Mariusz Iwaniuk

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4 years, 307 days

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These are replies submitted by Mariusz Iwaniuk

@vv 

I copy code,paste,execute,dosen't work?

If I convert to 1D  then works. 

I'm confused.

@shimaa sadk 

On Maple 2018.1 yours attached file works fine.Maple calculate  all worksheet about 30 seconds.

See result:


 

NULL

with(LinearAlgebra); with(VectorCalculus); with(Student[LinearAlgebra]); with(SignalProcessing); with(Statistics); with(IntegrationTools)

NULL

r[1] := int(lambda[1]*alpha^2*exp(lambda[1]*Z)/((exp(lambda[1]*Z)-1+alpha)^2*(exp(lambda[2]*Z)-1+alpha)), Z = 0 .. infinity)

int(lambda[1]*alpha^2*exp(lambda[1]*Z)/((exp(lambda[1]*Z)-1+alpha)^2*(exp(lambda[2]*Z)-1+alpha)), Z = 0 .. infinity)

(1)

NULL

``

NULL

R[1] := Expand(diff(r[1], lambda[1])); R[1, 1] := diff(r[1], lambda[1], lambda[1]); R[1, 2] := diff(r[1], lambda[1], lambda[2]); R[1, 3] := diff(r[1], lambda[1], alpha); R[2] := diff(r[1], lambda[2]); R[2, 1] := diff(r[1], lambda[2], lambda[1]); R[2, 2] := diff(r[1], lambda[2], lambda[2]); R[2, 3] := diff(r[1], lambda[2], alpha); R[3] := diff(r[1], alpha); R[3, 1] := diff(r[1], alpha, lambda[1]); R[3, 2] := diff(r[1], alpha, lambda[2]); R[3, 3] := diff(r[1], alpha, alpha)

NULL

lambda[1] := 3/10; lambda[2] := 2*(1/10); alpha := 4

aa[1] := 0; aa[2] := 0; aa[3] := 0; bb[1] := 0; bb[2] := 0; bb[3] := 0

r_r[1] := Re(simplify(r[1]))

(1/27)*(6*3^(1/6)+10*3^(5/6))*arctan((2/3)*3^(1/6)-(1/3)*3^(1/2))+(1/27)*(15*3^(1/3)-3*3^(2/3))*ln(1+3^(1/3))+(2/9)*Pi*3^(1/2)-(1/9)*3^(1/6)*Pi-(5/27)*3^(5/6)*Pi-(10/27)*3^(1/3)*ln(2)+(2/27)*3^(2/3)*ln(2)+(2/3)*ln(2)+1/3

(2)

R_R[1] := Re(evalf(R[1])); R_R[1, 1] := Re(evalf(R[1, 1])); R_R[1, 2] := Re(evalf(R[1, 2])); R_R[1, 3] := Re(evalf(R[1, 3])); R_R[2] := Re(evalf(R[2])); R_R[2, 1] := Re(evalf(R[2, 1])); R_R[2, 2] := Re(evalf(R[2, 2])); R_R[2, 3] := Re(evalf(R[2, 3])); R_R[3] := Re(evalf(R[3])); R_R[3, 1] := Re(evalf(R[3, 1])); R_R[3, 2] := Re(evalf(R[3, 2])); R_R[3, 3] := Re(evalf(R[3, 3]))

Re(Expand(-10*dilog(1+((1/3)*I)*3^(1/2))+(100/9)*ln(2)-(10/81)*3^(1/3)*dilog(1+((1/2)*I)*3^(1/6)-(1/6)*3^(2/3))+(20/3)*dilog(1-((1/2)*I)*3^(1/6)-(1/6)*3^(2/3))-(40/729)*ln(3)*Pi*3^(5/6)-(520/243)*ln(3)*Pi*3^(1/6)-(10/81)*3^(1/3)*dilog(1-((1/2)*I)*3^(1/6)-(1/6)*3^(2/3))-(100/81)*arctan((2/3)*3^(1/6)-(1/3)*3^(1/2))*3^(5/6)+(130/81)*3^(2/3)*dilog(1+((1/2)*I)*3^(1/6)-(1/6)*3^(2/3))+(130/81)*3^(2/3)*dilog(1-((1/2)*I)*3^(1/6)-(1/6)*3^(2/3))+(5/3)*3^(1/2)*ln(3)*Pi-((130/27)*I)*3^(1/6)*dilog(1+((1/2)*I)*3^(1/6)-(1/6)*3^(2/3))+((130/27)*I)*3^(1/6)*dilog(1-((1/2)*I)*3^(1/6)-(1/6)*3^(2/3))+((10/3)*I)*dilog(1+((1/3)*I)*3^(1/2))*3^(1/2)-((10/3)*I)*dilog(1-((1/3)*I)*3^(1/2))*3^(1/2)+((10/81)*I)*3^(5/6)*dilog(1-((1/2)*I)*3^(1/6)-(1/6)*3^(2/3))-((10/81)*I)*3^(5/6)*dilog(1+((1/2)*I)*3^(1/6)-(1/6)*3^(2/3))-10*dilog(1-((1/3)*I)*3^(1/2))-(25/18)*ln(3)^2+(20/3)*dilog(1+((1/2)*I)*3^(1/6)-(1/6)*3^(2/3))-(260/81)*dilog(1+(1/3)*3^(2/3))*3^(2/3)+(20/81)*dilog(1+(1/3)*3^(2/3))*3^(1/3)-(10/81)*ln(1-3^(1/3)+3^(2/3))*3^(2/3)-(100/81)*ln(1+3^(1/3))*3^(1/3)+(50/81)*ln(1-3^(1/3)+3^(2/3))*3^(1/3)+(20/27)*Pi*3^(1/2)-(25/54)*Pi^2+(20/3)*dilog(1+(1/3)*3^(2/3))+(20/81)*ln(1+3^(1/3))*3^(2/3)-(20/27)*arctan((2/3)*3^(1/6)-(1/3)*3^(1/2))*3^(1/6)-(520/729)*3^(2/3)*Pi^2+(40/729)*3^(1/3)*Pi^2+(50/81)*3^(5/6)*Pi+(10/27)*3^(1/6)*Pi))

