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f():=[];//Echo(CurrentFile()," line ",CurrentLine());
Testing("UnaryFunctionInverses");
Verify(Sin(ArcSin(a)),a); f();
Verify(Cos(ArcCos(a)),a); f();
//TODO ??? Verify(Tan(ArcTan(a)),a);
//TODO ??? this is not always the correct answer! Verify(ArcTan(Tan(a)),a);
Verify(Tan(Pi/2),Infinity); f();
Verify(Tan(Pi),0); f();
Verify( Limit(x,Infinity) Sin(x), Undefined ); f();
Verify( Limit(x,Infinity) Cos(x), Undefined ); f();
Verify( Limit(x,Infinity) Tan(x), Undefined ); f();
Verify( Limit(x,Infinity) Gamma(x), Infinity ); f();
Verify( Limit(x,Infinity) Abs(x), Infinity ); f();
Verify( Limit(x,Infinity) x!, Infinity); f();
Verify( Sin(x)/Cos(x), Tan(x) ); f();
Verify( TrigSimpCombine(Sin(x)^2 + Cos(x)^2), 1 ); f();
Verify( Sinh(x)-Cosh(x), Exp(-x)); f();
Verify( Sinh(x)+Cosh(x), Exp(x) ); f();
Verify( Sinh(x)/Cosh(x), Tanh(x) ); f();
Verify( Sinh(Infinity), Infinity); f();
Verify( Sinh(x)*Csch(x), 1); f();
Verify( 1/Coth(x), Tanh(x) ); f();
Verify(2+I*3,Complex(2,3)); f();
Verify(Magnitude(I+1),Sqrt(2)); f();
Verify(Re(2+I*3),2); f();
Verify(Im(2+I*3),3); f();
// Shouldn't these be in linalg.yts?
Verify(ZeroVector(3),{0,0,0}); f();
Verify(BaseVector(2,3),{0,1,0}); f();
Verify(Identity(3),{{1,0,0},{0,1,0},{0,0,1}}); f();
Testing("Derivatives");
Verify(Differentiate(x) a,0); f();
Verify(Differentiate(x) x,1); f();
Verify(Differentiate(x) (x+x),2); f();
Verify(Differentiate(x) (x*x),2*x); f();
Verify(Differentiate(x) Differentiate(x) Sin(x),-Sin(x)); f();
Testing("Limits");
Verify( Limit(x,0,Right) Ln(x)*Sin(x), 0 ); f();
KnownFailure( Limit(k,Infinity) ((k-phi)/k)^(k+1/2) = Exp(-phi) ); f();
// tests adapted from mpreduce
Verify(Limit(x,0)Sin(x)/x, 1);
Verify(Limit(x,0)Sin(x)^2/x, 0);
Verify(Limit(x,1)Sin(x)/x, Sin(1));
/* This is actually a tricky one - for complex it should bring
infinity but for real it's undefined. BTW, reduce seems
to get it wrong */
Verify(Limit(x,0)1/x, Undefined);
Verify(Limit(x,0)(Sin(x)-x)/x^3, -1/6);
Verify(Limit(x,Infinity)x*Sin(1/x), 1);
/* reduce seems to get it wrong */
Verify(Limit(x,0)Sin(x)/x^2, Undefined);
Verify(Limit(x,Infinity)x^2*Sin(1/x), Infinity);
// tests adapted from mpreduce
Verify(Limit(x,2)x^2-6*x+4,-4);
Verify(Limit(x,-1)(x+3)*(2*x-1)/(x^2+3*x-2), 3/2);
Verify(Limit(h,0)(Sqrt(4+h)-2)/h, 1/4);
Verify(Limit(x,4)(Sqrt(x)-2)/(4-x), -1/4);
Verify(Limit(x,2)(x^2-4)/(x-2), 4);
Verify(Limit(x,-1)1/(2*x-5), -1/7);
Verify(Limit(x,1)Sqrt(x)/(x+1), 1/2);
Verify(Limit(x,Infinity)(2*x+5)/(3*x-2), 2/3);
Verify(Limit(x,1)(1/(x+3)-2/(3*x+5))/(x-1), 1/32);
Verify(Limit(x,0)Sin(3*x)/x, 3);
