Markiyan Hirnyk

Markiyan Hirnyk

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These are questions asked by Markiyan Hirnyk

Let us consider the improper integral

int((abs(sin(2*x))-abs(sin(x)))/x, x = 0 .. infinity);

Si(Pi)-Si((1/2)*Pi)+sum(-(-1)^_k*Si(Pi*_k)+signum(sin((1/2)*Pi*_k))*Si((1/2)*Pi*_k)+Si(Pi*_k+Pi)*(-1)^_k-signum(cos((1/2)*Pi*_k))*Si((1/2)*Pi*_k+(1/2)*Pi), _k = 1 .. infinity)
                    

Mathematica 11 produces a similar expression and a warning

Integrate::isub: Warning: infinite subdivision of the integration domain has been used in computation of the definite integral \!\(\*SubsuperscriptBox[\(\[Integral]\), \(0\), \(\[Infinity]\)]\(\*FractionBox[\(\(-Abs[Sin[x]]\) + Abs[Sin[2\ x]]\), \(x\)] \[DifferentialD]x\)\). If the integral is not absolutely convergent, the result may be incorrect.

Up to Pedro Tamaroff http://math.stackexchange.com/questions/61828/proof-of-frullanis-theorem , the answer is 2/Pi*ln(2) because of 

J := int(abs(sin(2*x))-abs(sin(x)), x = 0 .. T) assuming T>2;
-1/2-signum(sin(T))*signum(cos(T))*cos(T)^2+(1/2)*signum(sin(T))*signum(cos(T))+cos(T)*signum(sin(T))+floor(2*T/Pi)

B := limit(J/T, T = infinity);
                               2 /Pi

K := x*(int((abs(sin(2*t))-abs(sin(t)))/t^2, t = x .. 1)) assuming x>0,x<1;

     2*sin(x)*cos(x)-2*Ci(2*x)*x+Ci(x)*x+sin(1)*x-sin(2)*x+2*Ci(2)*x-Ci(1)*x-sin(x)

                         
A := limit(K, x = 0, right);
                               0

Its numeric calculation results 

evalf(Int((abs(sin(2*x))-abs(sin(x)))/x, x = 0 .. infinity));
                        Float(undefined)

which seems not to be true.

The question is: how to obtain the reliable results for it with Maple, both symbolic and numeric? 

I have in mind all the real roots of the equation 2*tan(Pi*t^2)-tan(Pi*t)+tan(Pi*t)*tan(Pi*t^2)^2 = 0.

Maple fails with it:

>RealDomain:-solve(2*tan(Pi*t^2)-tan(Pi*t)+tan(Pi*t)*tan(Pi*t^2)^2 = 0, t);

RootOf(tan(_Z)*tan(_Z^2/Pi)^2-tan(_Z)+2*tan(_Z^2/Pi))/Pi

 Even its numerical solution has gaps.

>Digits := 15; a := Student[Calculus1]:-Roots(2*tan(Pi*t^2)-tan(Pi*t)+tan(Pi*t)*tan(Pi*t^2)^2 = 0, t = -2 .. 2);
Warning, some roots are returned as numeric approximations
 [-1.35078105935821, -1.18614066163451, -1.00000000000000, 0, 

   1.00000000000000, 1.28077640640442, 1.68614066163451,    1.85078105935821]

>nops(a);

8

>b := Student[Calculus1]:-Roots(2*tan(Pi*t^2)-tan(Pi*t)+tan(Pi*t)*tan(Pi*t^2)^2 = 0, t = -2 .. 2, numeric);
 [-1.35078105935821, -1.18614066163451, -1.00000000000000, 

   -0.780776406404415, 0., 1.00000000000000, 1.28077640640442, 

   1.68614066163451, 1.85078105935821, 2.00000000000000]
>nops(b);
                               10


whereas 

>plot(2*tan(Pi*t^2)-tan(Pi*t)+tan(Pi*t)*tan(Pi*t^2)^2, t = -2 .. 2);

shows 14 solutions.

