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I'd like to pay attention to the recent article "The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them?"

In particular, the authors consider the integral

int(abs(exp(2*Pi*Ix)+exp(2*Pi*I*y)),[x=0..1,y=0..1]),

stating "Both Mathematica and Maple return zero as the answer to this calculation. Yet this cannot be correct, because the integrand is clearly positive and nonzero in the indicated region". Unfortunately, they give only the Mathematica command to this end.

Of course, the integral under consideration is complicated so the the simple-minded trials

int(evalc(abs(exp((2*Pi*I)*x)+exp((2*Pi*I)*y))), [x = 0 .. 1, y = 0 .. 1]);

and

VectorCalculus:-int(evalc(abs(exp((2*Pi*I)*x)+exp((2*Pi*I)*y))), [x,y]=Rectangle( 0 .. 1, 0 .. 1));

fail. However,this can be found with Maple (I think with Mathematica too.) in such a way.

 

A := evalc(abs(exp((2*Pi*I)*x)+exp((2*Pi*I)*y)))

((cos(2*Pi*x)+cos(2*Pi*y))^2+(sin(2*Pi*x)+sin(2*Pi*y))^2)^(1/2)

 (1)

NULL

B := simplify(A, trig)

(2*cos(2*Pi*x)*cos(2*Pi*y)+2+2*sin(2*Pi*x)*sin(2*Pi*y))^(1/2)

 (2)

op(B)[1]

2*cos(2*Pi*x)*cos(2*Pi*y)+2+2*sin(2*Pi*x)*sin(2*Pi*y)

 (3)

combine(op(B)[1], x)

2*cos(2*Pi*x-2*Pi*y)+2

 (4)

C := eval(B, op(B)[1] = combine(op(B)[1], x))

(2*cos(2*Pi*x-2*Pi*y)+2)^(1/2)

 (5)

int(C, [x = 0 .. 1, y = 0 .. 1])

4/Pi

 (6)

``

 

Download int.mw

 

 
integral cases

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