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**Introduction**

*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.

**Summary**

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

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