Here, the roots of complex mathematics Holomorphic Functions are compared with a physical example of complex mathematical problem of minimal surface Soap Bubble. Holomorphic function is nothing but a type of complex valued function which is differentiable in a neighborhood of every point of its domain and a soap bubble is an extremely thin film of soapy water surrounded by air. Comparison between holomorphic function and soap bubble is revealed by the following mathematical study. If a holomorphic function is defined on a closed disk and on the boundary of disk, function is known then by using Cauchy integral formula, we can determine the function in the interior of disk. In the same way, if we have a soap bubble formed on a closed wire and shape of wire (

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International Indexed & Refereed Research Journal, ISSN 0974-2832, (Print), E-ISSN- 2320-5474, July, 2013 VOL-V * ISSUE -54
Introcuction
Complex analysis is one of the classical
branches in mathematics with its roots in the 19th cen-
turyandjustprior.Inmathematics,HolomorphicFunc-
tions are the central objects ofstudy in complex analy-
sis. The word "Holomorphic" was introduced by two
Cauchy's students, Briot (1817-1882) and Bouquet
(1819-1895) and derives from Greek words "holos"
meaning 'entire' and "morphic" meaning 'form or ap-
pearance'. Repeating use of holomorphic functions
created the "pictures of Fractal". Fractals are typically
self similar patterns where self similar means they are
"the same from near as from far". Pattern in nature is
also an example of fractal pattern, which display self
similarityoverextended,butfinite,scaleranges.Natu-
ral pattern includes symmetries, trees, spirals, mean-
ders, waves,foams, array, cracks and stripes etc. Early
Greeks philosophers, studied pattern with Plato,
Pythagoras and Empedocles attempting to explain or-
der in nature. The modern mathematics of visible pat-
tern developed gradually over time. In the 19th
century
BelgianPhysicistJosephPlateauexaminedsoapfilms,
leading him to formulate the concept of a minimal sur-
face. Soap films are thin layer of liquid surrounded by
air and a closed soap film is termed as soap bubble.
Thus soap bubble is an extremely thin film of soapy
waterenclosingtheair.Inpresentpaper,Iwouldliketo
compare soap bubble formed on a closed wire with
holomorphic function on a disk. Before comparing, I
describetheconceptofholomorphicfunctionandsoap
bubble in brief.
HolomorphicFunction
Holomorphic function is a complex valued
function of one or more complex variable that is com-
Research Paper -Mathematics
July, 2013
Comparison of Soap BubbleWith
HolomorphicFunctionInComplex Analysis
* Deepa Gupta
*LecturerinDepttof Mathematics,HinduGirlsCollege,Sonepat
Here, the roots of complex mathematics Holomorphic Functions are compared with a physical example of complex
mathematical problem of minimal surface Soap Bubble. Holomorphic function is nothing but a type of complex valued
function which is differentiable in a neighborhood of every point of its domain and a soap bubble is an extremely thin film
of soapy water surrounded by air. Comparison between holomorphic function and soap bubble is revealed by the following
mathematical study. "If a holomorphic function is defined on a closed disk and on the boundary of disk, function is known
then by using Cauchy integral formula, we can determine the function in the interior of disk. In the same way, if we have
a soap bubble formed on a closed wire and shape of wire (boundary) is known then we can find interior of soap bubble
or shape of soap bubble (minimal surface) by using mathematical tool calculus of variation and motion under curvature
A B S T R A C T
Keywords: Holomorphic function, surface tension, mean curvature, minimal surface.
plex differentiable in a neighborhood ofevery point in
its domain. A holomorphic function whose domain is
the whole complex plane is called an entire function.
The phrase "holomorphic at a point z0
" means not just
differentiableatz0butdifferentiableeverywherewithin
some neighborhood of z0
in the complex plane. For a
given complex valued function 'f' of single complex
variable, the derivative of 'f' at a point z0
in its domain
is defined by the limit
0
0
0
0
)()(lim
)('
zz
zfzf
zz
it
zf
−
−
→
=
The limit is taken as the complex number z
approaches z0
and must have same value for any se-
quence of complex values for z that approaches z0
on
the complex plane. If the limit exists, we say that 'f' is
complex differentiableat point z0
. Centraltoolin com-
plex analysis is the line integral and the line integral of
holomorphic function around a closed path is given by
Cauchy integral theorem which states that "The inte-
gral around a closed path of a holomorphic function is
always zero".
SoapBubble
Soap bubbles are physical example of the
complexmathematicalproblemofminimalsurface.They
will assume the shape of least surface area possible
containing a given volume due to surface tension.
Surface tension is a contractive tendency of surface of
liquid that allows it to resist an external force. The
cohesive forces among liquid molecules are respon-
sible for surface tension. In bulk of the liquid, each
molecule is pulled equallyin everydirection byneigh-
boring liquid molecules, resulting in to a net force of

