Analytical solutions of incomplete elliptic integrals motivated by the work of Ramanujan

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Analytical solutions of incomplete elliptic integrals motivated by the

work of Ramanujan

M. I. Qureshi

a

, C. W. Mohd

b

, M. P. Chaudhary

c,∗

, K. A. Quraishi

d

and I. H. Khan

d

a,b_{Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia, New Delhi-110025, India.}

c,d_{Former Researcher, Department of Applied Sciences and Humanities, Faculty of Engineering and Technology, Jamia Millia Islamia, New Delhi-110025, India.}

Abstract

In this paper, we obtain exact solutions of some unsolved incomplete elliptic integrals of first, second and

third kinds, given in Entry 7 of Chapter XVII of second notebook of Srinivasa Ramanujan. Furthermore,

we generalize these elliptic integrals in the forms of multiple series identities involving bounded multiple

sequences.

Keywords: Gaussian Hypergeometric Function; Incomplete Elliptic Integrals; Multiple Series Identities;

Srivastava-Daoust double Hypergeometric Function ; Kamp´e de F´eriet double Hypergeometric Function;

Sri-vastava’s Triple Hypergeometric Function.

1

Introduction and Preliminaries

Some Interesting Series Identities

We recall the following identities which are potentially useful in the series rearrangement techniques.

∞

∑

m=0 m−1

∑

r=0

Θ(m, r) =

∞

∑

m=0

∞

∑

r=0

Θ(m+r+1, r) (1.1)

∞

∑

m=0 2m−1

∑

r=0

Θ(m, r) =

∞

∑

m=0

∞

∑

r=0

Θ(m+r+1, r) +

∞

∑

m=0

∞

∑

r=0

Θ(m+r+1, 2r+m+1) (1.2)

∗

Corresponding author.

E-mail address:miqureshi [email protected](M. I. Qureshi),mpchaudhary [email protected](M. P. Chaudhary),[email protected]

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∞

∑

m=0

∞

∑

n=0 m+n−1

∑

r=0

Ψ(m, n, r) =

∞

∑

n=0

∞

∑

r=0

Ψ(0, n+r+1, r) +

∞

∑

m=0

∞

∑

n=0

∞

∑

r=0

Ψ(m+r+1, n, r)+

+

∞

∑

m=0

∞

∑

n=0

∞

∑

r=0

Ψ(m+1, n+r+1, r+m+1) (1.3)

∞

∑

m=0

∞

∑

n=0 m+n

∑

r=0

Ψ(m, n, r) =

∞

∑

m=0

∞

∑

n=0

∞

∑

r=0

Ψ(m+r, n, r) +

∞

∑

m=0

∞

∑

n=0

∞

∑

r=0

Ψ(m, n+r+1, r+m+1) (1.4)

where{Θ(m,r)}∞_m,r₌₀and{Ψ(m,n,r)}∞_m,n,r₌₀are suitably bounded double and triple sequences of essentially arbitrary(real or complex) parameters respectively.

Legendre’s Normal Forms of Incomplete Elliptic Integrals

Following integrals arise in the solutions of certain classes of vibration problems and problems involving

computations of the radiation field off axis from a uniform circular disc radiating according to an arbitrary

angular distribution law.

Following elliptic integrals (R.H.S.) have been represented in different notations (L.H.S.) by researchers

First Kind : F(x, φ) =

Z φ

0

dθ

q

(1−x2_sin2 θ)

(1.5)

Second Kind : E(x, φ) =

Z φ

0

q

(1−x2_sin2

θ)dθ (1.6)

Third Kind :

∏

(a, x, φ) =

Z φ

0

dθ

(1−asin2θ)

q

(1−x2_sin2_θ₎

(1.7)

where 0≤ x≤ 1, 0≤ φ≤ π

2, −∞<a<∞, a6=1

The integrands of elliptic integrals are periodic functions with a periodπ. Herex, φandaare called modulus,

amplitude and characteristic parameter respectively. In casex=sinδ,δis called modular angle.

Some Useful Indefinite Integrals

Whenm=0, 1, 2, 3,· · ·, then

Z

sin2m(cθ)dθ=

(

−(1₂)_msin(cθ)cos(cθ) (1)m c

m−1

∑

r=0

(1)_rsin2r(cθ) (3₂)_r

)

+

(

θ(1₂)m

(1)m

)

+Constant (1.8)

Z

cos2m(cθ)dθ=

(

(1₂)_msin(cθ)cos(cθ) (1)m c

m−1

∑

r=0

(1)_rcos2r(cθ) (3₂)_r

)

+

(

θ(1₂)m

(1)m

)

+Constant (1.9)

Z

sin2m+1(cθ)dθ= −(1)mcos(cθ)

(3₂)m c m

∑

r=0

(1₂)rsin2r(cθ)

(1)r

+Constant (1.10)

Z

cos2m+1(cθ)dθ= (1)msin(cθ)

(3₂)_m c m

∑

r=0

(1₂)rcos2r(cθ)

(1)_r +Constant (1.11)

Above formulas (1.8)-(1.11) can be verified form=0, 1, 2, 3,· · · and it is the convention that the empty sum −1

∑

r=0

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2

Seventh entry of Chapter Seventeenth of Second Notebook of

Ramanujan[9,pp.104-107,pp.112-113]

Ramanujan’s notebooks have been divided into several chapters and contains large numbers of important

and useful results on elliptic integrals. Many results on elliptic integrals have been proved by B. C. Berndt [8,

pp.104-113] and R. Y. Deniset al.[15].

We have also verified the following entries of Ramanujan by using the methods of B. C. Berndt and R. Y. Denis

et al.

