ANALYTICAL EXPRESSIONS FOR THE EXACT CURVED SURFACE AREA OF A HEMIELLIPSOID VIA MELLIN-BARNES TYPE CONTOUR INTEGRATION M.A. Pathan12 M. I. Qureshi3 Javid Majid3

2025-04-27 0 0 887.07KB 17 页 10玖币

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ANALYTICAL EXPRESSIONS FOR THE EXACT CURVED SURFACE AREA OF

A HEMIELLIPSOID VIA MELLIN-BARNES TYPE CONTOUR INTEGRATION

M.A. Pathan1,2, M. I. Qureshi3, Javid Majid3,∗

1Centre for Mathematical and Statistical Sciences (CMSS), Peechi,

Thrissur, Kerala-680653, India

2Department of Mathematics, Aligarh Muslim University,

Aligarh, U.P., India

3Department of Applied Sciences and Humanities

Faculty of Engineering and Technology

Jamia Millia Islamia (A Central University), New Delhi-110025, India.

Emails: mapathan@gmail.com, miqureshi delhi@yahoo.co.in

∗Corresponding author: javidmajid375@gmail.com

Abstract: In this article, we aim at obtaining the analytical expressions (not previously found

and not recorded in the literature) for the exact curved surface area of a hemiellpsoid in terms

of Appell’s double hypergeometric function of ﬁrst kind. The derivation is based on Mellin-Barnes

type contour integral representations of generalized hypergeometric function pFq(z), Meijer’s G-

function and analytic continuation formula for Gauss function. Moreover, we obtain some special

cases related to ellipsoid, Prolate spheroid and Oblate spheroid. The closed forms for the exact

curved surface area of a hemiellpsoid are also veriﬁed numerically by using Mathematica Program.

Keywords: Appell’s function of ﬁrst kind; Mellin-Barnes contour integral; Meijer’s G-function;

Hemiellipsoid, Ellipsoid, Prolate spheroid, Oblate spheroid; Mathematica Program.

2020 MSC: 33C20, 33C70, 97G30, 97G40.

1. Introduction and preliminaries

For the deﬁnition of Pochhammer symbols, power series form of generalized hypergeometric function

pFq(z) and several related results, we refer the beautiful monographs (see, e.g., [1,10,17,18,28,29,33])

•Some results recorded in the table of Prudnikov et al.[ [23], p.474, Entry(98) and p.479, Entry(210)]:

2F1



2,2;

2;

=1

2"1

(1 −z)+tanh−1(√z)

p(z)#;|z|<1,(1.1)

2F1



2,2;

2;

=3

2z1

(1 −z)−tanh−1(√z)

√z;|z|<1.(1.2)

•Analytic continuation formula [10, p.63, Eq.(2.1.4(17)), [17], p.249, Eq.(9.5.9), [23], p.454, En-

try(7.3.1(6)), [29], p.36, Eq.(1.8.1.11)]:

When|z|>1, then

2F1



a, b;

c;

=Γ(c) Γ(b−a)

Γ(b) Γ(c−a)(−z)−a2F1



a, 1 + a−c;

1 + a−b;

+

+Γ(c) Γ(a−b)

Γ(a) Γ(c−b)(−z)−b2F1



b, 1 + b−c;

1 + b−a;

,(1.3)

arXiv:2210.02858v1 [math.GM] 3 Oct 2022

where |arg(−z)|< π, |arg(1 −z)|< π and (a−b)6= 0,±1,±2,±3, ....

•Mellin-Barnes type contour integral representation of binomial function:

(1 −z)−a=1F0



−;

=1

(2πi) Γ(a)Z+i∞

−i∞

Γ(a+s)Γ(−s)(−z)sds :z6= 0,(1.4)

where |arg(−z)|< π, |z|<1, a ∈C\Z−

0and i=p(−1).