2. 9SHODH, SAMIKSHA AUR MULYANKAN
International Indexed & Refereed Research Journal, ISSN 0974-2832, (Print), E-ISSN- 2320-5474, July, 2013 VOL-V * ISSUE -54
zero. The molecules at the surface do not have other
molecules on all sides ofthem and therefore are pulled
inward. This creates some internal pressure and forces
liquid surface to contract to the minimal area.Another
way to view surface tension is in terms of energy. A
molecule in contact with a neighbor is in a lower state
of energy than if it were alone (not in contact with a
neighbor).Theinteriormoleculeshaveasmanyneigh-
bors as they can possibly have, but the boundary
molecules have missing neighbors (compared to inte-
riormolecules)andthereforehaveahigherenergy.For
the liquid to minimize its energy state, the number of
higherenergyboundarymoleculesmustbeminimized.
Theminimizedquantityofboundarymoleculesresults
in a minimized area. A true minimal surface is more
properly illustrated by a soap film which has equal
pressure on inside as outside, hence is a surface with
zero mean curvature. The term "minimal surface" is
used becausegiven a fixed boundarycurve, the area of
"minimal surface" is extremal with respect to other
surfaces with same boundary. The physical model of
area minimizing minimal surface can be made by dip-
pingawireframeintosoapsolution,framingasoapfilm
which is minimal surface whose boundary is the wire
frame.
ComparisonofSoapBubbleFormedonaClosedWire
withHolomorphicFunctiononaDisk:Interiorofsoap
bubbleformedonawireisdeterminedbyshapeofwire
(boundary) just as holomorphic function inside a disk
is determined by their behavior on the boundary of the
disk.Now,Iwillexplainbothdeterminationsseparately.
To determine holomorphic function inside a disk by
their behavior on the boundary of the disk: Let 'f' be a
holomorphicfunctiononadisk D={z:|z-z0
| <r}.Let
ybethecircleformingtheboundaryof D.Supposethat
function 'f' is given on the boundary of y and 'a' is
arbitrarypointintheinteriorofD.ThenbyusingCauchy
integralformula,onecanshowthattheintegralover is
equal to thesameintegral takenover anarbitrarysmall
circlearound'a'.Sincef(z)iscontinuous,wecanchoose
a circle small enough on which f(z) is close to f(a). On
the other hand, the integral where 'c' is any
circle centered at 'a'.
This can be calculated directly via a param-
eterization z(t) = a + ei t where 0 t 2 and is radius of
the circle. Letting 0 gives the desired estimate
But 'a' is arbitrary, so we can determine function 'f' at
every point of interior of D by using above formula
(that requires function 'f' on the boundary of disk D).
This formula is known as Cauchy's integral formula
named after "Augustin-Louis Cauchy".
Todetermineinteriorofsoapbubbleformedonaclosed
wirebyitsboundary(shapeofwire)orTodetermine
minimalsurfacewithgivenboundary:Nowstartwith
a piece of wire, connect the two ends together and dip
it in a bath of soapy water and then pull it out again,
what is the shape of the soap film that results . Physi-
cally, surface tension makes the resulting soap film
minimizeitsareawhilestillspanningthewireframei.e.
there are many different possible surface touching the
entire given wire, the main task is to find the one that
has the smallest total area. This minimal surface prob-
lemis also knownas plateau'sproblemnamed afterthe
nineteenth century French physicist Joseph Plateau
who conducted systematic experiments on such soap
films.Mathematicallythisproblemcanbehandledwith
the powerful tool calculus of variation developed by
Bernoulli,EulerandLagrange.Now,Iexplainhowmini-
mal surface area problem is handled with "calculus of
variation".
For simplicity, let following boundary curve
'C' denotes the closed wire. My aim is to find the mini-
mum surface area enclosed by the curve 'C'
Fig (i)
Now we shall assume that the boundary curve 'C'
projects down to a simple closed curve = that
bounds an open domain R2
in the (x,y) plane, as
shown in Fig (i). The space curve C R3
is then given
i
azC
2
1
=
−∫
0)()(
max)())((
2
1)()(
2
1
)(
)(
2
1
0
2
0
→−
=−
≤
−
≤
−
−
=−
− →∫∫∫ afzf
az
dt
aftzf
dz
az
afzf
i
afdz
az
zf
i CC
)(
)(
2
1)(
2
1
afdz
az
zf
i
dz
az
zf
i C
=
−
=
−
⇒ ∫∫
dz
az
zf
i
af ∫ −
=
)(
2
1
)(
∂Ω
Ω