Entry 7(i): If sinα= √

xsinβor_tantanα_β =

√

xcosβ

√

(1−xsin2_β₎, then Z α

0

dθ

q

(x−sin2θ)

=

Z β

0

dθ

q

(1−xsin2θ)

(2.1)

In a paper of Deniset al.[15, p.113(1)], a misprint condition is written in Entry 7(i).

Entry 7(ii): If tanα=p(1−x)tanβor sinα= √

(1−x)sinβ

√

(1−xsin2β), then

Z α

0

dθ

p

(1−xcos2_θ₎ =

Z β

0

dθ

q

(1−xsin2θ)

(2.2)

Entry 7(iii): If tanα=p(1−b)tanβor sinβ= √ sinα

(1−bcos2_α₎, then Z α

0

dθ

q

{1−(a₁−₋b_b)sin2θ}

=p(1−b)

Z β

0

dθ

q

(1−asin2θ) (1−bsin2θ)

(2.3)

Entry 7(iv): If tanα=p(1+x)tanβ, then

Z α

0

dθ

p

(1+xcos 2θ)

=

Z β

0

dθ

q

(1−x2_sin4 θ)

(2.4)

which is the correct form of a misprint result of Deniset al.[15, p.115(4)].

Entry 7(v): Degenerate form of addition theorem

If cotαcotβ=p(1−x), then

Z α

0

dθ

q

(1−xsin2θ)

+

Z β

0

dθ

q

(1−xsin2θ)

=π

2 2F1 

   

1 2, 12 ;

x

1 ;



   

(2.5)

Entry 7(vi): Classical duplication formula (Special Case of the converse of Entry 7(viii)a)

If cotαtan(β₂) =

q

(1−xsin2α), then

2 Z α

0

dθ

q

(1−xsin2θ)

=

Z β

0

dθ

q

(1−xsin2θ)

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Entry 7(vii): Jacobi’s imaginary transformation (For incomplete elliptic integral of first kind having imag-inary amplitude)

Ifα=ln{tan(π₄ +β₂)}oreα = ₁₋cos_sinβ_β = 1+_cossin_ββ or sinh(α) =tanβ, then

Z iα

0

dθ

q

(1−xsin2θ)

=i

Z β

0

dθ

q

(1−(1−x)sin2θ)

(2.7)

which is the correct form of a misprint result of Deniset al.[15, p.116(7)].

Entry 7(viii): Addition theorem

If

Z α

0

dθ

q

(1−xsin2θ)

+

Z β

0

dθ

q

(1−xsin2θ)

=

Z γ

0

dθ

q

(1−xsin2θ)

(2.8)

then four different sets of hypothesis (implications) are given by

(a) tanγ 2

= sinα

q

(1−xsin2β) +sinβ

q

(1−xsin2α)

cosα+cosβ

which is the correct form of a misprint condition of Deniset al.[15, p.117(i)].

(b) γ=tan−1{tanα

q

(1−xsin2β)}+tan−1{tanβ

q

(1−xsin2α)}

(c) Legendre’s canonical form of the addition theorem

cotαcotβ= cosγ

sinαsinβ

+

q

(1−xsin2γ)

(d)

√ x

2 =

p

{sin(s)sin(s−α)sin(s−β)sin(s−γ)}

sinα sinβsinγ ;where s

= α+β+γ

2

The four different sets of hypothesis (implications)(a)−(d)in Entry 7(viii) are both necessary and sufficient.

Entry 7(xii): Gauss’ transformation

If(1+xsin2α)sinβ= (1+x)sinα, then

(1+x)

Z α

0

dθ

q

(1−x2_sin2 θ)

=

Z β

0

dθ

q

{1− 4x

(1+x)2sin

2_θ} (2.9)

which is the correct form of misprint result of Deniset al.[15, p.118(9)].

Entry 7(xiii): Landen’s transformation (i.e. The first geometric representation)

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(1+x)

Z α

0

dθ

q

(1−x2_sin2 θ)

=2

Z β

0

dθ

q

{1− 4x

(1+x)2sin

2_θ} (2.10)

Entries 7(xii) and 7(xiii) are very similar in appearance.

3

A General Family of Multiple-Series Identities

Theorem 3.1. Let{Ωm}∞_m=0be a suitably bounded sequence of arbitrary complex numbers. Then

∞

∑

m=0

Ωm

Z γ

0

sin2m(cθ)dθ

_ym

m!

=−ysin(cγ)cos(cγ)

2c

∞

∑

m=0

∞

∑

r=0

Ωm+r+1

(3₂)m+r(1)rym(ysin2(cγ))r

(2)m+r(2)m+r(3₂)r

+γ

∞

∑

m=0

Ωm

(1₂)mym

(m!)2 (3.1) provided that each of the series involved is absolutely convergent.

∞

∑

m=0

Ωm

Z γ

0

cos2m(cθ)dθ

_ym

m!

= ysin(cγ)cos(cγ)

2c

∞

∑

m=0

∞

∑

r=0

Ωm+r+1

(3₂)m+r(1)rym(ycos2(cγ))r

(2)m+r(2)m+r(3₂)r

+γ

∞

∑

m=0

Ωm

(1₂)mym

(m!)2 (3.2) provided that each of the series involved is absolutely convergent.

