A class of hyperelliptic integrals are expressed through hypergeometric functions (like those of Gauss, Lauricella and Appell), namely multiple power series. Whenever they can on their own be reduced to elliptic ones via some algebraic transformation, we get a two-fold representation of the same mathematical object, and then several completely new π determinations through the above special functions and/or Euler integrals. All our π formulas have been succesfully tested by means of convenient Mathematica's packages and enter in a wide historical/sound context of π- formulas quite far from being exhausted. Due to their structure, the formulas' practical value does not lie in computing π, but in allowing, through pi, a benchmark for computing the involved special functions, particularly the less elementary ones.
G. Mingari Scarpello, D. Ritelli (2009). The hyperelliptic integrals and π. JOURNAL OF NUMBER THEORY, 129, 3094-3108 [10.1016/j.jnt.2009.06.002].
The hyperelliptic integrals and π
MINGARI SCARPELLO, GIOVANNI;RITELLI, DANIELE
2009
Abstract
A class of hyperelliptic integrals are expressed through hypergeometric functions (like those of Gauss, Lauricella and Appell), namely multiple power series. Whenever they can on their own be reduced to elliptic ones via some algebraic transformation, we get a two-fold representation of the same mathematical object, and then several completely new π determinations through the above special functions and/or Euler integrals. All our π formulas have been succesfully tested by means of convenient Mathematica's packages and enter in a wide historical/sound context of π- formulas quite far from being exhausted. Due to their structure, the formulas' practical value does not lie in computing π, but in allowing, through pi, a benchmark for computing the involved special functions, particularly the less elementary ones.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
