TY - JOUR
T1 - Complete p-elliptic integrals and a computation formula of πp for p= 4
AU - Takeuchi, Shingo
PY - 2018/6/1
Y1 - 2018/6/1
N2 - The complete p-elliptic integrals are generalizations of the complete elliptic integrals by the generalized trigonometric function sin pθ and its half-period πp. It is shown, only for p= 4 , that the generalized p-elliptic integrals yield a computation formula of πp in terms of the arithmetic–geometric mean. This is a πp-version of the celebrated formula of π, independently proved by Brent and Salamin in 1976.
AB - The complete p-elliptic integrals are generalizations of the complete elliptic integrals by the generalized trigonometric function sin pθ and its half-period πp. It is shown, only for p= 4 , that the generalized p-elliptic integrals yield a computation formula of πp in terms of the arithmetic–geometric mean. This is a πp-version of the celebrated formula of π, independently proved by Brent and Salamin in 1976.
KW - Arithmetic–geometric mean
KW - Brent–Salamin’s algorithm
KW - Complete elliptic integrals
KW - Generalized trigonometric functions
KW - p-Laplacian
UR - http://www.scopus.com/inward/record.url?scp=85045150378&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85045150378&partnerID=8YFLogxK
U2 - 10.1007/s11139-018-9993-y
DO - 10.1007/s11139-018-9993-y
M3 - Article
AN - SCOPUS:85045150378
VL - 46
SP - 309
EP - 321
JO - Ramanujan Journal
JF - Ramanujan Journal
SN - 1382-4090
IS - 2
ER -