∞

∑

m=0

Ωm

Z γ

0

sin4m(cθ)dθ

_ym

m! =γ

∞

∑

m=0

Ωm

(1₄)m(3₄)mym

(1₂)_m(m!)2 −

−3ysin(cγ)cos(cγ)

8c

∞

∑

m=0

∞

∑

r=0

Ωm+r+1

(5₄)_m+r(7₄)m+r(1)rym(ysin2(cγ))r

(2)m+r(2)m+r(3₂)m+r(3₂)r −

− ysin

3₍_cγ₎_cos₍_cγ₎

4c

∞

∑

m=0

∞

∑

r=0

Ωm+r+1

(5₄)m+r(₄7)m+r(2)m+2r(ysin2(cγ))m(ysin4(cγ))r

(2)m+r(2)m+r(3₂)m+r(5₂)m+2r

(3.3)

provided that each of the series involved is absolutely convergent.

Theorem 3.4. Let{Ω_m}∞_m₌₀be a suitably bounded sequence of arbitrary complex numbers. Then

∞

∑

m=0

Ωm

Z γ

0

cos4m(cθ)dθ

ym

m! =γ

∞

∑

m=0

Ωm

(1₄)_m(3₄)mym

(1₂)_m(m!)2 +

+3ysin(cγ)cos(cγ)

8c

∞

∑

m=0

∞

∑

r=0

Ωm+r+1

(5₄)m+r(7₄)m+r(1)rym(ycos2(cγ))r

(2)m+r(2)m+r(3₂)m+r(3₂)r

+

+ ysin(cγ)cos

3₍_cγ₎

4c

∞

∑

m=0

∞

∑

r=0

Ωm+r+1

(5₄)m+r(₄7)m+r(2)m+2r(ycos2(cγ))m(ycos4(cγ))r

(2)m+r(2)m+r(3₂)m+r(5₂)m+2r

(3.4)

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Theorem 3.5. Let{Λm,n}∞_m,n=0be a suitably bounded double sequence of arbitrary complex numbers. Then

∞

∑

m=0

∞

∑

n=0

Λm,n

Z γ

0

sin2m+2n(cθ)dθ

_ym_zn

m!n!

=γ

∞

∑

m=0

∞

∑

n=0

Λm,n

(1₂)m+nymzn

(m+n)!m!n! −

zsin(cγ)cos(cγ)

2c

∞

∑

n=0

∞

∑

r=0

Λ0,n+r+1

(3₂)n+r(1)r zn(zsin2(cγ))r

(2)_n+r(2)n+r(3₂)r −

− ysin(cγ)cos(cγ)

2c

∞

∑

m=0

∞

∑

n=0

∞

∑

r=0

Λm+r+1,n

(3₂)_m+n+r(1)rymzn(ysin2(cγ))r

(2)_m+n+r(2)m+r(3₂)r(1)n −

− yzsin

3₍_cγ₎_cos₍_cγ₎

4c

∞

∑

m=0

∞

∑

n=0

∞

∑

r=0

Λm+1,n+r+1

(5₂)m+n+r(2)m+r(ysin2(cγ))mzn(zsin2(cγ))r

(3)m+n+r(5₂)m+r(2)n+r(2)m

(3.5)

Theorem 3.6. Let{Λm,n}∞m,n=0be a suitably bounded double sequence of arbitrary complex numbers. Then

∞

∑

m=0

∞

∑

n=0

Λm,n

Z γ

0

cos2m+2n(cθ)dθ

ymzn

m!n!

=γ

∞

∑

m=0

∞

∑

n=0

Λm,n

(1₂)m+nymzn

(m+n)!m!n!+

zsin(cγ)cos(cγ)

2c

∞

∑

n=0

∞

∑

r=0

Λ0,n+r+1

(3₂)n+r(1)r zn(zcos2(cγ))r

(2)n+r(2)n+r(3₂)r

+

+ ysin(cγ)cos(cγ)

2c

∞

∑

m=0

∞

∑

n=0

∞

∑

r=0

Λm+r+1,n

(3₂)m+n+r(1)rymzn(ycos2(cγ))r

(2)m+n+r(2)m+r(3₂)r(1)n

+

+ yzsin(cγ)cos

3₍_cγ₎

4c

∞

∑

m=0

∞

∑

n=0

∞

∑

r=0

Λm+1,n+r+1

(5₂)_m+n+r(2)m+r(ycos2(cγ))mzn(zcos2(cγ))r

(3)_m+n+r(5₂)m+r(2)n+r(2)m

(3.6)

∞

∑

m=0

Ωm

Z γ

0

sinm(cθ)dθ

ym m! =γ ∞

∑

m=0

Ω2m

y2m

4m₍_m_!₎2 −

y2_sin₍_cγ₎_cos₍_cγ₎

4c

∞

∑

m=0

∞

∑

r=0

Ω2m+2r+2

(1)ry2m(ysin(cγ))2r

4m+r₍₂₎_m

+r(2)m+r(3₂)r

+

+y

c

∞

∑

m=0

Ω2m+1

y2m

4m₍3 2)m(32)m

−ycos(cγ) c

∞

∑

m=0

∞

∑

r=0

Ω2m+2r+1

(1₂)_ry2m(ysin(cγ))2r

4m+r₍3

2)m+r(32)m+r(1)r

(3.7)

∞

∑

m=0

Ωm

Z γ

0

cosm(cθ)dθ

ym m! =γ ∞

∑

m=0

Ω2m y 2m

4m₍_m_!₎2 +

y2sin(cγ)cos(cγ)

4c

∞

∑

m=0

∞

∑

r=0

Ω2m+2r+2

(1)ry2m(ycos(cγ))2r

4m+r₍₂₎

m+r(2)m+r(32)r

+

+ysin(cγ)

c

∞

∑

m=0

∞

∑

r=0

Ω2m+2r+1

(1₂)ry2m(ycos(cγ))2r

4m+r₍3

2)m+r(32)m+r(1)r

(3.8)

