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Happy New Year, dear friends !
Would it be possible for you advice me with method to solve this integral:
Xm-upper limit of integral, i.e number
α=const.
Happy New Year, dear friends !
Would it be possible for you advice me with method to solve this integral:
Xm-upper limit of integral, i.e number
α=const.
Hello,
I dont consider "closed form" to include any infinite sums, although some people do like to call it that. For example, an integral which has solution consisting of one or more erf() functions i do not consider closed form.
Does this mean that you don't consider results involving sine, cosine, tangent, exponential or log to be closed form, since they all involve infinite sums?
Does this mean that you don't consider results involving sine, cosine, tangent, exponential or log to be closed form, since they all involve infinite sums?
Overall, when you get to the piece that says ...
[latex] \int \frac{\ln (y-1) (\ln (y))^2}{y}dy[/latex]
I think you are in need of something special, whether it be a special integral or special series form.
In my view, what you showed is a valid closed form solution, albeit a complicated and somewhat obscure one for most of us engineers.
In my original comment, I was expressing the opinion that the form might be in terms of logarithmic integrals,
which I believe would not represent a closed form solution (unless some kind of series form can be shown for those, as well). However, you have shown the answer in terms of the polylog functions, which is not the same thing as the logarithmic integral. Hence, I was in error, and there is a closed form representation, and it is one that can be useful when using Mathematical or Matlab, since the polylog functions are readily available there.
Personally, I view the erf and logarithmic integral functions as not being closed form because they might typically be evaluated with numerical integration, rather than by a series formula. However, if a usable series representation can be shown for these integrals, I think it is proper to call it a closed form. This is just my opinion however, and I'm not sure what the mathematicians will say about it.
I'm curious to know the formal mathematical definition of "closed form". I think I'll do a search on it now.
EDIT: OK, so I did a search and this is what I found (see wiki link below). Personally, I find the distinction somewhat arbitrary. For example, sine and cosine are elementary enough to be classed as closed-form, but what about Bessel Functions? Those are not all that different? And, why not polylog functions?
Anyway, it's just semantics and does not really have too much impact on what we actually do. The term "analytical expression" or "analytical form" are more consistent with what many of use might consider to be a useful "closed form" solution.
https://en.wikipedia.org/wiki/Closed-form_expression
steveB, I continued integration to fix flaws:
View attachment 69465
Result is not clear as yet.
MrAl, In the Debye model ɑ=h/kT. h-Planck's constant, k-Boltzmann constant.
The Electrician, thanks for your support. I will try consider Lambert W-function.
I propose to postpone the issue until the 2013.
Happy New Year, dear colleagues !)
For what it is worth, the solution to the integral from 0 to xm=+infinity is:
G(4)*Z(4)/a^4
where
G is the Gamma function and Z is the Zeta function.
If anyone posts a solution here test it for convergence with the desired a and xm range of values you need. Some of the solutions spit out by integration machines dont converge for many input parameters. Be especially careful with the so called Li(a,b) functions.
The OP already gave this result (equivalently) in the first attachment to post #8
The question of convergence is properly applied to infinite series. For a function which is not an infinite series, we ask whether the function is defined and correct for a particular argument.
For example, if the problem were to evaluate ∫cos(x)dx from 0 to Pi, and the closed form symbolic solution we used for the integral were sin(x), we wouldn't ask whether sin(x) converges; we ask whether it is defined over the range of integration.
The polylog functions used in the symbolic solution I gave in post #4 are defined and return a finite result for any finite real argument. That solution gives a finite, correct, result for any finite "a" and "Xm" greater than zero. The question of convergence is not applicable.
My example numerical integration gives the same result as the symbolic expression for a selection of values of "a", and for a range of "Xm" which I tested. I gave a plot showing the agreement for a particular case in my post.
I also gave a series approximation. Even though it is a series, since it's not an infinite series, the question of convergence is not applicable; the only question is how good is the approximation, and over what range of argument.
With regard to "G(4)*Z(4)/a^4"...
"The OP already gave this result (equivalently) in the first attachment to post #8"
The OP gave the numerical result of the constant part when integrating over 0 to +infinity, i have shown the way it is calculated:
G(4)*Z(4)/a^4
So what's your point?
Also, i am trying to establish the required range of parameters from the OP but he's not easily giving up that information