## Generalized elliptic integrals and modular equations

Venue: | Pacific J. Math |

Citations: | 21 - 8 self |

### BibTeX

@ARTICLE{Anderson_generalizedelliptic,

author = {G. D. Anderson and S. -l. Qiu and M. K. Vamanamurthy and M. Vuorinen},

title = {Generalized elliptic integrals and modular equations},

journal = {Pacific J. Math},

year = {},

pages = {1--37}

}

### OpenURL

### Abstract

In geometric function theory, generalized elliptic integrals and functions arise from the Schwarz-Christoffel transformation of the upper half-plane onto a parallelogram and are naturally related to Gaussian hypergeometric functions. Certain combinations of these integrals also occur in analytic number theory in the study of Ramanujan’s modular equations and approximations to π. The authors study the monotoneity and convexity properties of these quantities and obtain sharp inequalities for them. 1. Introduction. In 1995 B. Berndt, S. Bhargava, and F. Garvan published an important paper [BBG] in which they studied generalized modular equations and gave proofs for numerous statements concerning these equations made by Ramanujan in his unpublished notebooks. No record of Ramanujan’s original

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(Show Context)
Citation Context ... study the monotoneity and convexity properties of these quantities and obtain sharp inequalities for them. 1. Introduction. In 1995 B. Berndt, S. Bhargava, and F. Garvan published an important paper =-=[BBG]-=- in which they studied generalized modular equations and gave proofs for numerous statements concerning these equations made by Ramanujan in his unpublished notebooks. No record of Ramanujan’s origina... |

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(Show Context)
Citation Context ...construct a conformal map of a parallelogram with angles πa, π(1 − a), 0 <a<1,onto a half-plane. This map is denoted by sna. For a= 1 2this map reduces to the well-known Jacobian elliptic function sn =-=[Bo]-=-. In Sections 3 and 4 we summarize some of the basic properties of the hypergeometric functions, obtaining a new proof for a formula due to Ramanujan. We also derive differentiation formulas for Ka, E... |

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Citation Context ...2 ) satisfies rr ′ 2 d2y dr2 + [(2c − 1) − (2a +2b+1)r2] dy − 4abry =0. dr14 G.D. ANDERSON, S.-L. QIU, M.K. VAMANAMURTHY AND M. VUORINEN For a review of the theory of this differential equation, see =-=[Var]-=-. In particular, the functions y = Ka(r) (or K ′ a(r)) and z = Ea(r) (or E ′ a(r)) satisfy the differential equations ⎧ ⎪⎨ rr ⎪⎩ ′ 2 d2y dr2 +(1−3r2) dy − 4a(1 − a)ry =0, dr rr ′ 2 d2z + r′ 2 dz dr2 d... |

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(Show Context)
Citation Context ...i function ψ(x), and the beta function B(x, y). For Re x>0, Re y>0, these functions are defined by (3.1) Γ(x) ≡ ∫∞ 0 e −t t x−1 dt, ψ(x) ≡ Γ′ (x) Γ(x)Γ(y) ,B(x, y) ≡ Γ(x) Γ(x + y) , respectively (cf. =-=[WW]-=-). It is well known that the gamma function satisfies the difference equation [WW, p. 237] (3.2) Γ(x +1)=xΓ(x), and the reflection property [WW, p. 239] Γ(x)Γ(1 − x) = π = B(x, 1 − x). sin πx (3.3) We... |

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(Show Context)
Citation Context ...d (3.8), it follows that this constant has the asserted value, and (5) follows. If, in addition, c = 1, then (6) and (7) follow from (1) and (2), respectively. □ 3.14. Elliott’s formula. E.B. Elliott =-=[El]-=- (cf. [Ba, p. 85]) proved the identity (3.15) ( ) 1 F + λ, −1 − ν;1+λ+µ;z F 2 2 + F −F ( ) 1 1 −λ, + ν;1+ν+µ;1−z 2 2 ( ) ( 1 1 + λ, − ν;1+λ+µ;z F − 2 2 1 ) 1 −λ, + ν;1+ν+µ;1−z 2 2 ( ) ( ) 1 1 1 1 +λ, ... |

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