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∞

∑

m=0

Ωm

Z γ

0

sin2m+1(cθ)dθ

_ym

m! =

1

c

∞

∑

m=0

Ωm y m

(3₂)_m −

cos(cγ)

c

∞

∑

m=0

∞

∑

r=0

Ωm+r

(1₂)r ym(ysin2(cγ))r

(3₂)m+r(1)r

(3.9)

∞

∑

m=0

Ωm

Z γ

0

cos2m+1(cθ)dθ

_ym

m! =

sin(cγ)

c

∞

∑

m=0

∞

∑

r=0

Ωm+r

(1₂)r ym(ycos2(cγ))r

(3₂)m+r(1)r

(3.10)

Theorem 3.11. Let{Λm,n}∞_m,n=0be a suitably bounded double sequence of arbitrary complex numbers. Then

∞

∑

m=0

∞

∑

n=0

Λm,n

Z γ

0

sin2m+2n+1(cθ)dθ

ymzn

m!n!

=−cos(cγ) c

∞

∑

m=0

∞

∑

n=0

∞

∑

r=0

Λm+r,n

(1)_m+n+r(1₂)_rymzn(ysin2(cγ)) r

(3₂)_m₊_n₊_r(1)_m₊_r(1)_n (1)_r −

− zcos(cγ)sin 2₍_cγ₎

3c

∞

∑

m=0

∞

∑

n=0

∞

∑

r=0

Λm,n+r+1

(2)_m₊_n₊_r(3₂)_m₊_r(ysin2(cγ))mzn(zsin2(cγ))r (5₂)_m₊_n₊_r(2)_n₊_r(2)_m₊_rm! +

+ 1

c

∞

∑

m=0

∞

∑

n=0

Λm,n

(1)_m₊_nymzn

(3₂)_m

+nm!n!

(3.11)

Theorem 3.12. Let{Λm,n}∞m,n=0be a suitably bounded double sequence of arbitrary complex numbers. Then

∞

∑

m=0

∞

∑

n=0

Λm,n

Z γ

0

cos2m+2n+1(cθ)dθ

ymzn

m!n!

= sin(cγ)

c

∞

∑

m=0

∞

∑

n=0

∞

∑

r=0

Λm+r,n

(1)m+n+r(1₂)_rymzn(ycos2(cγ))r

(3₂)_m

+n+r(1)m+r(1)n(1)r

+

+ zsin(cγ)cos

2₍_cγ₎

3c

∞

∑

m=0

∞

∑

n=0

∞

∑

r=0

Λm,n+r+1

(2)_m₊_n₊_r(3₂)_m₊_r(ycos2(cγ))mzn(zcos2(cγ))r

(5₂)_m₊_n₊_r(2)_m₊_r(2)_n₊_r m! (3.12)

Proof of (3.1): ∞

∑

m=0

Ωm

Z γ

0

sin2m(cθ)dθ

ym

m!

=−

∞

∑

m=0 m−1

∑

r=0

Ωmsin

(cγ)cos(cγ)(1₂)m(1)rym(sin2(cγ))r c(1)m(1)m(3₂)r

+γ

∞

∑

m=0

Ωm

(1₂)mym

(m!)2

=−

∑

∞ m=0

m

∑

r=0

Ωm+1

sin(cγ)cos(cγ)(1₂)_m₊₁(1)_r ym+1(sin2(cγ))r

c(1)_m₊₁(1)_m₊₁(3₂)_r +γ

∞

∑

m=0

Ωm

(1₂)mym

(m!)2

=−ysin(cγ)cos(cγ)

2c

∞

∑

m=0

∞

∑

r=0

Ωm+r+1

(3₂)m+r(1)rym(ysin2(cγ))r

(2)m+r(2)m+r(3₂)r

+γ

∞

∑

m=0

Ωm

(1₂)mym

(m!)2

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4

Hypergeometric Generalizations of Incomplete Elliptic Integrals of Ramanujan and

their solutions

Puttingc=1 in theorems (3.1) to (3.6), (3.9) to (3.12) and setting

Ωm=

(a1)m(a2)m(a3)m· · ·(aA)m

(b1)m(b2)m(b3)m· · ·(bB)m

= ((aA))m ((bB))m, Λm,n

= ((aA))m+n((dD))m((gG))n ((bB))m+n((eE))m((hH))n ,

in theorems (3.1) to (3.12), using some algebraic properties of Pochhammer symbol and interpreting the

mul-tiple power series in hypergeometric forms of Gauss [34, p. 42(1)], Kamp´e de F´eriet [34, p.63(16); see also 33],

Srivastava [34, p.69(39,40)] and Srivastava-Daoust [33, p.37(21, 22, 23); 34, pp.64-65(18, 19, 20)], we get the

analytical solutions of generalized incomplete elliptic integrals.

Z γ

0 A FB



   

(aj)Aj=1 ;

ysin2θ

(bj)B_j₌₁ ;



   

dθ=γA+1FB+1



   

1

2,(aj)Aj=1 ; y

1,(bj)B_j₌₁ ;



   

−

ysinγcosγ

A

∏

i=1

(ai)

2∏B

i=1

(bi)

F_BA₊+₂1_::₀1_;;₁2



   

3

2,(1+aj)Aj=1 :1 ;1, 1 ;

y,ysin2γ

2, 2,(1+bj)Bj=1: ;32 ;



   

(4.1)

Z γ

0 A FB



   

(aj)jA=1 ;

ycos2θ

(bj)B_j₌₁ ;



   

dθ=γA+1FB+1



   

1

2,(aj)Aj=1 ; y

1,(bj)B_j₌₁ ;



   

+

ysinγ cosγ

A

∏

i=1

(ai)