Verify(Limit(x,0)(1-Cos(x))/x^2, 1/2);
Verify(Limit(x,0)(6*x-Sin(2*x))/(2*x+3*Sin(4*x)), 2/7);
Verify(Limit(x,0)(1-2*Cos(x)+Cos(2*x))/x^2, -1);
Verify(Simplify(Limit(x,0)(3*Sin(Pi*x) - Sin(3*Pi*x))/x^3), 4*Pi^3);
Verify(Limit(x,0)(Cos(a*x)-Cos(b*x))/x^2, (-a^2 + b^2)/2);
Verify(Limit(x,0)(Exp(x)-1)/x, 1);
Verify(Limit(x,0)(a^x-b^x)/x, Ln(a) - Ln(b));
// tests adapted from mpreduce
Verify(Limit(x,0)Sinh(2*x)^2/Ln(1+x^2), 4);
// The limit seems to hang mathpiper
//Verify(Limit(x,Infinity)x^2*(Exp(1/x)-1)*(Ln(x+2)-Ln(x)),2);
// another tricky problem with the result depending on the sign of
// alpha; I'm not sure how we should deal with it
//Limit(x,Infinity)x^alpha*Ln(x+1)^2/Ln(x);
Verify(Limit(x,0)(2*Cosh(x)-2-x^2)/Ln(1+x^2)^2, 1/12);
Verify(Limit(x,0)(x*Sinh(x)-2+2*Cosh(x))/(x^4+2*x^2), 1);
Verify(Limit(x,0)(2*Sinh(x)-Tanh(x))/(Exp(x)-1), 1);
Verify(Limit(x,0)x*Tanh(x)/(Sqrt(1-x^2)-1), -2);
Verify(Limit(x,0)(2*Ln(1+x)+x^2-2*x)/x^3, 2/3);
Verify(Limit(x,0)(Exp(5*x)-2*x)^(1/x), Exp(3));
Verify(Limit(x,Infinity)Ln(Ln(x))/Ln(x)^2, 0);
// tests adapted from mpreduce
Verify(Limit(x,0)(Exp(x)-1)/x, 1);
Verify(Limit(x,1)((1-x)/Ln(x))^2, 1);
Verify(Limit(x,0)x/(Exp(x)-1), 1);
Verify(Limit(x,0)x/Ln(x), 0);
Verify(Limit(x,Infinity)Ln(1+x)/Ln(x), 1);
Verify(Limit(x,Infinity)Ln(x)/Sqrt(x), 0);
Verify(Limit(x,0,Right)Sqrt(x)/Sin(x), Infinity);
Verify(Limit(x,0)x*Ln(x), 0);
Verify(Limit(x,0)Ln(x)/Ln(2*x), 1);
Verify(Limit(x,0)x*Ln(x)*(1+x), 0);
Verify(Limit(x,Infinity)Ln(x)/x,0);
Verify(Limit(x,Infinity)Ln(x)/Sqrt(x),0);
Verify(Limit(x,Infinity)Ln(x), Infinity);
Verify(Limit(x,0)Ln(x+1)/Sin(x), 1);
// Seems to hang mathpiper
//Verify(Limit(x,0)Ln(x+1/x)*Sin(x), 0);
Verify(Limit(x,0)-Ln(1+x)*(x+2)/Sin(x), -2);
Verify(Limit(x,Infinity)Ln(x+1)^2/Sqrt(x), 0);
Verify(Limit(x,Infinity)(Ln(x+1)-Ln(x)), 0);
Verify(Limit(x,Infinity)-Ln(x+1)^2/Ln(Ln(x)), -Infinity);
// Verify argument checking
[
Verify(Limit(n, Infinity) 2 + 11/n, 2);
Local(n, exception);
n := 5;
exception := False;
ExceptionCatch(Limit(n, Infinity) 2 + 11/n, exception := True);
Verify(exception, True);
];
[
Local(z);
// This function satisfies Laplaces eqn: Differentiate(x,2)z + Differentiate(y,2)z = 0
z:= ArcTan((2*x*y)/(x^2 - y^2)); f();
Verify(Simplify((Differentiate(x,2) z) + Differentiate(y,2) z), 0 );
];
Testing("Pslq");
VerifyPslq(left,right):=
[
If(left=right,
Verify(True,True),
`Verify(@left,-(@right)));
];
VerifyPslq(Pslq({ Pi+2*Exp(1) , Pi , Exp(1) },20),{1,-1,-2}); f();
VerifyPslq(Pslq({ 2*Pi+3*Exp(1) , Pi , Exp(1) },20),{1,-2,-3}); f();
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