The q-analog of integration is given by
|
(1)
|
which reduces to
|
(2)
|
in the case
(Andrews 1986 p. 10).
Special cases include
A specific case gives
|
(7)
|
where
is the q-gamma function and is a doubly periodic sigma function. If , the integral reduces to
|
(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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