• #1


Homework Helper
Insights Author


This bit is what new thing you can learn reading this:) As for original content, I only have hope that the method of using the sets
$$C_N^n: = \left\{ { \vec x \in {\mathbb{R}^n}|{x_i} \ge 0\forall i,\sum\limits_{k = 1}^n {x_k^{2N}} < n - 1 } \right\}$$
and Dirichlet integrals to evaluate certain integrals of the type
$$\mathop {\lim }\limits_{N \to \infty } \int\limits_{C_N^n} {f\left( {\vec x} \right)d\mu } = \int\limits_{{{\left[ {0,1} \right]}^n}} {f\left( {\vec x} \right)d\mu }$$
might be original material as I have never seen it my reading.


The main purpose of this paper is to derive the formulas in Sections 4 and 5. Section 4 hold n-fold iterated integral representations of some special functions (where n is a positive integer), though somewhat dense, all the material up to this and including Section 3 is just advanced Calc 3 level material; Sections 4&5 are the analysis content, section 5 contains fractional integrals as analytic continuations of the previous section's formulas on the variable n which gets continued to a complex-valued parameter.

Just what is a fractional integral? Ever hear of n-fold integrals? What would it mean to allow complex numbers for the order of integration? This is what fractional integrals are, a generalization of integration.

Section 1​

Gives a glimpse of the gamma and beta functions. As for special functions, the gamma function is the least special function and should be the first special function one meets: it is the analytic continuation of the factorial. It arises in many venues: mathematical and engineering to be sure, and others as well. The beta function is defined in terms of gamma functions.

Section 2

It Covers Dirichlet Integrals which are handy for evaluating certain n-dimensional integrals over a general class of domains in terms of the gamma function.

Section 3

The purpose of this section is not readily apparent, trust me we'll need this in section 4 and it works like magic in combination with the Dirichlet integrals we studied in the last section, this section is dedicated mostly to defining a sequence of sets that point-wise converges to an orthotope (the sets ##C_N^n## above) which will be used in section 4 to evaluate certain multiple integrals over the unit hypercube.

Section 4

Involves some formulas for certain special functions represented as n-fold iterated integrals over the unit hypercube evaluated in a unique fashion. These special functions are the Lerch Transcendent, Legendre Chi, Polygamma, Polylogarithm of Order n, Hurwitz Zeta, Dirichlet Beta, Dirichlet Eta, and the Dirichlet Lambda functions (all of these depend on a positive integer n being the order of integration).

Section 5

Expands the previous section’s special functions represented as multiple integrals to fractional integrals which provide analytic continuations of the prior section's identities to complex orders of integration (n is continued to a complex-valued variable,) in particular, the Hadamard fractional integral operator is employed to this end.

The Integrals of Dirichlet

The proofs for Dirichlet Integrals have been allowed to follow as corollaries of the more general theorem of Louisville in Appendix A, and this has been done to drastically reduce the reading chore. In this section instead of proofs we simply state the the two most used corollaries in the remainder of this work.

Dirichlet integrals as I learned them from an Advanced Calculus book are just that formula evaluating the integral to Gamma functions, they are not a type of integral like Riemann integral, more just a formula that would go on a table of integrals. Content is the 4+-dimensional version of volume (some writers use hypervolume instead of content).

A result due to Dirichlet is given by

Corollary 2.2: Dirichlet Integrals (Modified Domain 1)

If ##t,{\alpha _p},{\beta _q},\Re \left[ {{\gamma _r}} \right] > 0\forall p,q,r## and ##V_t^n: = \left\{ {\left( {{z_1},{z_2}, \ldots ,{z_n}} \right) \in {\mathbb{R}^n}|{z_j} \geq 0\forall j,\sum\limits_{k = 1}^n {{{\left( {\frac{{{z_k}}}{{{\alpha _k}}}} \right)}^{{\beta _k}}} \leq t} } \right\}##, then

$$\iint {\mathop \cdots \limits_{V_t^n} \int {\prod\limits_{\lambda = 1}^n {\left( {z_\lambda ^{{\gamma _\lambda } - 1}} \right)} d{z_n} \ldots d{z_2}d{z_1}} } = {t^{\sum\limits_{p = 1}^n {\frac{{{\gamma _p}}}{{{\beta _p}}}} }}{{\prod\limits_{q = 1}^n {\left[ {\frac{{\alpha _q^{{\gamma _q}}}}{{{\beta _q}}}\Gamma \left( {\frac{{{\gamma _q}}}{{{\beta _q}}}} \right)} \right]} } \mathord{\left/{\vphantom {{\prod\limits_{q = 1}^n {\left[ {\frac{{\alpha _q^{{\gamma _q}}}}{{{\beta _q}}}\Gamma \left( {\frac{{{\gamma _q}}}{{{\beta _q}}}} \right)} \right]} } {\Gamma \left( {1 + \sum\limits_{k = 1}^n {\frac{{{\gamma _k}}}{{{\beta _k}}}} } \right)}}} \right. } {\Gamma \left( {1 + \sum\limits_{k = 1}^n {\frac{{{\gamma _k}}}{{{\beta _k}}}} } \right)}}$$

