));
 
 //////////
 // inverse tangent (arc tangent)
 //////////
 
-ex atan_evalf(ex const & x)
+static ex atan_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
         TYPECHECK(x,numeric)
@@ -415,34 +539,37 @@ ex atan_evalf(ex const & x)
     return atan(ex_to_numeric(x)); // -> numeric atan(numeric)
 }
 
-ex atan_eval(ex const & x)
+static ex atan_eval(const ex & x)
 {
     if (x.info(info_flags::numeric)) {
         // atan(0) -> 0
-        if (x.is_equal(exZERO()))
-            return exZERO();
+        if (x.is_equal(_ex0()))
+            return _ex0();
         // atan(float) -> float
-        if (!x.info(info_flags::rational))
+        if (!x.info(info_flags::crational))
             return atan_evalf(x);
     }
     
     return atan(x).hold();
 }    
 
-ex atan_diff(ex const & x, unsigned diff_param)
+static ex atan_deriv(const ex & x, unsigned deriv_param)
 {
-    ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
 
-    return power(1+x*x, -1);
+    // d/dx atan(x) -> 1/(1+x^2)
+    return power(_ex1()+power(x,_ex2()), _ex_1());
 }
 
-REGISTER_FUNCTION(atan, atan_eval, atan_evalf, atan_diff, NULL);
+REGISTER_FUNCTION(atan, eval_func(atan_eval).
+                        evalf_func(atan_evalf).
+                        derivative_func(atan_deriv));
 
 //////////
 // inverse tangent (atan2(y,x))
 //////////
 
-ex atan2_evalf(ex const & y, ex const & x)
+static ex atan2_evalf(const ex & y, const ex & x)
 {
     BEGIN_TYPECHECK
         TYPECHECK(y,numeric)
@@ -452,35 +579,37 @@ ex atan2_evalf(ex const & y, ex const & x)
     return atan(ex_to_numeric(y),ex_to_numeric(x)); // -> numeric atan(numeric)
 }
 
-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::rational) &&
-        x.info(info_flags::numeric) && !x.info(info_flags::rational)) {
+    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);
     }
     
     return atan2(y,x).hold();
 }    
 
-ex atan2_diff(ex const & y, ex const & x, unsigned diff_param)
+static ex atan2_deriv(const ex & y, const ex & x, unsigned deriv_param)
 {
-    ASSERT(diff_param<2);
-
-    if (diff_param==0) {
+    GINAC_ASSERT(deriv_param<2);
+    
+    if (deriv_param==0) {
         // d/dy atan(y,x)
-        return power(x*(1+y*y/(x*x)),-1);
+        return x*power(power(x,_ex2())+power(y,_ex2()),_ex_1());
     }
     // d/dx atan(y,x)
-    return -y*power(x*x+y*y,-1);
+    return -y*power(power(x,_ex2())+power(y,_ex2()),_ex_1());
 }
 
-REGISTER_FUNCTION(atan2, atan2_eval, atan2_evalf, atan2_diff, NULL);
+REGISTER_FUNCTION(atan2, eval_func(atan2_eval).
+                         evalf_func(atan2_evalf).
+                         derivative_func(atan2_deriv));
 
 //////////
 // hyperbolic sine (trigonometric function)
 //////////
 
-ex sinh_evalf(ex const & x)
+static ex sinh_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -489,47 +618,52 @@ ex sinh_evalf(ex const & x)
     return sinh(ex_to_numeric(x)); // -> numeric sinh(numeric)
 }
 
-ex sinh_eval(ex const & x)
+static ex sinh_eval(const ex & x)
 {
     if (x.info(info_flags::numeric)) {
-        // sinh(0) -> 0
-        if (x.is_zero())
-            return exZERO();
-        // sinh(float) -> float
-        if (!x.info(info_flags::rational))
+        if (x.is_zero())  // sinh(0) -> 0
+            return _ex0();        
+        if (!x.info(info_flags::crational))  // sinh(float) -> float
             return sinh_evalf(x);
     }
     