•Appell’s function of ﬁrst kind [33, p.53, Eq.(4)] is deﬁned as:

F1a;b, c;d;x, y =F1:1;1

1:0;0 



a:b;c;

x, y

d:−;−;

=∞

m,n=0

(a)m+n(b)m(c)nxmyn

(d)m+nm!n!(1.5)

=∞

m=0

(a)m(b)mxm

(d)mm!2F1



a+m, c;

d+m;

=∞

n=0

(a)n(c)nyn

(d)nn!2F1



a+n, b;

d+n;

(1.6)

•Convergence conditions of Appell’s double series F1:

(i) Appell’s series F1is convergent when |x|<1,|y|<1; a, b, c, d ∈C\Z−

(ii) Appell’s series F1is absolutely convergent when |x|= 1,|y|= 1; a, b, c, d ∈C\Z−

0;R(a+

b−d)<0,R(a+c−d)<0 and R(a+b+c−d)<0.

(iii) Appell’s series F1is conditionally convergent when |x|= 1,|y|= 1; x6= 1, y 6= 1; a, b, c, d ∈

C\Z−

0;R(a+b−d)<1,R(a+c−d)<1 and R(a+b+c−d)<2.

(iv) Appell’s series F1is a polynomial If ais a negative integer; b, c, d ∈C\Z−

(v) Appell’s series F1is a polynomial If band care negative integers; a, d ∈C\Z−

•Mellin-Barnes type contour integral representation of Meijer’s G-function ( [33, p.45, Eq.(1)], see

also [10, 18]):

When p≤qand 1 ≤m≤q, 0≤n≤p, then

Gm,n

p,q z

α1, α2, α3, ..., αn;αn+1, ..., αp

β1, β2, β3, ..., βm;βm+1, ..., βq=1

2πi Z+i∞

−i∞Qm

j=1 Γ(βj−s)Qn

j=1 Γ(1 −αj+s)

j=m+1 Γ(1 −βj+s)Qp

j=n+1 Γ(αj−s)(z)sds

2πi Z+i∞

−i∞

Γ(β1−s)...Γ(βm−s)Γ(1 −α1+s)...Γ(1 −αn+s)

Γ(1 −βm+1 +s)...Γ(1 −βq+s)Γ(αn+1 −s)...Γ(αp−s)(z)sds, (1.7)

where z6= 0,(αi−βj)6= positive integers, i= 1,2,3, ..., n;j= 1,2,3, ..., m. For details of contours,

see [10, p.207, [18], p.144].

•Convergence conditions of Meijer’s G-function:

When Λ = m+n−p+q

2, ν =Pq

j=1 βj−Pp

j=1 αj,then

(i) The integral (1.7) is convergent when |arg(z)|<Λπand Λ >0.

(ii) If |arg(z)|= Λπand Λ ≥0, then the integral (1.7) is absolutely convergent when p=qand

R(ν)<−1.

(iii) If |arg(z)|= Λπand Λ ≥0, then the integral (1.7) is also absolutely convergent, when

p6=q, (q−p)σ > R(ν)+1−q−p

2and s=σ+ik, where σand kare real. σis chosen so

that for k→ ±∞.

For other two types of contours, following will be convergence conditions of the integral (1.7):

(iv) The integral (1.7) is convergent if q≥1 and either p < q, 0<|z|<∞or p=q, 0<|z|<1.

(v) The integral (1.7) is convergent if p≥1 and either p > q, 0<|z|<∞or p=q, |z|>1.

•Relations between Meijer’s G- function and 2F1(z) [21, p.61, [36], p.77, Eq.(1)]:

G2 2

2 2 z

1−a, 1−b;−

0, c −a−b;−=Γ(a)Γ(b)Γ(c−a)Γ(c−b)

Γ(c)2F1



a, b;

1−z

c;

,

where |1−z|<1 and c−a, c −b6= 0,−1,−2, ...