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International Indexed & Refereed Research Journal, ISSN 0974-2832, (Print), E-ISSN- 2320-5474, July, 2013 VOL-V * ISSUE -54
(i) Calculus of variations with Applications by A.S.Gupta, Prentice Hall of India, New Delhi, 1997.
(ii) H.A. Priestly Introduction to Complex Analysis, clarehdon Press Oxford, 1990.
(iii) Mark j.Ablowitz and A.S Fokas, complex variables: Introduction and Applications, Cambridge University Press, South Asian
Edition, 1998.
(iv) J.B. Conway, Function of one complex variable, Springer International Student-Edition, Narosa Publishing House, 1980
R E F E R E N C E
by z = g (x,y) for (x,y) = . For reasonable boundary
curve'C',weexpectthatthegraphofafunctionz=u(x,y)
parameterized by (x,y) . According to the basic
calculus formula, the surface area of such a graph is
given by double integral
€∂Ω
€∂Ω
[ ] dxdy
y
u
x
u
uJ ∫∫Ω






∂
∂
+





∂
∂
+=
22
1
To find theminimal surface, then, weseek the function
z=u(x,y)thatminimizesthesurfaceareintegral(i)when
subject to the Dirichlet boundary conditions
u(x,y)=g(x,y) for (x,y) Also this minimal surface
u(x,y) must satisfy the Euler-Lagrange equation
(1+uy)2 uxx
2ux
uy
uxy
+ (1+ux
)2 uyy
=0
Another way to find the minimal surface that spans the
given wire frame is motion under curvature. The cur-
vature measures how fast a curve bends at any spot.
For example, a circle has a constant curvature because
italwaysisturningatthesamerate;asmallercirclehas
a higher constant curvature because it turns faster.
Now suppose each piece of curve moves perpendicu-
lar to the curve with speed proportional to the curva-
ture.Sincethecurvaturecanbeeitherpositiveornega-
tive (depending on whether the curve is turning clock-
wise or counter clockwise). Some parts of the curve
moveoutwardwhileothermovesinwards.Thisiscalled
"motion under curvature"
For obtaining the minimal surface the "mean curva-
ture" should be zero. This means they are equally
convexand concaveatallpoints.Fromaboveitisclear
that, one can find a curve in space which is the bound-
ary curve of several different minimal surfaces.