2 ∏B

i=1

(bi)

F_BA₊+₂1_::₀1_;₁;2



   

3

2,(1+aj)Aj=1 :1 ;1 , 1 ;

y,ycos2_γ

2, 2,(1+bj)Bj=1: ; 32 ;



   

(4.2)

Z γ

0 A

FB



   

(aj)Aj=1 ;

ysin4θ

(bj)Bj=1 ;



   

dθ=γA+2FB+2



   

1

4,34,(aj)jA=1 ; y

1,1₂,(bj)Bj=1 ;



   

−

3y sinγcosγ A

∏

i=1

(ai)

8∏B

i=1

(bi)

F_BA₊+₃2_::₀1_;₁;2



   

5

4,74,(1+aj)Aj=1 : 1; 1, 1 ;

y,ysin2γ

2, 2,3₂,(1+bj)Bj=1: ; 32 ;



   

−

ysin3γ cosγ

A

∏

i=1

(ai)

4∏B

i=1

(bi)

F_BA₊+₄3_:₀:1_;;₀1



   

[(1+aj):1, 1]jA=1, [54:1, 1], [74:1, 1], [2:1, 2] :

(9)

[1:1];[1:1] ;

ysin2γ,ysin4γ

; ;



   

(4.3)

Z γ

0 A

FB



   

(aj)Aj=1 ;

ycos4θ

(bj)Bj=1 ;



   

dθ=γA+2FB+2



   

1

4,34,(aj)Aj=1 ; y

1,1₂,(bj)Bj=1 ;



   

+

3y sinγcosγ A

∏

i=1

(ai)

8 ∏B

i=1

(bi)

F_BA₊+₃2_:₀:1_;;₁2



   

5

4,74,(1+aj)Aj=1 : 1;1, 1 ;

y,ycos2γ

2, 2,3₂,(1+bj)B_j₌₁: ;3₂ ;



   

+

ysinγcos3γ

A

∏

i=1

(ai)

4∏B

i=1

(bi)

F_BA₊+₄3_:₀:1_;;₀1



   

[(1+aj):1, 1]A_j₌₁, [5₄:1, 1], [7₄:1, 1], [2:1, 2] :

[(1+bj):1, 1]jB=1,[2:1, 1],[2:1, 1],[32:1, 1],[52:1, 2] :

[1:1];[1:1] ;

ycos2γ,ycos4γ

; ;



   

(4.4)

Z γ

0

F_BA_::_ED_;_H;G



   

(aj)Aj=1:(dj)Dj=1;(gj)Gj=1 ;

ysin2θ,zsin2θ

(bj)B_j₌₁:(ej)_jE₌₁;(hj)H_j₌₁ ;



   

dθ

=γ F_BA₊+₁1_:_E:D_;_H;G



   

(aj)A_j₌₁, 1₂:(dj)D_j=1;(gj)Gj=1 ;

y,z

(bj)B_j₌₁, 1:(ej)E_j₌₁;(hj)H_j₌₁ ;



   

−

zsinγcosγ

A

∏

i=1

(ai) _∏G

i=1

(gi)

2∏B

i=1

(bi) H

∏

i=1

(hi)

×

× F_BA₊+_HG₊+₂1_::₀1_;;₁2



   

3

2,(1+aj)Aj=1,(1+gj)Gj=1 : 1;1, 1 ;

z,zsin2γ

2, 2, (1+bj)Bj=1,(1+hj)Hj=1: ; 32 ;



   

−

y sinγcosγ

A

∏

i=1

(ai) _∏D

i=1

(di)

2∏B

i=1

(bi) E

∏

i=1

(ei)

×

×F(3)



   

3

2,(1+aj)jA=1:: ; ;(1+dj)Dj=1 :1;(gj)Gj=1;1, 1 ;

y,z,ysin2γ

2,(1+bj)B_j₌₁:: ; ;(1+ej)E_j₌₁, 2: ;(hj)H_j₌₁;3₂ ;



   

(10)

−

y zsin3γ cosγ

A

∏

i=1

(ai) A

∏

i=1

(1+ai) D

∏

i=1

(di) G

∏

i=1

(gi)

4 ∏B

i=1

(bi) ∏B

i=1

(1+bi) ∏E

i=1

(ei) ∏H

i=1

(hi)

×

×F(3)



   

5

2,(2+aj)Aj=1:: ;(1+gj)Gj=1 ;2:1,(1+dj)Dj=1;1;1 ;

ysin2γ,z,zsin2γ

3,(2+bj)B_j₌₁:: ;(1+hj)H_j₌₁, 2;5₂:2,(1+ej)E_j₌₁; ; ;



   

(4.5)

Z γ

0

F_BA_::_ED_;_H;G



   

(aj)Aj=1:(dj)Dj=1;(gj)Gj=1 ;

ycos2_θ_,_z_cos2_θ

(bj)Bj=1:(ej)Ej=1;(hj)Hj=1 ;



   

dθ

=γF_BA₊+₁1_:_E:D_;_H;G



   

(aj)jA=1, 12:(dj)Dj=1;(gj)Gj=1 ;

y,z

(bj)Bj=1, 1:(ej)Ej=1;(hj)jH=1 ;



   

+

zsinγcosγ

A

∏

i=1

(ai) G

∏

i=1

(gi)

2 ∏B

i=1

(bi) ∏H

i=1

(hi)

×

×F_BA₊+_HG₊+₂1_::₀1_;;₁2



   

3

2,(1+aj)Aj=1,(1+gj)Gj=1 : 1;1, 1 ;

z,zcos2_γ

2, 2, (1+bj)Bj=1,(1+hj)Hj=1: ; 32 ;