+    if ((x/Pi).info(info_flags::numeric) &&
+        ex_to_numeric(x/Pi).real().is_zero())  // sinh(I*x) -> I*sin(x)
+        return I*sin(x/I);
+    
     if (is_ex_exactly_of_type(x, function)) {
-        ex t=x.op(0);
+        ex t = x.op(0);
         // sinh(asinh(x)) -> x
         if (is_ex_the_function(x, asinh))
             return t;
         // sinh(acosh(x)) -> (x-1)^(1/2) * (x+1)^(1/2)
         if (is_ex_the_function(x, acosh))
-            return power(t-exONE(),exHALF())*power(t+exONE(),exHALF());
+            return power(t-_ex1(),_ex1_2())*power(t+_ex1(),_ex1_2());
         // sinh(atanh(x)) -> x*(1-x^2)^(-1/2)
         if (is_ex_the_function(x, atanh))
-            return t*power(exONE()-power(t,exTWO()),exMINUSHALF());
+            return t*power(_ex1()-power(t,_ex2()),_ex_1_2());
     }
     
     return sinh(x).hold();
 }
 
-ex sinh_diff(ex const & x, unsigned diff_param)
+static ex sinh_deriv(const ex & x, unsigned deriv_param)
 {
-    ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
+    // d/dx sinh(x) -> cosh(x)
     return cosh(x);
 }
 
-REGISTER_FUNCTION(sinh, sinh_eval, sinh_evalf, sinh_diff, NULL);
+REGISTER_FUNCTION(sinh, eval_func(sinh_eval).
+                        evalf_func(sinh_evalf).
+                        derivative_func(sinh_deriv));
 
 //////////
 // hyperbolic cosine (trigonometric function)
 //////////
 
-ex cosh_evalf(ex const & x)
+static ex cosh_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -538,47 +672,53 @@ ex cosh_evalf(ex const & x)
     return cosh(ex_to_numeric(x)); // -> numeric cosh(numeric)
 }
 
-ex cosh_eval(ex const & x)
+static ex cosh_eval(const ex & x)
 {
     if (x.info(info_flags::numeric)) {
-        // cosh(0) -> 1
-        if (x.is_zero())
-            return exONE();
-        // cosh(float) -> float
-        if (!x.info(info_flags::rational))
+        if (x.is_zero())  // cosh(0) -> 1
+            return _ex1();
+        if (!x.info(info_flags::crational))  // cosh(float) -> float
             return cosh_evalf(x);
     }
     
+    if ((x/Pi).info(info_flags::numeric) &&
+        ex_to_numeric(x/Pi).real().is_zero())  // cosh(I*x) -> cos(x)
+        return cos(x/I);
+    
     if (is_ex_exactly_of_type(x, function)) {
-        ex t=x.op(0);
+        ex t = x.op(0);
         // cosh(acosh(x)) -> x
         if (is_ex_the_function(x, acosh))
             return t;
         // cosh(asinh(x)) -> (1+x^2)^(1/2)
         if (is_ex_the_function(x, asinh))
-            return power(exONE()+power(t,exTWO()),exHALF());
+            return power(_ex1()+power(t,_ex2()),_ex1_2());
         // cosh(atanh(x)) -> (1-x^2)^(-1/2)
         if (is_ex_the_function(x, atanh))
-            return power(exONE()-power(t,exTWO()),exMINUSHALF());
+            return power(_ex1()-power(t,_ex2()),_ex_1_2());
     }
     
     return cosh(x).hold();
 }
 
-ex cosh_diff(ex const & x, unsigned diff_param)
+static ex cosh_deriv(const ex & x, unsigned deriv_param)
 {
-    ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
+    // d/dx cosh(x) -> sinh(x)
     return sinh(x);
 }
 
-REGISTER_FUNCTION(cosh, cosh_eval, cosh_evalf, cosh_diff, NULL);
+REGISTER_FUNCTION(cosh, eval_func(cosh_eval).
+                        evalf_func(cosh_evalf).
+                        derivative_func(cosh_deriv));
+
 
