APPENDIX A

Volume, Surface Area, andIntegration on Spheres

Volume of the Ball and Surface Area of theSphere

In this section we compute the volume of the unit ball and surfacearea of the unit sphere in Rn. Recall that B = Bn denotes the unitball in Rn and that V = Vn denotes volume measure in Rn. We beginby evaluating the constant V(B), which appears in several formulasthroughout the book.

A.l Proposition: The volume oi the unit ball in Rn equals

trn / 2

(n/2)!

2(n+l)/2tr(n-l)/2

1 ·3 ·5" ·n

iEn is even,

ifn is odd.

PROOF: Assume n > 2, denote a typical point in Rn by (x, y),where x E R2 and y E Rn-2, and express the volume Vn(Bn) as aniterated integral:

214 Appendix A. Volume, Surface Area, and Integration on Spheres

The inner integral on the last line equals the (n - 2)-dimensionalvolume of a ball in Rn-2 with radius (1_lxI2) l j 2. Thus

Vn(Bn) = Vn-2(Bn-2) r (1_lxI2)(n- 2)j2dV2(X) .1B2

Switching to the usual polar coordinates in R2, we get

Vn(Bn) = Vn-2(Bn-2) 111" r1(1 - r2)(n-2)j2r dr dO

-11"10211"

= -Vn-2(Bn-2).n

The last formula can be easily used to prove the desired formula forVn(Bn) by induction in steps of 2, starting with V2(B2) = 11" andV3(B3) = 411"/3. 0

Readers familiär with the gamma function should be able torewrite the formula given by A.1 as a single expression that holdswhether n is even or odd (see Exercise 6 of this appendix) .

Turning now to surface-area measure, we let Sn denote theunit sphere in Rn ." Unnormalized surface-area measure on Sn willbe denoted by Sn and normalized surface-area measure on Sn will bedenoted by O'n . We presume some familiarity with surface-area mea­sure for smooth manifolds. The arguments we give in the remainderof this appendix will be more intuitive than rigorous; the readershould have no trouble filling in the missing details.

Let us find the relationship between Vn(Bn) and sn(Sn)' Wedo this with an old trick from calculus: For h ~ 0 we have

((1+ h)" - l)Vn(Bn) = Vn((l + h)Bn) - Vn(Bn)

~ sn(Sn)h .

•A more common notation is s-», which emphasizes that the sphere hasdimension n - 1 as a manifold. We use Sn to emphasize that the sphere livesin Rn .

Slice Integration on Spheres 215

Dividing by h and letting h ---+ 0, we obtain nVn(Bn) = Bn(Sn)' Werecord this result in the following proposition.

A.2 Proposition: The unnormalized surface area of the unitsphere in Rn equals nVn(Bn).

Slice Integration on Spheres

The map 'I/J: Bn-1 ---+ Sn defined by

'I/J(x) = (x, VI -lxI 2 )

parameterizes the upper hemisphere of Sn . The correspondingchange of variables is given by the formula

A.3 d (ol.( )) _ dVn-l(X)Sn 'f/ X - ~I.

V I- IX I-

Equation A.3 is found in most calculus texts in the cases n = 2,3.Consider now the map 'lt : Bn-k x Sk ---+ Sn defined by

'lt(x,() = (x, VI -lxl2 () .

Here 1 ::; k < n. The map 'lt is one-to-one, and the range of 'lt isSn minus a set that has sn-measure 0 (namely, the set of points onSn whose last k coordinates vanish). We wish to find the change ofvariables formula associated with 'lt .

Observe that Bn-k x Sk is an (n - 1)-dimensional submani­fold of Rn whose element of surface area is d(Vn-k x Bk). For fixedx, 'lt changes (k - 1)-dimensional area on {x} x Sk by the factor(1 - IxI2){k- l )/ 2. For fixed (, A.3 shows that 'lt changes (n - k)­dimensional area on Bn-k x {O by the factor (1-lxI2)- 1/2. Fur­thermore, the submanifolds 'lt ({x} x Sk) and 'lt (Bn- k x {O) areperpendicular at their point of intersection, as is easily checked. Thelast statement implies that 'lt changes (n - 1)-dimensional measureon Bn-k x Sk by the product of the factors above. In other words,

The last equation leads to a useful formula (A.5) for integrat­ing over a sphere by iterating an integral over lower-dimensional

216 Appendix A. Volume, Surface Area, and Integration on Spheres

spherical slices. This formula is an immediate consequence of A.2and A.4. (We state A.5 in terms of normalized surface-area measurebecause that is what we have used most often.)

