# Closed form or simplification of a multiple definit integral of a product of a weight averaged parameters

I am trying to obtain a closed form solution of this definite integral, or in a form at least which simplify its numerical treatment. $$\int_{x_1=0}^1...\int_{x_N=0}^1 \prod_r \left( \frac {x_r f_r} {\sum_g x_g f_g}\right) ^{n_r} dx_1...dx_N$$ where $n_r$ are positive integers, and $f_r\ge0$ constants.

I tried to introduce a contour transformation, by $s=x_1 f_1 + ... + x_n f_n, y_2=x_2 f_2,..., y_N=x_N f_N$, |det Jacobian|=constant, which "reduce" to $$\int_{s=0}^{\sum_g f_g}ds\int_{y_2=0}^{?_2}dy_2...\int_{y_N=0}^{?_N}dy_N \prod_r \left( \frac {y_r} s\right) ^{n_r}$$

where $y_r$ are constrained (integral upper limits $?_r$) to total $s$ and be no greater than $f_r$, which seems no advance at all to me .

Your multiple integral $I$ can be converted into a single integral (to start with - more simplifications are perhaps possible). Use the identity $$x^{-s}=\frac{1}{\Gamma(s)}\int_0^\infty d\xi\ \xi^{s-1}e^{-\xi x}$$ with $s=\sum_r n_r$ to 'exponentiate' the denominator and obtain $$I = \left(\prod_r f_r^{n_r}\right)\frac{1}{\Gamma(\sum_r n_r)}\int_0^\infty d\xi\ \xi^{\sum_r n_r-1}\prod_{r=1}^N \left[\int_0^1 dx\ x^{n_r}e^{-\xi f_r x}\right]$$ $$= \left(\prod_r f_r^{n_r}\right)\frac{1}{\Gamma(\sum_r n_r)}\int_0^\infty d\xi\ \xi^{\sum_r n_r-1}\prod_{r=1}^N \left[(f_r \xi )^{-n_r-1}\gamma(n_r+1,\xi f_r)\right]\$$ $$=\left(\prod_r\frac{1}{f_r}\right)\frac{1}{\Gamma(\sum_r n_r)}\int_0^\infty d\xi \frac{\prod_{r=1}^N \gamma(n_r+1,\xi f_r)}{\xi^{N+1}}\ ,$$ where $\gamma(a,x)=\int_0^x dt\ t^{a-1}e^{-t}$ is the lower incomplete Gamma function http://mathworld.wolfram.com/IncompleteGammaFunction.html
Further simplifications are perhaps possible, noting that the lower incomplete Gamma function whose first argument is an integer (as in this case) has an elementary representation as $(1-\text{a finite sum})$ (see the link above).
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9938176274299622, "perplexity": 357.42431046880034}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526517.67/warc/CC-MAIN-20190720132039-20190720154039-00097.warc.gz"}