/* TruncRadian truncates the radian r so it is between 0 and 2*Pi. * It calculates r mod 2*Pi using the required precision. */ TruncRadian(_r) <-- [ Local(twopi); r:=N(Eval(r),2*GetPrecision()); twopi:=N(2*Pi,2*GetPrecision()); N(r-MathFloor(r/twopi)*twopi); ]; HoldArg("TruncRadian",r); /* Standard analytic functions */ 2 # Sin(x_IsNumber)_(Numeric) <-- [ If(x<0 Or x>7, x:=TruncRadian(x)); MathSin(x); ]; 4 # Sin(ArcSin(_x)) <-- x; 5 # Sin(- _x) <-- -Sin(x); 5 # Sin(x_IsNegativeNumber * _y) <-- -Sin((-x)*y); 5 # Sin(_y * x_IsNegativeNumber) <-- -Sin(y*(-x)); 110 # Sin(Complex(_r,_i)) <-- (Exp(I*Complex(r,i)) - Exp(- I*Complex(r,i))) / (I*2) ; 1 # SinMap( _n )_(Not(IsRationalOrNumber(n))) <-- UnList({Atom("Sin"),n*Pi}); 2 # SinMap( _n )_(n<0) <-- -SinMap(-n); 2 # SinMap( _n )_(n>2) <-- SinMap(Mod(n,2)); 3 # SinMap( _n )_(n>1) <-- SinMap(n-2); 4 # SinMap( _n )_(n>1/2) <-- SinMap(1-n); 5 # SinMap( n_IsInteger ) <-- 0; 5 # SinMap( 1/6 ) <-- 1/2; 5 # SinMap( 1/4 ) <-- Sqrt(2)/2; 5 # SinMap( 1/3 ) <-- Sqrt(3)/2; 5 # SinMap( 1/2 ) <-- 1; 10 # SinMap(_n) <-- UnList({Atom("Sin"),n*Pi}); 200 # Sin(_v)_(CanBeUni(v,Pi) And Degree(v,Pi) < 2 And Coef(v,Pi,0) = 0) <-- [ SinMap(Coef(v,Pi,1)); ]; /*TODO remove? 200 # Sin(0) <-- 0; 200 # Sin(Pi) <-- 0; 200 # Sin(n_IsInteger * Pi) <-- 0; 200 # Sin(((n_IsPositiveInteger * Pi)/2))_IsEven(Div(n,2)) <-- 1; 200 # Sin(((n_IsPositiveInteger * Pi)/2))_IsOdd(Div(n,2)) <-- -1; 200 # Sin(( Pi)/2) <-- 1; 200 # Sin((-Pi)/2) <-- -1; 200 # Sin(((n_IsNegativeInteger * Pi)/2)) <-- -Sin(Pi*((-n)/2)); 200 # Sin(((n_IsNegativeInteger * Pi)/2)) <-- -Sin(Pi*((-n)/2)); */ /* TODO very ugly! */ /*TODO remove? 200 # Sin((Pi)/4) <-- Sqrt(2)/2; 200 # Cos((Pi)/4) <-- Sqrt(2)/2; */ 2 # Cos(x_IsNumber)_(Numeric) <-- [ If(x<0 Or x>7, x:=TruncRadian(x)); MathCos(x); ]; 4 # Cos(ArcCos(_x)) <-- x; 5 # Cos(- _x) <-- Cos(x); 5 # Cos(x_IsNegativeNumber * _y) <-- Cos((-x)*y); 5 # Cos(_y * x_IsNegativeNumber) <-- Cos(y*(-x)); 110 # Cos(Complex(_r,_i)) <-- (Exp(I*Complex(r,i)) + Exp(- I*Complex(r,i))) / (2) ; 1 # CosMap( _n )_(Not(IsRationalOrNumber(n))) <-- UnList({Atom("Cos"),n*Pi}); 2 # CosMap( _n )_(n<0) <-- CosMap(-n); 2 # CosMap( _n )_(n>2) <-- CosMap(Mod(n,2)); 3 # CosMap( _n )_(n>1) <-- CosMap(2-n); 4 # CosMap( _n )_(n>1/2) <-- -CosMap(1-n); 5 # CosMap( 0 ) <-- 1; 5 # CosMap( 1/6 ) <-- Sqrt(3)/2; 5 # CosMap( 1/4 ) <-- Sqrt(2)/2; 5 # CosMap( 1/3 ) <-- 1/2; 5 # CosMap( 1/2 ) <-- 0; 10 # CosMap(_n) <-- UnList({Atom("Cos"),n*Pi}); 200 # Cos(_v)_(CanBeUni(v,Pi) And Degree(v,Pi) < 2 And Coef(v,Pi,0) = 0) <-- CosMap(Coef(v,Pi,1)); /*TODO remove? 