 //////////
 // hyperbolic tangent (trigonometric function)
 //////////
 
-ex tanh_evalf(ex const & x)
+static ex tanh_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -587,47 +727,65 @@ ex tanh_evalf(ex const & x)
     return tanh(ex_to_numeric(x)); // -> numeric tanh(numeric)
 }
 
-ex tanh_eval(ex const & x)
+static ex tanh_eval(const ex & x)
 {
     if (x.info(info_flags::numeric)) {
-        // tanh(0) -> 0
-        if (x.is_zero())
-            return exZERO();
-        // tanh(float) -> float
-        if (!x.info(info_flags::rational))
+        if (x.is_zero())  // tanh(0) -> 0
+            return _ex0();
+        if (!x.info(info_flags::crational))  // tanh(float) -> float
             return tanh_evalf(x);
     }
     
+    if ((x/Pi).info(info_flags::numeric) &&
+        ex_to_numeric(x/Pi).real().is_zero())  // tanh(I*x) -> I*tan(x);
+        return I*tan(x/I);
+    
     if (is_ex_exactly_of_type(x, function)) {
-        ex t=x.op(0);
+        ex t = x.op(0);
         // tanh(atanh(x)) -> x
         if (is_ex_the_function(x, atanh))
             return t;
         // tanh(asinh(x)) -> x*(1+x^2)^(-1/2)
         if (is_ex_the_function(x, asinh))
-            return t*power(exONE()+power(t,exTWO()),exMINUSHALF());
+            return t*power(_ex1()+power(t,_ex2()),_ex_1_2());
         // tanh(acosh(x)) -> (x-1)^(1/2)*(x+1)^(1/2)/x
         if (is_ex_the_function(x, acosh))
-            return power(t-exONE(),exHALF())*power(t+exONE(),exHALF())*power(t,exMINUSONE());
+            return power(t-_ex1(),_ex1_2())*power(t+_ex1(),_ex1_2())*power(t,_ex_1());
     }
     
     return tanh(x).hold();
 }
 
-ex tanh_diff(ex const & x, unsigned diff_param)
+static ex tanh_deriv(const ex & x, unsigned deriv_param)
 {
-    ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
-    return exONE()-power(tanh(x),exTWO());
+    // d/dx tanh(x) -> 1-tanh(x)^2
+    return _ex1()-power(tanh(x),_ex2());
+}
+
+static ex tanh_series(const ex &x, const relational &rel, int order)
+{
+    // method:
+    // Taylor series where there is no pole falls back to tanh_deriv.
+    // On a pole simply expand sinh(x)/cosh(x).
+    const ex x_pt = x.subs(rel);
+    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);
 }
 
-REGISTER_FUNCTION(tanh, tanh_eval, tanh_evalf, tanh_diff, NULL);
+REGISTER_FUNCTION(tanh, eval_func(tanh_eval).
+                        evalf_func(tanh_evalf).
+                        derivative_func(tanh_deriv).
+                        series_func(tanh_series));
 
 //////////
 // inverse hyperbolic sine (trigonometric function)
 //////////
 
-ex asinh_evalf(ex const & x)
+static ex asinh_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -636,34 +794,37 @@ ex asinh_evalf(ex const & x)
     return asinh(ex_to_numeric(x)); // -> numeric asinh(numeric)
 }
 
-ex asinh_eval(ex const & x)
+static ex asinh_eval(const ex & x)
 {
     if (x.info(info_flags::numeric)) {
         // asinh(0) -> 0
         if (x.is_zero())
-            return exZERO();
+            return _ex0();
         // asinh(float) -> float
-        if (!x.info(info_flags::rational))
+        if (!x.info(info_flags::crational))
             return asinh_evalf(x);
     }
     
     return asinh(x).hold();
 }
 
-ex asinh_diff(ex const & x, unsigned diff_param)
+static ex asinh_deriv(const ex & x, unsigned deriv_param)
 {
-    ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
-    return power(1+power(x,exTWO()),exMINUSHALF());
+    // d/dx asinh(x) -> 1/sqrt(1+x^2)
+    return power(_ex1()+power(x,_ex2()),_ex_1_2());
 }
 