G2 2

2 2 z

a1, a2;−

b1, b2;−=Γ(1 −a1+b1)Γ(1 −a1+b2)Γ(1 −a2+b1)Γ(1 −a2+b2)zb1

Γ(2 −a1−a2+b1+b2)×

×2F1



1−a1+b1,1−a2+b1;

1−z

2−a1−a2+b1+b2;

;|1−z|<1.(1.8)

•When the two dimensional curve i.e, generating curve lying in x-yplane (suppose y=f(x)) is

revolved about x-axis, then equation of generated three dimensional surface will be y2+z2= [f(x)]2.

•When the two dimensional curve i.e, generating curve lying in x-yplane (suppose x=F(y)) is

revolved about y-axis, then equation of generated three dimensional surface will be x2+z2= [F(y)]2.

Similarly we can write the equation of generated three dimensional surface, when the curve lies in

y-zplane and z-xplane.

•The equation of an ellipse is

a2+y2

b2= 1; a > b > 0.(1.9)

•When the above ellipse represented by (1.9) is revolved about major axis (i.e, x-axis), then equation

of generated 3-D surface called Prolate spheroid, will be

a2+y2+z2

b2= 1; a > b > 0.(1.10)

Figure 1. Prolate Spheroid.

•When the ellipse represented by (1.9) is revolved about minor axis (i.e, y-axis), then equation of

generated 3-D surface called Oblate spheroid, will be

Figure 2. Oblate Spheroid.

x2+z2

a2+y2

b2= 1; a > b > 0.(1.11)

•The sphere x2+y2+z2=c2, prolate spheroid and oblate spheroid are the particular cases of the

ellipsoid

a2+y2

b2+z2

c2= 1; a > b > 0.(1.12)

•Suppose φ(x, y) = 0 is the projection of the curved surface of three dimensional ﬁgure z=f(x, y)

over the x-yplane, then curved surface area is given by

S=Z Z

|{z}

over the area

φ(x,y)=0

t(1 + ∂z

∂x 2

+∂z

∂y 2)dx dy. (1.13)

•Suppose ψ(y, z) = 0 is the projection of the curved surface of three dimensional ﬁgure x=g(y, z)

over the y-zplane, then curved surface area is given by

S=Z Z

|{z}

over the area

ψ(y,z)=0

t(1 + ∂x

∂y 2

+∂x

∂z 2)dy dz. (1.14)

•A deﬁnite integral Zπ

θ=0

sinαθcosβθ dθ =

Γα+1

2Γβ+1

2

2Γ α+β+2

2,(1.15)

where R(α)>−1,R(β)>−1.

Motivated by the work of Andrews [2, 3], Bakshi et al. [5], Burchnall et al. [6] and others

[4, 7–9, 11–16, 19, 20, 22, 24–27, 30–32, 34, 35, 37, 38], we evaluated some important deﬁnite integrals

Rπ

θ=−πcos2θ

β2+sin2θ

λ2s

dθ and R1

r=0

r2s+1

(1−r2)sdr with suitable convergence conditions in section 2, by

using Mellin-Barnes type contour integral representation of binomial function 1F0(z), Meijer’s G-

function, classical Beta function of two variables and series manipulation technique. These integrals

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ANALYTICALEXPRESSIONSFORTHEEXACTCURVEDSURFACEAREAOFAHEMIELLIPSOIDVIAMELLIN-BARNESTYPECONTOURINTEGRATIONM.A.Pathan1;2,M.I.Qureshi3,JavidMajid3;1CentreforMathematicalandStatisticalSciences(CMSS),Peechi,Thrissur,Kerala-680653,India2DepartmentofMathematics,AligarhMuslimUniversity,Aligarh,U.P.,India3Dep...

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ANALYTICAL EXPRESSIONS FOR THE EXACT CURVED SURFACE AREA OF A HEMIELLIPSOID VIA MELLIN-BARNES TYPE CONTOUR INTEGRATION M.A. Pathan12 M. I. Qureshi3 Javid Majid3

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