Mittag-Leffler series for generating continuum sum?

[bo198214]

Does it refer to some other book/paper/etc. within it when it discusses the formula and its coefficients? Could you chase after that?

Its a self contained proof, no further references.

What about the 1977 one?

I dont possess it and didnt find it in the web.

[mike3]

Does it refer to some other book/paper/etc. within it when it discusses the formula and its coefficients? Could you chase after that?

Its a self contained proof, no further references.

Mmm. And I suppose it's just an "existence" as opposed to a "construction" proof, then? Yet the site says:

"The degrees of the polynomials and their coefficients , are independent of the form of f(z) and can be evaluated once and for all."

and references sources [2] and [3] which are the 1977 thing you mentioned you didn't have, and one that's in French. And the 1965 one didn't say how to get the values...

What about the 1977 one?

I dont possess it and didnt find it in the web.

I thought you went to a library to get this stuff. I suppose not, then, or the library did not have it?

So it seems the trail has run cold on this one, then? Or perhaps the answer is there, but it's in French

[bo198214]

I thought you went to a library to get this stuff. I suppose not, then, or the library did not have it?

Mike, you didnt read what I wrote (again!, why do I give all the extra information?), I clearly said that I found the djvu on the web.
And if you are really interested then you type "A.I. Markushevich, Theory of functions of a complex variable, djvu" in the google field and get it too and can read the 1965 edition yourself. In its construction it mentions the Runge polynomial approximation, perhaps there you can find the coefficients.

See, for me it takes off 2 hours from my sparse time to go to a library. And this for a book I am not even sure whether it contains the needed information.

[mike3]

Mike, you didnt read what I wrote (again!, why do I give all the extra information?), I clearly said that I found the djvu on the web.
And if you are really interested then you type "A.I. Markushevich, Theory of functions of a complex variable, djvu" in the google field and get it too and can read the 1965 edition yourself. In its construction it mentions the Runge polynomial approximation, perhaps there you can find the coefficients.

See, for me it takes off 2 hours from my sparse time to go to a library. And this for a book I am not even sure whether it contains the needed information.

Ah. I guess I just missed the bit where you said "internet".

I looked at it and I'm not sure what to make of what's there. The locations of the singularities seem to be involved in the proof, but given the "universality" of the coefficients mentioned, it appears they may not be needed for their construction (as different functions have different singularities).

It still doesn't seem to give anything useful for the construction. I also found a limited book preview for the 1977 edition on Google, the second page of the proof is omitted and it seems the section ends right after that page. Given the length of the proof (2 pages) in the 1965 book, I'm also not sure if this 1977 book would contain the method either, so yeah, I could understand your concern about taking the time to go to the library! Though I wonder then why it was referenced on the website, but perhaps maybe that's just because the book may, like the 1965 edition, just mention the fact they are universal, not actually construct them, and the actual construction may be in the French-language paper by Borel, or something it references.

[bo198214]

I looked at it and I'm not sure what to make of what's there. The locations of the singularities seem to be involved in the proof, but given the "universality" of the coefficients mentioned, it appears they may not be needed for their construction (as different functions have different singularities).

Yes, there you are right (and i was wrong). If I remember correctly you only need the Runge approximation for 1/(z-w) or so. So if you have the coefficients of the Runge approximation maybe you can deduce the Star-expansion coefficients following the proof.

But I really wonder why the coefficients are not given in the Springer online reference. I was inclined to think that they are quite complicated.

Though I wonder then why it was referenced on the website

Indeed I wonder too. It mentions Painlevé having explicitely found the coefficients, but the given references are not by Painlevé, not even mentioned in [2]. I think it is just an oversight, forgetting/or being to lazy to give the proper references.

In wikipedia the Mittag-Leffler expansion is also mentioned but with rather soft text-book references, which surely dont contain the coefficient computations.

[mike3]

Even if it is fairly complicated, one should still be able to find it somewhere, no? I find it strange that this would be so obscure, given the relevance to a fundamental part of complex analysis -- analytic continuation. How would one go about finding the right reference?

[mike3]

Yes, there you are right (and i was wrong). If I remember correctly you only need the Runge approximation for 1/(z-w) or so. So if you have the coefficients of the Runge approximation maybe you can deduce the Star-expansion coefficients following the proof.

Hmm. I was emailed a snippet of the 1977 book for Complex analysis (the one by Markushevich mentioned on the website), showing the missing other page of the proof you couldn't see in Google's book preview, with which had this:

(the comment in "[]"s is mine)

The function \( \frac{1}{1 - w} \) is single-valued and analytic on G [described as "the domain G bounded by the segment u >= 1, v = 0 (the part of the real axis going from 1 to \( \infty \)", which sounds like the Mittag-Leffler star of that function], and hence, by Runge's theorem (Theorem 3.5) there exists a sequence of polynomials

\( P_n(w) = c_0^{(n)} + c_1^{(n)} w + ... + c_{k_n}^{(n)} w^{k_n} \)

such that

\( lim_{n \rightarrow \infty} P_n(w) = \frac{1}{1 - w} \)

where the convergence is uniform inside G, in particular on the compact set \( E(F, L) \subset G \).

It mentions "Runge's theorem", does that have to do with "Runge approximations"? Can this be useful in determining the polynomials (and so the magic coefficients)? If so, how would one do it?

[bo198214]

The function \( \frac{1}{1 - w} \) is single-valued and analytic on G [described as "the domain G bounded by the segment u >= 1, v = 0 (the part of the real axis going from 1 to \( \infty \)", which sounds like the Mittag-Leffler star of that function], and hence, by Runge's theorem (Theorem 3.5) there exists a sequence of polynomials

\( P_n(w) = c_0^{(n)} + c_1^{(n)} w + ... + c_{k_n}^{(n)} w^{k_n} \)

such that

\( lim_{n \rightarrow \infty} P_n(w) = \frac{1}{1 - w} \)

where the convergence is uniform inside G, in particular on the compact set \( E(F, L) \subset G \).

yes thats the same as in the 1967 edition.
There you will also find Runge's theorem.

[mike3]

yes thats the same as in the 1967 edition.
There you will also find Runge's theorem.

I just found that, but how do you use that to actually construct the necessary sequence of polynomials?

[bo198214]

So I had a look at
E. Borel: Leçons sur les séries divergentes (1901)
In chapter V, page 156 he explains the Mittag-Leffler expansion.
Mostly I fighted my way through the text with an online translator.
I found the following interesting formulas, where \( \phi \) is the function we search the expansion of:
\( g_n(x)=\sum_{\lambda_1=0}^{n^{2n}}\sum_{\lambda_2=0}^{n^{2n-2}}
\dots\sum_{\lambda_n=0}^{n^2} \frac{\phi^{(\lambda_1+\dots+\lambda_n)}(0)}{\lambda_1!\dots\lambda_n!} \left(\frac{x}{n}\right)^{\lambda_1+\dots+\lambda_n}
\)
\( G_0(x)=g_0(x)=\phi(0) \)
\( G_n(x)=g_n(x)-g_{n-1}(x) \) for \( n>0 \),
\( \phi(x)=\sum_{n=0}^\infty G_n(x) \)