-REGISTER_FUNCTION(asinh, asinh_eval, asinh_evalf, asinh_diff, NULL);
+REGISTER_FUNCTION(asinh, eval_func(asinh_eval).
+                         evalf_func(asinh_evalf).
+                         derivative_func(asinh_deriv));
 
 //////////
 // inverse hyperbolic cosine (trigonometric function)
 //////////
 
-ex acosh_evalf(ex const & x)
+static ex acosh_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -672,40 +833,43 @@ ex acosh_evalf(ex const & x)
     return acosh(ex_to_numeric(x)); // -> numeric acosh(numeric)
 }
 
-ex acosh_eval(ex const & 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(exONE()))
-            return exZERO();
+        if (x.is_equal(_ex1()))
+            return _ex0();
         // acosh(-1) -> Pi*I
-        if (x.is_equal(exMINUSONE()))
+        if (x.is_equal(_ex_1()))
             return Pi*I;
         // acosh(float) -> float
-        if (!x.info(info_flags::rational))
+        if (!x.info(info_flags::crational))
             return acosh_evalf(x);
     }
     
     return acosh(x).hold();
 }
 
-ex acosh_diff(ex const & x, unsigned diff_param)
+static ex acosh_deriv(const ex & x, unsigned deriv_param)
 {
-    ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
-    return power(x-1,exMINUSHALF())*power(x+1,exMINUSHALF());
+    // d/dx acosh(x) -> 1/(sqrt(x-1)*sqrt(x+1))
+    return power(x+_ex_1(),_ex_1_2())*power(x+_ex1(),_ex_1_2());
 }
 
-REGISTER_FUNCTION(acosh, acosh_eval, acosh_evalf, acosh_diff, NULL);
+REGISTER_FUNCTION(acosh, eval_func(acosh_eval).
+                         evalf_func(acosh_evalf).
+                         derivative_func(acosh_deriv));
 
 //////////
 // inverse hyperbolic tangent (trigonometric function)
 //////////
 
-ex atanh_evalf(ex const & x)
+static ex atanh_evalf(const ex & x)
 {
     BEGIN_TYPECHECK
        TYPECHECK(x,numeric)
@@ -714,30 +878,35 @@ ex atanh_evalf(ex const & x)
     return atanh(ex_to_numeric(x)); // -> numeric atanh(numeric)
 }
 
-ex atanh_eval(ex const & x)
+static ex atanh_eval(const ex & x)
 {
     if (x.info(info_flags::numeric)) {
         // atanh(0) -> 0
         if (x.is_zero())
-            return exZERO();
+            return _ex0();
         // atanh({+|-}1) -> throw
-        if (x.is_equal(exONE()) || x.is_equal(exONE()))
-            throw (std::domain_error("atanh_eval(): infinity"));
+        if (x.is_equal(_ex1()) || x.is_equal(_ex_1()))
+            throw (std::domain_error("atanh_eval(): logarithmic pole"));
         // atanh(float) -> float
-        if (!x.info(info_flags::rational))
+        if (!x.info(info_flags::crational))
             return atanh_evalf(x);
     }
     
     return atanh(x).hold();
 }
 
-ex atanh_diff(ex const & x, unsigned diff_param)
+static ex atanh_deriv(const ex & x, unsigned deriv_param)
 {
-    ASSERT(diff_param==0);
+    GINAC_ASSERT(deriv_param==0);
     
-    return power(exONE()-power(x,exTWO()),exMINUSONE());
+    // d/dx atanh(x) -> 1/(1-x^2)
+    return power(_ex1()-power(x,_ex2()),_ex_1());
 }
 
-REGISTER_FUNCTION(atanh, atanh_eval, atanh_evalf, atanh_diff, NULL);
+REGISTER_FUNCTION(atanh, eval_func(atanh_eval).
+                         evalf_func(atanh_evalf).
+                         derivative_func(atanh_deriv));
 
+#ifndef NO_NAMESPACE_GINAC
 } // namespace GiNaC
+#endif // ndef NO_NAMESPACE_GINAC