A.5 Theorem: Let f be a Borel measurable, integrable functionon Sn. If 1 ::; k < n, then JS

nf dun equals

Slice Integration: Special Cases

Some cases of Theorem A.5 deserve special mention. We begin bychoosing k = n -1, which is the largest permissible value of k. Thiscorresponds to decomposing Sn into spheres of one less dimensionby intersecting Sn with the family of hyperplanes orthogonal to thefirst coordinate axis. The ball Bn-k is just the unit interval (-1, 1),and so for xE Bn-k, we can write x2 instead of Ix/2 . Thus we obtainthe following corollary of A.5.

A.6 Corollary: Let f be a Borel measurable, integrable functionon Sn. Then JS

nf dun equals

n - 1 V(Bn - 1) 11( 2)n231 (V 2) ( )-- V(B ) 1 - X f x, 1 - x ( dun-l ( dx.

n n -1 Sn-l

At the other extreme we can choose k = 1. This correspondsto decomposing Sn into pairs of points by intersecting Sn with thefamily of lines parallel to the nth coordinate axis. The sphere SI isthe two-point set {-1, 1}, and dul is counting measure on this set,normalized so that each point has measure 1/2. Thus we obtain thefollowing corollary of A.5.

A.7 Corollary: Let f be a Borel measurable, integrable functionon Sn. Then JSn f dunequals

1 r f(x, Jf=lXF) + f(x, -Jf=lXF) dlt. ()nV(Bn) JBn-l Jf=lXF n-l X .

Slice Integration: Special Cases 217

Let us now try k = 2 (assuming n > 2). Thus in A.5 the term(1 _lxI2) (k- 2)/ 2 disappears. The variable ( in the formula given byA.5 now ranges over the unit circle in R2, so we can replace ( by(cosO,sinO), which makes du2(() equal to dO/(21r). Thus we obtainthe following corollary of A.5.

A.8 Corollary (n> 2): Let! be a Borel measurable, integrablefunction on Sn. Then Isn! dun equals

V~B) r 11\" !(x,V1-\xI2cosO,Vl-lxI2sinO)dOdVn_2(x).n n JBn - 2 -1\"

An important special case of the last result occurs when n = 3.In this case Bn- 2 is just the interval (-1,1), and we get the followingcorollary.

A.9 Corollary: Let! be a Borel measurable, integrable functionon S3' Then Is

3! dU3 equals

I 1111\"

- !(x, VI - x2cos 0, VI - x2sin 0) dO dx.41r -1 -1\"

218 Appendix A. Volume, Surface Area, and Integration on Spheres

Exercises

1. Prove that

kn (lxi ~ l)P dV(x) < 00

if and only if p > n.

2. (a) Consider the region on the unit sphere in R 3lying betweentwo parallel planes that intersect the sphere. Show that thearea of this region depends only on the distance between thetwo planes. (This result was discovered by the ancient Greeks.)

(b) Show that the result in part (a) does not hold in Rn ifn =1= 3 and "planes" are replaced by "hyperplanes" .

3. Let f be a Borel measurable, integrable function on the unitsphere 84 in R4. Define a function q; mapping the reetangularbox [-1,1] x [-1,1] x [-11",11"] to 84 by setting q;(x,y,O) equalto

(x, J1- x2y, J1- x2J1- y2 cosO, J1- x2J1- y2 sinO).

Prove that

f f du4=2\jl J1_x2 jl j11" f(q;(x,y,O))dOdydx.} 84 11" -1 -1 - 11"

4. Without writing down anything or using a computer, evaluate

5. Let m be a positive integer. Use A.6 to give an explicit formulafor

Exercises 219

6. For readers familiar with the gamma function T: Prove thatthe volume of the unit ball in Rn equals

7fn/2

r(j +1)'

APPENDIXB

Mathematiea and HarmonieFunetion Theory

Using Mathematica,· a symbolic processing program, the authorshave written routines to manipulate many of the expressions thatarise in the study of harmonie functions. These routines allow theuser to make symbolic calculations that would take a prohibitiveamount of time if done without a computer. For example, Poissonintegrals of polynomials can be computed exactly.

Our routines for symbolic manipulation of harmonie functionsare distributed free of charge by electronic mail. They are designedto work on any computer that runs Mathematica. Requests for oursoftware package should be sent to axler@math.msu.edu. Comments,suggestions, and bug reports should also be sent to the same elec­tronic address.

