integral, explicit form

Question

I want to estimate the ratio of integrals:

$$ \frac{\int \frac{4 a T^3}{\frac{4 a T^4}{3}+\Lambda_1 \left(\frac{4}{T^3}+1\right)}}{\int \frac{4 a T^3}{\frac{4 a T^4}{3}+\Lambda_0 \left(\frac{4}{T^3}+1\right)}} $$

where $\Lambda_1$=$\Lambda$ (constant) and $\Lambda_1$=0 for some temperature $T_0$ (arbitrary constants are set to zero).

When I try to integrate the function:

f[T_] := (4 a T^3)/(4/3 a T^4 + Λ (1 + 4/T^3))
Integrate[ f[T], T]

I get the result like this (with # and &) :

12a RootSum[ 12 Λ + 3 Λ #1^3 + 4 a #1^7 &, (Log[T - #1] #1^4)/(9 Λ + 28 a #1^4) &]

What am I doing wrong? How do I get an explicit expression for the integral? How do I solve this problem ?

(1) The documentation provides some helpful hints. (2) Typically there are seven discrete poles of the integrand in the Complex plane, so the best you can hope for--except for some special values of $a$ and $\Lambda$--is precisely a sum over those poles, which is what RootSum accomplishes. What, then, would a more "explicit expression" look like? — whuber, Jan 22 '13 at 19:24
How can I, at least, approximate this function for $T\rightarrow\infty$? — molkee, Jan 22 '13 at 19:51
The integral diverges logarithmically at $\infty$, because eventually the $4aT^4/3$ term in the denominator overwhelms the other term, giving an integral proportional to $\int dT/T$. — whuber, Jan 22 '13 at 20:14
A comment on the mathematical problem itself: an indefinite integral is defined only up to an arbitrary constant, so for the ratio to be well defined you need additional conditions. — Szabolcs, Jan 22 '13 at 20:37
@Szabolcs A sufficient condition is that one take the same constant for both. — Daniel Lichtblau, Jan 22 '13 at 20:50
@Szabolcs To correct myself, I mean it is a sufficient condition whenever the indefinite integral itself goes to infinity in the variable. In this case that happens so we're fine using indefinite integrals. — Daniel Lichtblau, Jan 22 '13 at 21:04

score 4 · Answer 1 · answered Jan 22 '13 at 20:56

Can set it up as below. You need not look at the intermediate results, hence can pretend you never ran into those RootSum things.

f[t_, lam_] := (4 a*t^3)/(4/3 a*t^4 + lam*(1 + 4/t^3)); 
f2[t_, lam_] := Integrate[f[t, lam], t];

quot = f2[t, lam0]/f2[t, lam1];

Here is the asymptotic behavior of the quotient as t --> infinity.

Limit[quot, t -> Infinity, Assumptions -> Element[a, Reals]]

(* Out[144]= 1 *)

score 2 · Accepted Answer · edited Apr 13 '17 at 12:55

It would be more appropriate to use this definition :

f[T_, a_, Λ_] := (4 a T^3)/(4/3 a T^4 + Λ (1 + 4/T^3))

In general you cannot get an explicit expresion (i.e. in terms of radicals) when roots of higher order polynomials are taken into account. There is a fundamental mathematical barrier see e.g. Abel's Impossibility Theorem, Galois's Theorem. Only for special values of a and Λ you could get it. Root and RootSum objects are symbolic representations of certain well defined mathematical concepts. So for example you can evaluate them with arbitrary numerical accuracy. For a relation between RootSum and Root you can use Normal, e.g. Normal @ Integrate[ f[T, a, Λ], T]. More on Root objects you can find here How do I work with Root objects ?

You can find the limit of your ratio when T goes to Infinity :

Limit[  Integrate[ f[T, a, Λ1], T, Assumptions :> {a > 0, Λ1 > 0}]/
          Integrate[ f[T, a, Λ2], T, Assumptions :> {a > 0, Λ2 > 0}] ,
       T -> Infinity]

If you have values a, Λ1 and Λ2 you can compute the ratio for any temerature T :

ratio[ T_?NumericQ, a_?NumericQ, Λ1_?NumericQ, Λ2_?NumericQ] :=  
  NIntegrate[f[x, a, Λ1], {x, 0, T}]/  NIntegrate[f[x, a, Λ2], {x, 0, T}]
ratio[300, 10, 20, 15]

0.990688

Plot[ ratio[T, 10, 20, 15], {T, 0, 300}]

enter image description here

One can see that the ratio is monotonic and tends to 1 quite rapidly.

integral, explicit form

2 Answers2