   

+

ysinγ cosγ

A

∏

i=1

(ai) D

∏

i=1

(di)

2 ∏B

i=1

(bi) E

∏

i=1

(ei)

×

×F(3)



   

3

2,(1+aj)jA=1:: ; ;(1+dj)Dj=1 :1;(gj)Gj=1; 1, 1 ;

y,z,ycos2_γ

2, (1+bj)jB=1:: ; ;(1+ej)Ej=1, 2: ;(hj)Hj=1; 32 ;



   

+

y zsinγcos3γ

A

∏

i=1

(ai) _∏A

i=1

(1+ai) _∏D

i=1

(di) _∏G

i=1

(gi)

4 ∏B

i=1

(bi) B

∏

i=1

(1+bi) E

∏

i=1

(ei) H

∏

i=1

(hi)

F(3)



   

5

2,(2+aj)Aj=1:: ;(1+gj)Gj=1 ;2:

3,(2+bj)B_j₌₁:: ;(1+hj)H_j₌₁, 2 ;5₂:

1,(1+dj)Dj=1; 1 ; 1 ;

ycos2γ,z,zcos2γ

2,(1+ej)Ej=1; ; ;



   

(4.6)

Z γ

0 A FB



   

(aj)jA=1 ;

ysin(cθ)

(bj)B_j₌₁ ;



   

(11)

=γ2AF2B+1



   

(a₂j)_jA₌₁,(1+₂aj)A_j₌₁ ;

y2 4(1+B−A)

1,(b₂j)B_j₌₁,(1+₂bj)B_j₌₁ ;



   

−

ycos(cγ)

A

∏

i=1

(ai)

c ∏B

i=1

(bi) ×

×F_2B2A₊₂:_:1₀;_;1₀



   

(1+₂aj)_jA₌₁,(2+₂aj)A_j₌₁ :1;1₂ ;

y2

4(1+B−A),

y2_sin2₍_c

γ)

4(1+B−A)

3 2, 32,(

1+bj 2 )Bj=1,(

2+bj

2 )Bj=1: ; ;



   

−

y2sin(cγ)cos(cγ)_∏A

i=1

(ai)_∏A

i=1

(1+ai)

4c ∏B

i=1

(bi) B

∏

i=1

(1+bi)

×

×F_2B2A₊₂:_:1₀;_;2₁



   

(2+₂aj)_jA₌₁,(3+₂aj)_jA₌₁ :1;1, 1 ;

y2

4(1+B−A),

y2_sin2₍_c

γ)

4(1+B−A)

2, 2,(2+₂bj)B_j₌₁,(3+₂bj)B_j₌₁: ; 3₂ ;



   

+

y ∏A

i=1

(ai)

c∏B

i=1

(bi)

2A+1F2B+2



   

1,(1+₂aj)A_j₌₁,(2+₂aj)A_j₌₁ ;

y2

4(1+B−A)

3 2, 32,(

1+bj 2 )Bj=1,(

2+bj 2 )Bj=1 ;



   

(4.7)

Z γ

0 A FB



   

(aj)Aj=1 ;

ycos(cθ)

(bj)B_j₌₁ ;



   

dθ

=γ2AF2B+1



   

(a₂j)_jA₌₁,(1+₂aj)_jA₌₁ ;

y2

4(1+B−A)

1,(b₂j)B_j₌₁,(1+₂bj)B_j₌₁ ;



   

+

ysin(cγ)

A

∏

i=1

(ai)

c ∏B

i=1

(bi) ×

×F_2B2A₊₂:_:1₀;_;1₀



   

(1+₂aj)A_j₌₁,(2+₂aj)A_j₌₁ :1; 1₂ ;

y2

4(1+B−A),

y2_cos2₍_c_γ₎

4(1+B−A)

3 2, 32,(

1+bj 2 )Bj=1,(

2+bj

2 )Bj=1: ; ;



   

+

y2sin(cγ)cos(cγ)_∏A

i=1

(ai) A

∏

i=1

(1+ai)

4c ∏B

i=1

(bi) B

∏

i=1

(1+bi)

×

×F_2B2A₊₂:_:1₀;_;2₁



   

(2+₂aj)A_j₌₁,(3+₂aj)A_j₌₁ :1;1, 1 ;

y2

4(1+B−A),

y2_cos2₍_c_γ₎

4(1+B−A)

2, 2, (2+₂bj)B_j₌₁,(3+₂bj)B_j₌₁: ;3₂ ;



   

(12)

Z γ

0

sinθAFB



   

(aj)jA=1 ;

ysin2θ

(bj)Bj=1 ;



   

dθ=A+1FB+1



   

1,(aj)Aj=1 ; y

3

2,(bj)Bj=1 ;



   

−

− cosγF_BA₊₁:_:1₀;_;1₀



   

(aj)Aj=1 :1 ; 12 ;

y, ysin2γ

3

2,(bj)Bj=1: ; ;



   

(4.9)

Z γ

0

cosθAFB



   

(aj)Aj=1 ;

ycos2_θ

(bj)Bj=1 ;



   

dθ=sinγF_BA₊₁:_:1₀_;;1₀



   

(aj)Aj=1 :1 ; 12 ;

y,ycos2_γ

3

2,(bj)Bj=1: ; ;



   

(4.10)

Z γ

0

sinθF_BA_::_ED_;_H;G



   

(aj)jA=1:(dj)Dj=1;(gj)Gj=1 ;

ysin2θ,zsin2θ

(bj)B_j₌₁:(ej)_jE₌₁;(hj)H_j₌₁ ;



   

dθ

=F_BA₊+₁1_::_ED_;_H;G



   