•Mathematica is a registered trademark of Wolfram Research.

References

[1] Sheldon Axler, Harmonie funetions from a eomplex analysis view­point, American Mathematical Monthly 93 (1986), 246-258.

[2] John B. Conway, F'unctions oi one complex variable, Springer-Verlag,New York, 1973.

[3] P. Fatou, Series trigonometriques et series de Taylor, Acta Mathe­matica 30 (1906), 335-400.

[4] L. L. Helms, Introduction to Potential Theory, Wiley-Interscienee,New York, 1969.

[5] Oliver Dimon Kellogg, Foundations oEPotential Theory, Springer­Verlag, Berlin, 1929.

[6] Steven G. Krantz, F'unction Theory oE Several Complex Variables,John Wiley, New York, 1982.

[7] Edward Nelson, A proof of Liouville's Theorem, Proceedings oE theAmerican Mathematical Society 12 (1961), 995.

[8] Walter Rudin, Principles oEMathematical Analysis, third edition,MeGraw-Hill, New York, 1976.

[9] Walter Rudin, Real and Complex Analysis, third edition, MeGraw­Hill, New York, 1987.

[10] Elias M. Stein and Guido Weiss, Fourier Analysis on EuclideanSpaces, Prineeton University Press, Prineeton, 1971.

224 References

[l1J William Thomson (Lord Kelvin), Extraits de deux lettres adresseesa. M. Liouville, Journal de Mathematiques Pures et Appliques 12(1847), 256-264.

[12J William Thomson (Lord Kelvin) and Peter Guthrie Tait, Treatise onNatural Philosophy, Cambridge University Press, Cambridge, 1879.

[13J John Wermer, Potential Tbeoty, Lecture Notes in Mathematics 408,Springer-Verlag, New York, 1974.

Symbol Index

A[u],50 ds,4A,189 du, 5

Da, 15B(a, r), 5 o; 31, 45B(a,r),5 d(a, E), 34B,5 dm,81B,5 dx,126Bn,5 dY,170bP(O), 151 dsn, 214bm,189 dUn, 214

Ck(0),2E*,60

COO(0),2C(E),2 H,32Cn,126 Hn,32

Co(Rn-1), 128 1im(R n), 73Cc(Rn-l),131 1im(S ), 73C~, 170 hm,79Cm,191 hP(B),103

hP(H),132

Dm, 3 hOO(0),176

Dn , 4dV,4 K[u],61dVn , 4 K[u],135

226 Symbol Index

L2(S) ,75 S-,106

V(S) ,98 S,134V(Rn - 1) , 128 supp , 170

M(S) ,97SI' 200Sn, 214

M[JL], 112 Sn, 214M(Rn - 1) , 128

n, 1Ur, 3U,106

n ,4 Un, 106N,88 U y , 125No [u], 111

O(n ),74V, 4Vn,4

P( x , () , 12 ib, 163P [f], 12,98PE(x ,(),65 xO

, 19

Pdf], 65 x*, 60Pm(Rn),76 XE, 66, 67P[JL],97 X *,101PH(Z,t) , 127

ZT/ ' 78PH[JL], 128PH[f] ,129 Z((, Tl ), 78

PB,137 Zm(( ,Tl) , 78

P(x, y), 157 Zm(x , y) , 154

PH(Z,w), 163 Ixl,lPA(x ,( ), 190 la l, 19PA[f], 191 IIJLII,97P[f] ,200 Il fl lp , 98

Q,l54 IlullhP , 103, 132Ilfllp , 128

R(y) ,19 Ilullp , 151R n U{oo} ,59

0 ,1R [u], 111Rn , 152 A, 1

Res(u ,a), 187 'V,40',5

S, 4 a ,15S ,5 XE, 18S+,106 a l,19

Symbol Index

ra(a), 38OE, 66ffi, 74( , ), 75( , )m, 76[m/2],76J.Lf,98Oa((),110

~((, 6), 1123~, 114J..,118~, 134r~(a) , 142ä,181Un , 214r,219

227

Index

annular region, 183approximate identity, 13, 126Arzela-Ascoli Theorem, 35

Baire's Theorem, 42balls internally tangent to B, 167barrier, 201barrier funetion, 201barrier problem, 203basis for 'Hm (Rn), n: (S), 92Bergman space, 151Bergman, Stefan, 151Bloch space, 42, 167boundary data, 15bounded harmonie function, 31bounded harmonie function on B,

