Can This Integral Equation Be Solved Analytically?

[gursimran]

Is it possible to solve this integral ??

Homework Statement

Is it possible to solve this integral equation analytically. Actually I wanted to curve fit some data in MATLAB using a equation of which this is a part. I tried various techniques all in vain. PLease help..

Homework Equations

int((x^4)*(e^2)/((e^x)-1)^2)

The Attempt at a Solution

 

[Ray Vickson]

Homework Statement

Is it possible to solve this integral equation analytically. Actually I wanted to curve fit some data in MATLAB using a equation of which this is a part. I tried various techniques all in vain. PLease help..

Homework Equations

int((x^4)*(e^2)/((e^x)-1)^2)

The Attempt at a Solution

It involves "non-elementary" functions. Here is what I get using Maple:
f:=x^4/(exp(x)-1)^2:
J:=int(f,x);
Maple's answer is
J:= -x^4/(exp(x)-1) + x^5 /5 + (12x^2 - 4x^3) polylog(2,exp(x)) +
(12x^2 - 24) polylog(3,exp(x)) + (24 - 24x) polylog(4,exp(x)) +
24 polylog(5,exp(x)) - x^4 + (4x^3 - x^4) log(1-exp(x))
Here, Maple uses the non-elementary function
polylog(a,z) = sum_{n=1..infinity} z^n / n^a .


RGV

 

[gursimran]

Thanks for replying, I read the article in non elementary intergrals, however there isn't satisfactory explanation given. I'm doing undergraduate and have never heard of non elementary integrals. Is that mean that you can't solve these integrals analytically without computers and how this is different from numerical integration. Even in numerical integration we can find integration and then fit a polynomial or any other taylor series. So we won't call that a integral.

BTW I tried to compute this in MATLAB symbolic toolkit, it says its indeterminable. How can mathematica solve it. can mathematica solve integrals better than matlab?

It involves "non-elementary" functions. Here is what I get using Maple:
f:=x^4/(exp(x)-1)^2:
J:=int(f,x);
Maple's answer is
J:= -x^4/(exp(x)-1) + x^5 /5 + (12x^2 - 4x^3) polylog(2,exp(x)) +
(12x^2 - 24) polylog(3,exp(x)) + (24 - 24x) polylog(4,exp(x)) +
24 polylog(5,exp(x)) - x^4 + (4x^3 - x^4) log(1-exp(x))
Here, Maple uses the non-elementary function
polylog(a,z) = sum_{n=1..infinity} z^n / n^a .


RGV

 

[Ray Vickson]

Thanks for replying, I read the article in non elementary intergrals, however there isn't satisfactory explanation given. I'm doing undergraduate and have never heard of non elementary integrals. Is that mean that you can't solve these integrals analytically without computers and how this is different from numerical integration. Even in numerical integration we can find integration and then fit a polynomial or any other taylor series. So we won't call that a integral.

BTW I tried to compute this in MATLAB symbolic toolkit, it says its indeterminable. How can mathematica solve it. can mathematica solve integrals better than matlab?

I don't have Mathematica (I use Maple instead), but since the integral is not elementary, Mathematica will not be able to do anything very different. Perhaps it would express the answer using non-elementary functions different from Maple's polylog, but the result will still be in terms of non-elementary functions.

Non-elementary integrals are just integrals for which we have no finite, closed-form formula. They appear all over the place, in many types of problems. For example, the integrals of sin(x)/x or exp(x)/x are non-elementary; the integral of exp(-x^2) is non-elementary; the integral of sqrt((1-x^2)*(1-k*x^2)) is non-elementary, etc. Nevertheless, whether or not an integral is elementary, we still need methods of computing it numerically, and many ways exist of doing that (such as good approximations, series expansions, or straight numerical methods, etc.) Some scientific calculators have buttons that give the integral of exp(-x^2), for example, and all spreadsheets have similar capabilities. Programs like Maple or Mathematica can compute a wide variety of such functions, and are so easy to use that the distinction between "elementary" and "non-elementary" almost disappears.

RGV

 

[gursimran]

Thanks a lot, for your help

I don't have Mathematica (I use Maple instead), but since the integral is not elementary, Mathematica will not be able to do anything very different. Perhaps it would express the answer using non-elementary functions different from Maple's polylog, but the result will still be in terms of non-elementary functions.

Non-elementary integrals are just integrals for which we have no finite, closed-form formula. They appear all over the place, in many types of problems. For example, the integrals of sin(x)/x or exp(x)/x are non-elementary; the integral of exp(-x^2) is non-elementary; the integral of sqrt((1-x^2)*(1-k*x^2)) is non-elementary, etc. Nevertheless, whether or not an integral is elementary, we still need methods of computing it numerically, and many ways exist of doing that (such as good approximations, series expansions, or straight numerical methods, etc.) Some scientific calculators have buttons that give the integral of exp(-x^2), for example, and all spreadsheets have similar capabilities. Programs like Maple or Mathematica can compute a wide variety of such functions, and are so easy to use that the distinction between "elementary" and "non-elementary" almost disappears.

RGV