Continue reading...
Last edited:
  • Like
Likes Greg Bernhardt

Answers and Replies

  • #2
Do you have a specific question? ##\lim_{N\to \infty} C_N^n=[0,1)## is straightforward.
  • #3
Yes, it is, I had used that particular definition of ##C_N^n## with the ##\leq n-1## (opposed to ##\leq n##) to accommodate and singular point of an a few integrands in section 4, whereas the latter would converge to ##\left[ 0,1\right]##. These two definitions cover all of the integrals in the text.
  • #4
I updated the Insight to include more exercises in sections 4 and 5 and added a few answers to section 1 at the end.
  • #5
A few days ago I added the rest of the solutions to the exercises in section 1 (at the end of the document).
  • #6
I just wanted to say that God gave me every bit of skills, inspiration, people who helped me flesh this note out over the years I worked on it. Some people that helped me after having worked on this note throughout college and finally having had my analysis prof Akemann from UCSB read what was the beginnings of this note, of which he said what I was working with wasn’t well defined, and some years past. A couple of years ago God told me the finish my paper (I learned the other day that this work is called a note, because papers are published-I had been calling it my paper for quite a long time tho) so I came to PF and I got help from @fresh_42 and @FactChecker with and actually quite a lot of other math people here on PF helped (that would be a long list but I will just say search threads started by me and containing the word paper and from these search results you can see all the help I got, also there’s an Insights and Blog dev sub forum which is hidden by default with a few more threads with several advisors who also helped me).

Everything just came together those last few months I was writing this. I believe I had better defined what kind of integrals I was working over the limit of a sequence of nested sets using dominated convergence theorem which a suggestion iirc I got here on PF, I had the desire to extend the results I had derived (the n-fold integral representations of the Lerch Transcendent family of functions) and my results had an integer valued parameter n (the number of iterated integrals) I had wanted to analytically continue to a complex variable and I had the vague general notion the I could use fractional integrals to do this but hadn’t the foggiest idea what were or how to evaluate and integrals of this type until in my reading I stumbled upon the Hadamard fractional integral in a paper that was way over my head but which contained a simple result i could understand, the formula for iterated integral interpolation of the Hadamard fractional integral made to the results I mentioned earlier with the integer n parameter and these combined with the hypercube to simplex transformation @FactChecker had so graciously supplied me with worked out perfectly to derive the fractional integral representations of some special functions had no clue how to obtain just a few months earlier (Lerch transcendent family of functions).

Sorry this turned into kind of long winded credit where credit is due. I do feel that God gets the credit here, having worked all things to this end.

I’m currently rewriting this note, and seek a co-author if any of you are interested? I will say that I’m neural diverse, and if you contact me being interested in being a co-author I will tell you point-blank what my issues are, I just don’t feel comfortable doing so here publicly as I’m not knowing how these types of issues are handled in math circles yet.

Sorry for the essay.

Last edited:
  • Like
Likes jedishrfu and Greg Bernhardt
  • #8
Update: I added generous hints to Section 5 exercise 4) parts a) thru f), namely the desired fractional integral is stated: still remains to be shown that the fractional integral is equal to the series definition.
  • #9
The overhaul is through it's main phase, though I'm still going to post some more Answers to Exercises - Section 5. If previously you glanced at this Insight Article and found it rather too lengthy for you taste, please give it another go being as I have streamlined this work with the recommendation that you simply scan through sections 1 & 2. Sections 3 & 4 however are important for the proofs, and section 5 is the main results in terms of fractional integral representations of the Lerch Transcendent family of special functions.

Looking for co-author of this note that it might be elevated to the status of a paper. If you are interested, please say so here in the comments or send me a private message.
  • #10
I have just this minute added complete solutions to Exercises in Section 4, I do now believe there's only one problem whom does not have a solution provided now. Not bad!

Suggested for: A Path to Fractional Integral Representations of Some Special Functions