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Zaninetti Lorenzo

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An integral done by Maple 11 but not by ...

Dr. Zaninetti Lorenzo 40 Maple 2021
integration
April 13 2022
0 4

primes_integrale_exp.mwThe integral in x of 

exp(-sqrt(x^2 + c))

was done by Maple 11 but return unevaluated in Maple 2021,

see attached worksheet 

new definitions in Latex 2021...

Dr. Zaninetti Lorenzo 40 Maple 2021
latex
July 13 2021
1 0

The conversion of a maple formula in latex version 2021 produces some

commands which are not defined , see   primes_latex.mw.

Perhaps more informations should be given  in the news about the command latex in Maple 2021

An Integral to be done from 1911...

Dr. Zaninetti Lorenzo 40 Maple
integration
May 18 2020
1 1

The following spreadsheet contains an integral to be done

in the framework of the Maxwell–Jüttner distribution and therefore date back to 1911

integrale_juttner_primes.mw
 

> 

restart;
> 

with(IntegrationTools):

the following integrand represents the scaling of the Maxwell–Jüttner distribution (1911) in 3D

where x is the Lorentz factor which is equal or greater than 1

> 

integrandx:= x*sqrt(x^2 - 1)*exp(-x);

x*(x^2-1)^(1/2)*exp(-x)

 (1)
> 

C:=1/int(integrandx,x=1..infinity);

1/(BesselK(0, 1)+2*BesselK(1, 1))

 (2)
> 

pdf:=C*integrandx;

x*(x^2-1)^(1/2)*exp(-x)/(BesselK(0, 1)+2*BesselK(1, 1))

 (3)

the above pdf represents the probability density function of the Maxwell–Jüttner distribution

> 

 

> 

DF:=int(C*integrandx,x=1..xx) assuming xx>1;

int(x*(x^2-1)^(1/2)*exp(-x)/(BesselK(0, 1)+2*BesselK(1, 1)), x = 1 .. xx)

 (4)

the above DF represents the inert form of the distribution function but the integral at the moment

does not exists

> 

 

> 

 


 

Download integrale_juttner_primes.mw

 

eval and subexpressions...

Dr. Zaninetti Lorenzo 40 Maple 2019
eval substitution
July 05 2019
1 0

The command eval allows to simplify a complicated expression

in a more compact form for a later output in LATEX

In the example which follows I was able to insert A and B but non C in the expression.

There is already a post on this kind of topic but I failed to understand the details.

Perhaps a Maple worksheet of answer on this topic  would be useful !

bye Lorenzo
 

> 

restart;
> 

expression:=exp(-b*x^c/2)*((x^(-(3*c)/2 + a/2 + 1/2)*(c + a + 1)*b^(-(3*c + a + 1)/(2*c)) + c*x^(a/2 + 1/2 - c/2)*b^(-(c + a + 1)/(2*c)))*c*WhittakerM((-c + a + 1)/(2*c), (2*c + a + 1)/(2*c), b*x^c) + b^(-(3*c + a + 1)/(2*c))*x^(-(3*c)/2 + a/2 + 1/2)*WhittakerM((c + a + 1)/(2*c), (2*c + a + 1)/(2*c), b*x^c)*(c + a + 1)^2)/((a + 1)*(c + a + 1)*(2*c + a + 1));

exp(-(1/2)*b*x^c)*((x^(-(3/2)*c+(1/2)*a+1/2)*(c+a+1)*b^(-(1/2)*(3*c+a+1)/c)+c*x^((1/2)*a+1/2-(1/2)*c)*b^(-(1/2)*(c+a+1)/c))*c*WhittakerM((1/2)*(-c+a+1)/c, (1/2)*(2*c+a+1)/c, b*x^c)+b^(-(1/2)*(3*c+a+1)/c)*x^(-(3/2)*c+(1/2)*a+1/2)*WhittakerM((1/2)*(c+a+1)/c, (1/2)*(2*c+a+1)/c, b*x^c)*(c+a+1)^2)/((a+1)*(c+a+1)*(2*c+a+1))

 (1)

``

 (2)
> 

expression_ABC:=eval(expression,[x^(-(3*c)/2 + a/2 + 1/2)*(c + a + 1)*b^(-(3*c + a + 1)/(2*c))=A,c*x^(a/2 + 1/2 - c/2)*b^(-(c + a + 1)/(2*c))=B,((a + 1)*(c + a + 1)*(2*c + a + 1))=C]);

exp(-(1/2)*b*x^c)*((A+B)*c*WhittakerM((1/2)*(-c+a+1)/c, (1/2)*(2*c+a+1)/c, b*x^c)+b^(-(1/2)*(3*c+a+1)/c)*x^(-(3/2)*c+(1/2)*a+1/2)*WhittakerM((1/2)*(c+a+1)/c, (1/2)*(2*c+a+1)/c, b*x^c)*(c+a+1)^2)/((a+1)*(c+a+1)*(2*c+a+1))

 (3)
> 

 

> 

 

> 

 

> 

 


 

Download maple_primes_eval.mw

 

An integral from dark energy cosmology...

Dr. Zaninetti Lorenzo 40 Maple
March 24 2019
2 0

In dark matter cosmology the following integral in x  does not yet have an analytical solution. I report the integral in

Maple 2018 notation

int(1/sqrt((1+x)^3*a+(1-a)*(1+x)^(3+3*b)), x)

If someone has some smart idea is welcome

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