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q-Integral

The q-analog of integration is given by

int_0^1f(x)d(q,x)=(1-q)sum_(i=0)^inftyf(q^i)q^i,
(1)

which reduces to

int_0^1f(x)dx
(2)

in the case q->1^- (Andrews 1986 p. 10).

Special cases include

int_0^1xd(q,x)=1/(1+q)
(3)
int_0^1x^2d(q,x)=1/(1+q+q^2)
(4)
int_0^1x^nd(q,x)=(q-1)/(q^(n+1)-1)
(5)
int_0^1lnxd(q,x)=(qlnq)/(1-q).
(6)

A specific case gives

int_0^infty(x^(a-1))/(1-x)d(q,x)=([Gamma_q(1/2)]^2)/(sigma_q(a)),
(7)

where Gamma_q is the q-gamma function and sigma_q is a doubly periodic sigma function. If q=1, the integral reduces to

int_0^infty(x^(a-1))/(1-x)dx=pi/(sin(pia)).
(8)

See also

q-Analog, q-Beta Function, q-Gamma Function

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References

Andrews, G. E. q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., 1986.Jackson, F. H. "q-Definite Integrals." Quart. J. Math. 41, 163, 1910.Jackson, F. H. "The q-Integral Analogous to Borel's Integral." Mess. Math. 47, 57-64, 1917.

Referenced on Wolfram|Alpha

q-Integral

Cite this as:

Weisstein, Eric W. "q-Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/q-Integral.html

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