40, 105Böcher's Theorem, 50, 57, 175

Cauchy 's Estimates, 33computer package, 74, 87, 93, 96,

107, 158, 192, 195, 221cone, 206conformal map, 60conjugate index, 98convex region, 204eovering lemma, 114

decomposition theorem, 172, 173decomposition theorem for

holomorphie funetions, 181degree, 22, 73

dilate,2direct sum, 75Dirichlet problem, 12, 197Dirichlet problem for annular

regions, 189Dirichlet problem for annular

regions (n = 2), 195Dirichlet problem for convex

regions, 204Dirichlet problem for H, 128Dirichlet problem for smooth

regions, 204divergenee theorem, 4dual space, 101

equieontinuity, 102essential singularity, 185essential singularity (n = 2), 193essential singularity at 00, 194exterior Dirichlet problem, 65exterior Poisson integral, 65exterior Poisson kemel, 65extemal ball condition, 203extemal cone eondition, 206extemal segment condition, 211extremal function, 106extreme point, 122

Fatou Theorem, 110, 140finitely connected, 178Fourier series, 73-75 , 81fundamental pole, 185

230

fundamental pole (n = 2), 193fundamental pole at 00, 194fundamental solution of the

Laplacian, 171

gamma function, 219generalized annular Dirichlet

problem, 195generalized Dirichlet problem, 96Green's identity, 4

Hardy, G. H., 103Hardy-Littlewood maximal

funetion, 112harmonie,!harmonie at 00, 63harmonie Bergman space, 151harmonie Bloch space, 42, 167harmonie conjugate, 178harmonie functions, limits of, 15, 49harmonie Hardy space, 103, 132harmonie measure, 211harmonie motion, 24harmonics, 24Harnack's Inequality, 48Harnack's Inequality for B, 47, 56Harnack's Principle, 49holomorphie at 00, 70homogeneous expansion, 23, 84homogeneous harmonie polynomial,

24,73homogeneous polynomial, 22, 76Hopf Lemma, 28

inversion, 60inversion map, 134isolated singularity, 32, 185isolated singularity at 00, 61isolated singularity of positive

harmonie funetion, 50isolated zero, 8

Kelvin transform, 59, 61, 135

Laplace's equation, 1Laplacian, 1Laurent series, 171, 183law of eosines, 111Lebesgue decomposition, 118Lebesgue Differentiation Theorem,

146Lebesgue point , 123limits along rays, 39Liouville's Theorem, 31

Index"

Liouville's Theorem for positiveharmonie functions, 45, 56

local defining function, 204Local Fatou Theorem, 142loca1ly connected, 209loca1ly integrable, 17logarithmie conjugation theorem,

179

Mathematica, 221maximum principle, 6, 36maximum principle for

subharmonie funetions , 198maximum principle, local, 22mean-value property, 5mean-value property, eonverse of,

16mean-value property, volume

version, 6minimum principle, 6Morera's Theorem, 187multi-index, 15

non-extendability of harmoniefunctions, 206

nontangential approach region, 38,110

nontangentiallimit, 38, 111, 140nontangential maximal funetion,

111nontangentially bounded, 142normal family, 34north pole, 88

one-point compactification, 59one-radius theorem, 28open mapping, 27open mapping theorem, 159operator norm, 101order of a pole, 185order of a pole (n = 2), 193orthogonal projection, 154, 157orthogonal transformation, 3, 74orthonormal basis for b2(B2), 167

parallel orthogonal to 1], 80Perron family, 200Perron function, 200Picard's Theorem, 187point evaluation, 151point of density, 145Poisson integral, 12Poisson integral for annular region,

191

Index

Poisson integral for H, 128Poisson integral of a measure, 97Poisson integral of a polynomial,

83,87Poisson kernel, 9, 12, 84, 138, 157Poisson kernel, expansion into

zonal harmonies, 84Poisson kernel for annular region,

189Poisson kernel for H, 126, 138, 162Poisson modifieation, 200Poisson's equation, 171polar coordinates, integration in, 6polar coordinates, Laplacian in, 26pole, 185pole (n = 2), 193positive harmonie funet ion, 45positive harmonie function on B,