200 # Cos(0) <-- 1; 200 # Cos(Pi) <-- -1; 200 # Cos(n_IsEven * Pi) <-- 1; 200 # Cos(n_IsOdd * Pi) <-- -1; 200 # Cos((n_IsInteger*Pi) / 2) <-- 0; 200 # Cos(Pi / 2) <-- 0; 200 # Cos(-Pi / 2) <-- 0; */ 100 # Sin(_x)/Tan(_x) <-- Cos(x); 100 # Tan(_x)/Sin(_x) <-- (1/Cos(x)); 100 # Tan(_x)*Cos(_x) <-- Sin(x); 100 # Cos(_x)*Tan(_x) <-- Sin(x); 100 # Sin(_x)/Cos(_x) <-- Tan(x); 2 # Exp(x_IsNumber)_(Numeric) <-- MathExp(x); 4 # Exp(Ln(_x)) <-- x; 110 # Exp(Complex(_r,_i)) <-- Exp(r)*(Cos(i) + I*Sin(i)); 200 # Exp(0) <-- 1; 200 # Exp(-Infinity) <-- 0; 200 # Exp(Infinity) <-- Infinity; 2 # Tan(x_IsNumber)_(Numeric) <-- [ If(x<0 Or x>7, x:=TruncRadian(x)); MathTan(x); ]; 4 # Tan(ArcTan(_x)) <-- x; 110 # Tan(Complex(_r,_i)) <-- Sin(Complex(r,i))/Cos(Complex(r,i)); 200 # Tan(0) <-- 0; 200 # Tan(n_IsInteger*Pi) <-- 0; 200 # Tan(Pi*n_IsInteger) <-- 0; 200 # Tan((n_IsInteger/2)*Pi) <-- Infinity; 200 # Tan((n_IsInteger/2)*Pi) <-- Infinity; 200 # Tan(Pi*(n_IsInteger/2)) <-- Infinity; 200 # Tan(Pi*(n_IsInteger/2)) <-- Infinity; 2 # Ln(0) <-- -Infinity; 2 # Ln(1) <-- 0; 2 # Ln(Infinity) <-- Infinity; /* 2 # Ln(-Infinity) <-- 0; */ 2 # Ln(x_IsNegativeNumber)_(Numeric) <-- Complex(MathLog(-x), Pi); 3 # Ln(x_IsNumber)_(Numeric) <-- MathLog(x); 4 # Ln(Exp(_x)) <-- x; 3 # Ln(Complex(_r,_i)) <-- Complex( (1/2)*Ln(r^2 + i^2) , ArcTan( i / r) ); 4 # Ln(x_IsNegativeNumber) <-- Complex(Ln(-x), Pi); 2 # ArcSin(x_IsNumber)_(Numeric) <-- MathArcSin(x); 4 # ArcSin(Sin(_x)) <-- x; 110 # ArcSin(Complex(_r,_i)) <-- (- I) * Ln((I*Complex(r,i)) + ((1-(Complex(r,i)^2))^(1/2))); 200 # ArcSin(0) <-- 0; 200 # ArcSin(1) <-- (1/2)*Pi; 200 # ArcSin(-1) <-- (-1/2)*Pi; 2 # ArcCos(x_IsNumber)_(Numeric) <-- MathArcCos(x); 4 # ArcCos(Cos(_x)) <-- x; 110 # ArcCos(Complex(_r,_i)) <-- (- I)*Ln(Complex(r,i) + (Complex(r,i)^2 - 1)^(1/2)); /* TODO check! */ 200 # ArcCos(0) <-- (1/2)*Pi; 200 # ArcCos(1) <-- 0; 200 # ArcCos(_n)_(n = -1) <-- Pi; 2 # ArcTan(x_IsNumber)_(Numeric) <-- MathArcTan(x); 4 # ArcTan(Tan(_x)) <-- x; 110 # ArcTan(Complex(_r,_i)) <-- (- I*0.5)*Ln(Complex(1,Complex(r,i))/ Complex(1, - Complex(r,i))); 200 # ArcTan(1) <-- Pi/4; 200 # ArcTan(0) <-- 0; 200 # ArcTan(_n)_(n = -1) <-- -Pi/4; 200 # ArcTan(Infinity) <-- Pi/2; 200 # ArcTan(-Infinity) <-- -Pi/2; /* Threading of standard analytic functions */ Sin(xlist_IsList) <-- MapSingle("Sin",xlist); Cos(xlist_IsList) <-- MapSingle("Cos",xlist); Tan(xlist_IsList) <-- MapSingle("Tan",xlist); ArcSin(xlist_IsList) <-- MapSingle("ArcSin",xlist); ArcCos(xlist_IsList) <-- MapSingle("ArcCos",xlist); ArcTan(xlist_IsList) <-- MapSingle("ArcTan",xlist); Exp(xlist_IsList) <-- MapSingle("Exp",xlist); Ln(xlist_IsList) <-- MapSingle("Ln",xlist);