## 20 Reputation

15 years, 219 days

## An Integral to be done from 1911...

Maple

The following spreadsheet contains an integral to be done

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

 > 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);
 (1)
 > C:=1/int(integrandx,x=1..infinity);
 (2)
 > pdf:=C*integrandx;
 (3)

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

 >
 > DF:=int(C*integrandx,x=1..xx) assuming xx>1;
 (4)

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

does not exists

 >
 >

## eval and subexpressions...

Maple 2019

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));
 (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]);
 (3)
 >
 >
 >
 >

## An integral from dark energy cosmology...

Maple

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

 Page 1 of 1
﻿