The output of the command

>identify(a);

[1/4-(1/4)*sqrt(41), 1/4-(1/4)*sqrt(33), -1, 0, 1, 1/4+(1/4)*sqrt(17), 1/4+(1/4)*sqrt(33), 1/4+(1/4)*sqrt(41)]

suggests a closed-form expression for the roots.

Can somebody of Maple users execute the following command

restart; pdsolve({diff(w(x, y, z), x)+diff(w(x, y, z), y, y)+2*(diff(v(x, y, z), x)-(diff(u(x, y, z), y))-2*w(x, y, z)) = diff(w(x, y, z), z, z), 3*(diff(u(x, y, z), x, x))+2*(diff(u(x, y, z), y, y))+2*(diff(v(x, y, z), x, y))+2*(diff(w(x, y, z), y)) = diff(u(x, y, z), z, z), 3*(diff(v(x, y, z), y, y))+2*(diff(v(x, y, z), x, x))+2*(diff(u(x, y, z), x, y))-2*(diff(w(x, y, z), x)) = diff(v(x, y, z), z, z)}, {u(x, y, z), v(x, y, z), w(x, y, z)})

restart; pdsolve({diff(w(x, y, z), x)+diff(w(x, y, z), y, y)+2*(diff(v(x, y, z), x)-(diff(u(x, y, z), y))-2*w(x, y, z)) = diff(w(x, y, z), z, z), 3*(diff(u(x, y, z), x, x))+2*(diff(u(x, y, z), y, y))+2*(diff(v(x, y, z), x, y))+2*(diff(w(x, y, z), y)) = diff(u(x, y, z), z, z), 3*(diff(v(x, y, z), y, y))+2*(diff(v(x, y, z), x, x))+2*(diff(u(x, y, z), x, y))-2*(diff(w(x, y, z), x)) = diff(v(x, y, z), z, z)}, {u(x, y, z), v(x, y, z), w(x, y, z)})

Error, (in simplify/normal) Maple was unable to allocate enough memory to complete this computation.  Please see ?alloc

 

 

 

in Maple 2016.1.1 on a powerful comp and report the obtained result as an answer to the question?
 That would be very kind of her/him. Thanks in advance. 

Download pdsolve.mw

I mean the root of the equation

GAMMA(n-1/n)*GAMMA(1/n)/(n*GAMMA(n)) = 1

belonging to RealRange(Open(1),4). It should be noticed there are solutions outside this interval. Here is my try.

 

``

solve({n > 1, GAMMA(n-1/n)*GAMMA(1/n)/(n*GAMMA(n)) = 1, n < 4}, [n])``

[]

(1)

`in`(which*is*wrong, view*of)

simplify(eval(GAMMA(n-1/n)*GAMMA(1/n)/(n*GAMMA(n)), n = (1/2)*sqrt(5)+1/2))

1

(2)

Also

Student[Calculus1]:-Roots(A = 1, n = 1 .. 4)

[1.618033989]

(3)

There is a substitute

fsolve(GAMMA(n-1/n)*GAMMA(1/n)/(n*GAMMA(n)) = 1, n = 1 .. 4)

1.618033989

(4)

NULL

identify(%)

(1/2)*5^(1/2)+1/2

(5)

``

There is a shade of hope that GAMMA(n-1/n)*GAMMA(1/n)/(n*GAMMA(n))  can be simplified.

Download solution.mw

 PS. An SCR was submitted by me.

of the implicit function sin(x+y)+sin(x) = y at x = Pi , y=0 of order 15? Here is one of the difficulties

restart; eval(implicitdiff(sin(x+y)+sin(x) = y, y, x$15), [x = Pi, y = 0]);
Error, (in expand/bigprod) Maple was unable to allocate enough memory to complete this computation. Please see ?alloc

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