1, (aj)A_j₌₁:(dj)Dj=1;(gj)Gj=1 ;

y,z

3

2,(bj)Bj=1:(ej)Ej=1;(hj)Hj=1 ;



   

−

− cosγF(3)



   

1, (aj)Aj=1:: ; ;(dj)Dj=1 :1;(gj)Gj=1; 12 ;

y,z,ysin2γ 3

2,(bj)Bj=1:: ; ;1,(ej)Ej=1: ;(hj)jH=1; ;



   

−

zsin2γcosγ

A

∏

i=1

(ai) G

∏

i=1

(gi)

3 ∏B

i=1

(bi) H

∏

i=1

(hi)

×

×F(3)



   

2,(1+aj)Aj=1:: ;(1+gj)Gj=1 ; 32:(dj)Dj=1; 1 ; 1 ;

ysin2γ,z,zsin2γ

5

2,(1+bj)jB=1:: ;2,(1+hj)Hj=1;2:(ej)Ej=1; ; ;



   

(4.11)

Z γ

0

cosθF_BA_::_ED_;_H;G



   

(aj)_jA₌₁:(dj)D_j=1;(gj)Gj=1 ;

ycos2θ,zcos2θ

(bj)B_j₌₁:(ej)_jE₌₁;(hj)H_j₌₁ ;



   

dθ

=sinγF(3)



   

1,(aj)jA=1:: ; ;(dj)Dj=1 :1;(gj)Gj=1; 12 ;

y,z,ycos2_γ

3

2,(bj)Bj=1:: ; ;1,(ej)Ej=1: ;(hj)Hj=1; ;



   

+

zsinγcos2γ

A

∏

i=1

(ai) ∏G

i=1

(gi)

3 ∏B

i=1

(bi) _∏H

i=1

(hi)

(13)

×F(3)



   

2, (1+aj)jA=1:: ; (1+gj)jG=1 ;32:(dj)Dj=1; 1; 1 ;

ycos2γ,z,zcos2γ

5

2,(1+bj)Bj=1:: ;2, (1+hj)Hj=1;2:(ej)Ej=1 ; ; ;



   

(4.12)

provided that each of the series as well as associated integrals involved are convergent.

5

Solutions of Ramanujan’s incomplete elliptic integrals

SettingA=1,B=0 anda1= 1₂in (4.1) and (4.2) respectively, we get

F(√y,γ) =

Z γ

0

dθ

q

(1−ysin2θ)

=γ2F1



   

1 2,12 ;

y

1 ;



   

−ysinγ cosγ

4 ×

×F₂2_::₀1_;;₁2



   

3

2,32:1;1, 1 ;

y,ysin2γ

2, 2: ; 3₂ ;



   

;|y|<1 (5.1)

which is the exact solution of incomplete elliptic integral of first kind.

Z γ

0

dθ

p

(1−ycos2_θ₎ =γ2F1



   

1 2,12 ;

y

1 ;



   

+ysinγcosγ

4 ×

×F₂2_::₀1_;;₁2



   

3

2,32:1;1, 1 ;

y,ycos2γ

2, 2: ; 3₂ ;



   

;|y|<1 (5.2)

Puttingγ= π₂ in (5.1) and (5.2), we get

K(√y) =

Z π₂

0

dθ

q

(1−ysin2θ)

=

Z π₂

0

dθ

p

(1−ycos2_θ₎ = π

2 2F1 

   

1 2,12 ;

y

1 ;



   

;|y|<1 (5.3)

which is well-known complete elliptic integral of first kind.

PuttingA=1,B=0 anda1=−12in (4.1)and (4.2)respectively, we get

E(√y,γ) =

Z γ

0

q

(1−ysin2θ)dθ=γ2F1



   

1 2,−12 ;

y

1 ;



   

+ysinγcosγ

(14)

×F₂2_::₀1_;;₁2



   

1

2,32:1;1, 1 ;

y,ysin2γ

2, 2: ; 3₂ ;



   

;|y|<1 (5.4)

which is the exact solution of incomplete elliptic integral of second kind.

Z γ

0

q

(1−ycos2_θ₎_d_θ₌_γ_2F1



   

1 2,−12 ;

y

1 ;



   

−ysinγcosγ

4 ×

×F₂2_::₀1_;;₁2



   

1

2,32:1;1, 1 ;

y,ycos2γ

2, 2: ; 3₂ ;



   

;|y|<1 (5.5)

Settingγ= π₂ in (5.4) and (5.5), we get

E(√y) =

Z π₂

0

q

(1−ysin2θ)dθ=

Z π₂

0

q

(1−ycos2_θ₎_d_θ₌ π

2 2F1 

   

−1₂,1₂ ;

y

1 ;



   

;|y|<1 (5.6)

which is well-known complete elliptic integral of second kind.

PuttingA=1,B=0, a1= ₂1, γ=βandy=x2in (4.3), we get

Z β

0

dθ

q

(1−x2_sin4 θ)

=β2F1



   

1 4,34 ;

x2

1 ;



   

− 3x

2 _sin_β_cos_β

16 ×

×F₂2_::₀1_;;₁2



   

5

4,74:1;1, 1 ;

x2_,_x2 _sin2 β

2, 2: ; 3₂ ;



   

− x 2 _sin3

βcosβ

8 ×

×F₃3_::₀1_;;₀1



   

[5₄:1, 1],[7₄:1, 1],[2:1, 2]: [1:1]; [1:1] ;

x2sin2β,x2sin4β

[2:1, 1],[2:1, 1],[5₂:1, 2] : ; ;



   

;|x|<1 (5.7)

which is the exact solution of unsolved-incomplete elliptic integral given in Ramanujan’s Entry 7(iv).

