What is the term for this family of improper integrals?











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What is the name of the integrals of this form?




$$int_{0}^{infty} frac{sinleft(frac{x}{1}right)sinleft(frac{x}{3}right)cdotssinleft(frac{x}{2n + 1}right)}{left(frac{x}{1}right)left(frac{x}{3}right)cdotsleft(frac{x}{2n + 1}right)} :dx$$







integration terminology improper-integrals







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    @Travis - Thanks for the edit too :-)
    – DavidG
    2 days ago















up vote
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down vote

favorite
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What is the name of the integrals of this form?




$$int_{0}^{infty} frac{sinleft(frac{x}{1}right)sinleft(frac{x}{3}right)cdotssinleft(frac{x}{2n + 1}right)}{left(frac{x}{1}right)left(frac{x}{3}right)cdotsleft(frac{x}{2n + 1}right)} :dx$$







integration terminology improper-integrals







share|cite|improve this question




















  • 1




    @Travis - Thanks for the edit too :-)
    – DavidG
    2 days ago













up vote
7
down vote

favorite
2









up vote
7
down vote

favorite
2






2





What is the name of the integrals of this form?




$$int_{0}^{infty} frac{sinleft(frac{x}{1}right)sinleft(frac{x}{3}right)cdotssinleft(frac{x}{2n + 1}right)}{left(frac{x}{1}right)left(frac{x}{3}right)cdotsleft(frac{x}{2n + 1}right)} :dx$$







integration terminology improper-integrals







share|cite|improve this question















What is the name of the integrals of this form?




$$int_{0}^{infty} frac{sinleft(frac{x}{1}right)sinleft(frac{x}{3}right)cdotssinleft(frac{x}{2n + 1}right)}{left(frac{x}{1}right)left(frac{x}{3}right)cdotsleft(frac{x}{2n + 1}right)} :dx$$







integration terminology improper-integrals




integration terminology improper-integrals






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edited 2 days ago









Travis

58.4k765141




58.4k765141










asked 2 days ago









DavidG

671413




671413








  • 1




    @Travis - Thanks for the edit too :-)
    – DavidG
    2 days ago














  • 1




    @Travis - Thanks for the edit too :-)
    – DavidG
    2 days ago








1




1




@Travis - Thanks for the edit too :-)
– DavidG
2 days ago




@Travis - Thanks for the edit too :-)
– DavidG
2 days ago










1 Answer
1






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up vote
7
down vote



accepted










These are examples of Borwein integrals. See:




Borwein, D.; Borwein, J.M., "Some Remarkable Properties of $operatorname{Sinc}$ and Related Integrals", Ramanujan J. 5 (2001), 73-89.




John Baez' Azimuth blog has an illuminating post discussing these integrals, and Greg Egan gave an intuitive explanation in terms of Fourier transforms for the pattern-breaking phenomenon often mentioned when these integrals come up.






share|cite|improve this answer























  • Thanks @Travis!
    – DavidG
    2 days ago










  • Cheers!$!!!$
    – Travis
    2 days ago











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1 Answer
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active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
7
down vote



accepted










These are examples of Borwein integrals. See:




Borwein, D.; Borwein, J.M., "Some Remarkable Properties of $operatorname{Sinc}$ and Related Integrals", Ramanujan J. 5 (2001), 73-89.




John Baez' Azimuth blog has an illuminating post discussing these integrals, and Greg Egan gave an intuitive explanation in terms of Fourier transforms for the pattern-breaking phenomenon often mentioned when these integrals come up.






share|cite|improve this answer























  • Thanks @Travis!
    – DavidG
    2 days ago










  • Cheers!$!!!$
    – Travis
    2 days ago















up vote
7
down vote



accepted










These are examples of Borwein integrals. See:




Borwein, D.; Borwein, J.M., "Some Remarkable Properties of $operatorname{Sinc}$ and Related Integrals", Ramanujan J. 5 (2001), 73-89.




John Baez' Azimuth blog has an illuminating post discussing these integrals, and Greg Egan gave an intuitive explanation in terms of Fourier transforms for the pattern-breaking phenomenon often mentioned when these integrals come up.






share|cite|improve this answer























  • Thanks @Travis!
    – DavidG
    2 days ago










  • Cheers!$!!!$
    – Travis
    2 days ago













up vote
7
down vote



accepted







up vote
7
down vote



accepted






These are examples of Borwein integrals. See:




Borwein, D.; Borwein, J.M., "Some Remarkable Properties of $operatorname{Sinc}$ and Related Integrals", Ramanujan J. 5 (2001), 73-89.




John Baez' Azimuth blog has an illuminating post discussing these integrals, and Greg Egan gave an intuitive explanation in terms of Fourier transforms for the pattern-breaking phenomenon often mentioned when these integrals come up.






share|cite|improve this answer














These are examples of Borwein integrals. See:




Borwein, D.; Borwein, J.M., "Some Remarkable Properties of $operatorname{Sinc}$ and Related Integrals", Ramanujan J. 5 (2001), 73-89.




John Baez' Azimuth blog has an illuminating post discussing these integrals, and Greg Egan gave an intuitive explanation in terms of Fourier transforms for the pattern-breaking phenomenon often mentioned when these integrals come up.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 2 days ago

























answered 2 days ago









Travis

58.4k765141




58.4k765141












  • Thanks @Travis!
    – DavidG
    2 days ago










  • Cheers!$!!!$
    – Travis
    2 days ago


















  • Thanks @Travis!
    – DavidG
    2 days ago










  • Cheers!$!!!$
    – Travis
    2 days ago
















Thanks @Travis!
– DavidG
2 days ago




Thanks @Travis!
– DavidG
2 days ago












Cheers!$!!!$
– Travis
2 days ago




Cheers!$!!!$
– Travis
2 days ago


















 

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