 

-5.223468552

 

2.445216045

 

0.58653935e-1

 

-1.616996031

 

2.445216045

 

4.417156107

 

-0.87980902e-1

 

0.88621008e-2

 

0.58653935e-1

 

-0.87980902e-1

 

-0.16512086e-2

(3)

NULL


 

Download aquestion_(2).mw

 

If you have problem then  try this worksheet below:

aquestion_(1)_ver2.mw

@shimaa sadk 

For general integral with general constans :

int(lambda[1]*alpha^2*exp(lambda[1]*Z)/((exp(lambda[1]*Z)-1+alpha)^2*(exp(lambda[2]*Z)-1+alpha)), Z = 0 .. infinity)

Maple 2018.1 can't find symbolic solution.

r[1] := proc (lambda1, lambda2, alpha) options operator, arrow; lambda1*alpha^2*exp(lambda1*Z)/((exp(lambda1*Z)-1+alpha)^2*(exp(lambda2*Z)-1+alpha)) end proc;
R[1] := int(eval(diff(r[1](lambda1, lambda2, alpha), lambda1), [lambda1 = 3/10, lambda2 = 2*(1/10), alpha = 4]), Z = 0 .. infinity, numeric);


#R[1] := 1.077997357

 

See attached file for more information:

aquestion_(1).mw

@spalinowy 

I gave you only a comment,my point of view. Wait for the answer.

And so at the end you'll end up with numerical calculations.

You have a Transcendental equations that why Maple can't solve(Any CAS can't solve).Transcendental equations have no analytical solution.

Try numeric solver like fsolve, DirectSearch package,or something else.

 

See:https://en.wikipedia.org/wiki/Transcendental_equation

        https://www.mapleprimes.com/questions/88377-Direct-Search-Optimization-Package

@digerdiga 

I updated my answer.Its the same answer like done by  the user Carl Love.Probably a better simplification can not be done.

Ei_ver2.mw

@digerdiga 

restart;

assume(R::real, z::real); is(-R+sqrt(R^2+z^2) < 0);

#false

@digerdiga 

`assuming`([convert(signum(0, R-sqrt(R^2+z^2), 1), piecewise)], [R-sqrt(R^2+z^2) < 0]);

# -1

or:

`assuming`([signum(0, R-sqrt(R^2+z^2), 1)], [R-sqrt(R^2+z^2) < 0])

#-1

Maple have some limitation about regarding the resolution of the case and falsehood.

restart:

assume(R > 0, z > 0, R in real, z in real);

coulditbe(R-sqrt(R^2+z^2) < 0);#FAIL

is(R-sqrt(R^2+z^2) < 0);#FAIL

@one pound 

From Wikipedia, using property: Ei(1,z) = - Ei(-z) and faster converging series by Ramanujan:

For example, for z = 10,and for  40 terms  an answer correct to 16 significant figures.

series.mw

@kuwait1 

A numeric check gives a big numbers in domain x=(-5,5) and z=(-5,5).Integral is divergent.

r := 2.5;
Q := proc (x, z) options operator, arrow; evalf(Int(1/(x*r^2*cos(x-y)+z*r^2*sin(z-y))^5, y = 0 .. Pi, method = _Gquad, epsilon = 1/100000)) end proc;
seq(seq(lprint(Q(x, z)), x = -5 .. 5, .25), z = -5 .. 5, .25);

-0.3091653839e55
0.7846201518e57
0.2353821795e58
-0.7841865263e57
-0.7756899663e57
0.1123370081e58
...

@ssllys You are welcome :)

@mmcdara . Thanks for checking :)

@pr0t3ctabl3 

All  my code in answer works on Maple 2017.3 and 2018. I don't have earlier version of Maple, so I will not check it.

@pr0t3ctabl3 

I edited the answer.It should work now.

All code was exectuted in Maple 2018.

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