SettingA=B=E=H=0, D=G=1, d1=g1= 1₂, γ=β, y=a, z=bin (4.5), we get

Z β

0

dθ

q

(1−asin2θ) (1−bsin2θ)

=βF1₁:_:1₀;_;1₀



   

1

2: 12; 12 ; a,b

1: ; ;



   

−

−b sinβcosβ

4 F

2:1;2 2:0;1



   

3

2, 32: 1;1, 1 ;

b,bsin2β

2, 2: ; 3₂ ;



   

(15)

−asinβcosβ

4 F

(3)



   

3

2:: ; ;32 :1;12;1, 1 ;

a,b,asin2β

2:: ; ;2: ; ; 3₂ ;



   

−

− a bsin

3_β_cos_β

16 F

(3)



   

5

2:: ;32;2:1,32;1;1 ;

asin2β,b,bsin2β

3:: ;2 ;5₂:2; ; ;



   

;max{|a|,|b|}<1 (5.8)

which is the exact solution of unsolved-incomplete elliptic integral given in Ramanujan’s Entry 7(iii). Here

F_1:0;01:1;1[·]is Appell’s function of first kind, in the notation of Srivastava and Panda.

SettingA=B=E=H=0, D=G=1, d1=1, g1= 1₂, γ=φ, y=a, z=b2in (4.5), we get

∏

(a, b, φ) =

Z φ

0

dθ

(1−asin2θ)

q

(1−b2_sin2 θ)

=φF₁1_::₀1_;;₀1



   

1

2:1; 12 ; a,b2

1: ; ;



   

−

−b 2 _sin

φcosφ

4 F

2:1;2 2:0;1



   

3

2, 32 :1 ; 1, 1 ;

b2,b2sin2φ

2, 2 : ;3₂ ;



   

−

− asinφcosφ

2 F

(3)



   

3

2:: ; ; :1 ; 12 ;1, 1 ;

a,b2_,_a_sin2 φ

2:: ; ; : ; ;3₂ ;



   

−

− a b

2 _sin3_φ_cos_φ

8 F

(3)



   

5

2 :: ;32;2:1 ;1 ;1 ;

asin2φ,b2,b2sin2φ

3:: ;2;5₂ : ; ; ;



   

(5.9)

where max{|a|,|b|}<1

which is the exact solution of incomplete elliptic integral of third kind.

In (5.9), puttingφ= π₂ we get the exact solution of complete elliptic integral of third kind.

∏

a, b, π 2

=

Z π₂

0

dθ

(1−asin2θ)

q

(1−b2_sin2_θ₎

= π

2 F1 1

2;1, 1 2;1;a,b

2

;max{|a|,|b|}<1 (5.10)

whereF1

₁

2;1,12;1;a,b2

is Appell’s function of first kind[34,p.53(1.6.4)].

These solutions are not found in Ramanujan’s notebooks[29-31], Five notebooks of B. C. Berndt [6-10],

Three volumes of R. P. Agarwal [2-4] and other literature [5; 17; 18; 20; 22(pp.100-106); 24; 25; 26; 28; 32; 35] on

(16)

SettingA=1, B=0, a1= 12, γ=α, y=−x, c=2 in (4.8), we get

Z α

0

dθ

p

(1+xcos 2θ)

=α2F1



   

1 4, 34 ;

x2

1, ;



   

− xsin(2α)

4 ×

×F₂2_::₀1_;;₀1



   

3

4, 54:1;12 ;

x2,x2cos2(2α) 3

2, 32: ; ;



   

+

+3x

2_sin₍₄_α₎

64 F

2:1;2 2:0;1



   

5

4, 74:1;1, 1 ;

x2_,_x2_cos2₍₂_α₎

2, 2: ; 3₂ ;



   

;|x|<1 (5.11)

Further on using (5.2) in above integral, we get an elegant formula in the following form:

p

(1−x)

Z α

0

dθ

p

(1+xcos 2θ)

=

Z α

0

dθ

q

{1−(_x2x₋₁)cos2_θ₎_}

=α2F1



   

1 2,12 ;

2x x−1

1 ;



   

+ xsin 2α

4(x−1) F

2:1;2 2:0;1



   

3

2,32:1;1, 1 ; 2x x−1, 2xcos

2_α

x−1

2, 2: ;3₂ ;



   

;

2x

x−1

<1 (5.12)

which is the exact solution of Ramanujan’s unsolved-incomplete elliptic integral given in Entry 7(iv).

6

Special cases

In (4.1), setA=2,B=1,a1=b,a2=1−b,b1=1₂ andγ= π₂, we obtain

Z π₂

0

cos{(2b−1)sin−1(√ysinθ)}

q

(1−ysin2θ)

dθ= π

2 2F1 

   

b, 1−b ;

y

1 ;



   

(6.1)

which is the known result of Ramanujan [8, p.88(Entry 1)].

In (4.1), putA=2,B=1,a1=13,a2= 23,b1= 12andγ= π2, we obtain

Z π₂

0 2 F1



   

1 3, 23 ;

ysin2θ

1

2 ;



   

dθ= π

2 2F1 

   

1 3, 23 ;

y

1 ;



   

(6.2)

(17)

In (5.7), setβ= π₂ andx2=y, we have

Z π₂

0

dθ

q

(1−y sin4θ)

= π

2 2F1 

   

1 4,34 ;

y

1 ;



   

(6.3)

which is another known result of B.C.Berndt[8, p.110].

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(19)

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Received: March 14, 2014;Accepted: August 23, 2014

UNIVERSITY PRESS