54,105positive harmonie function on H,

136positive harmonie function on

R 2\ {O}, 46

positive harmonie function onRn \ {O}, 54

power series, 18principal part, 185principal part (n = 2), 193produet rule, 13punetured ball, 192

radial derivative, 123radial funet ion, 27, 51, 57radial harmonie funetion, 51, 57radial limit , 162radial maximal funetion , 111real analytie, 19reflection about a hyperplane, 66refleetion about a sphere, 67removable sets, 176removable singularity, 32, 166, 185removable singularity at 00, 61, 194reproducing kernel, 152reproducing kernel for B, 154reproducing kernel for H, 162residue, 187residue (n = 2), 195residue theorem, 187residue theorem (n = 2), 195Riemann-Lebesgue Lemma, 160Riesz Representation Theorem, 97rotation, 4

Schwarz Lemma, 105

231

Schwarz Lemma for V'u, 108Schwarz Lemma for h2

, 123Schwarz Reflection Principle, 66separable normed linear space, 102simply eonnnected, 178singular measure, 118singularity at 00, 194slice integration, 215smooth boundary, 204software for harmonie funetions,

74, 87, 93, 96, 107, 158, 192,195,221

spherieal average, 50spherieal cap, 112spherieal coordinates, Laplacian in,

26spherieal harmonie, 25, 73spherieal harmonies via

differentiation, 89Stone-Weierstrass Theorem, 77, 94,

190, 191subharmonie, 198submean-value property, 198support, 170surface area of S , 215symmetry about a hyperplane, 66symmetry about a sphere, 67symmetry lemma, 10

total variation norm, 97translate, 2

uniform boundedness principle, 120uniformly eontinuous funetion, 130,

147uniformly integrable, 121upper half-space , 125

volume of B , 213

weak* eonvergenee, 101, 131

zonal harmonie, 153zonal harmonie expansion of

Poisson kernel, 84zonal harmonie, formula for, 85zonal harmonie, geometrie

eharacterization, 87zonal harmonie with pole 1], 78

Graduate Texts in Mathematics

continu.dfrom pageü

48 SAOlS/WU. General Relativity for Mathematicians.49 GRUENBERO/WEIR. Linear Geometry. 2nd ed.50 EDWARDS. Fermat's Last Theorem.51 KLINGENBERO. A Course in Differential Geometry.52 HARTSHORNE. AlgebraicGeometry.53 MANIN. A Course in Mathematical Logic.54 GRAVER!WATKINS. Combinatorics with Emphasis on the Theory of Graphs.55 BROWNIPEARCY. Introduction to OperatorTheory I: Elements of Functional Analysis.56 MASSEY. AlgebraicTopology: An Introduction.57 CROWEU/FOX. Introduction to Knot Theory.58 KOBLITZ. p-adic Numbers, p-adic Analysis, and Zeta-Functions. 2nd ed,59 LANG. Cyclotomic Fields.60 ARNOLD. Mathematical Methods in Classical Mechanics. 2nd ed.61 WHITEHEAD. Elements of Homotopy Theory.62 KARGAPOLOV/MERLZJAKOV. Fundamentals of the Theory of Groups.63 BOLLOBAS. GraphTheory.64 EDWARDS. FourierSeries. Vol. I. 2nd ed.65 WELLS. Differential Analysison Complex Manifolds. 2nd ed.66 WATERHOUSE. Introduction to Affine Group Schemes.67 SERRE. Local Fields.68 WEIDMANN. Linear Operators in HilbertSpaces.69 LANG. Cyclotomic Fields 11.70 MASSEY. SingularHomology Theory.71 FARKAS/KRA. Riemann Surfaces. 2nd ed.72 STILLWELL. Classical Topology and Combinatorial GroupTheory.73 HUNGERFORD. Algebra.74 DAVENPORT. Multiplicative NumberTheory. 2nd ed.75 HOCHSCHlLD. BasicTheory of AlgebraicGroups and Lie Algebras.76 IITAKA. Algebraic Geometry.77 HECKE. Lectures on the Theory of Algebraic Numbers.78 BURRIS/SANKAPPANAVAR. A Course in Universal Algebra.79 WALTERS. An Introduction to ErgodieTheory.80 ROBINSON. A Course in the Theory of Groups.81 FORSTER. Lectures on Riemann Surfaces.82 BOTT!TU. Differential Forms in Algebraic Topology.83 WASHINGTON. Introduction to Cyclotomic Fields.84 IRELAND/ROSEN. A Classical Introduction to Modern NumberTheory. 2nd ed.85 EDWARDS. FourierSeries. Vol. 11. 2nd ed.86 vAN LINT. Introduction to Coding Theory. 2nd ed.87 BROWN. Cohomology of Groups.88 PIERCE. Associative Algebras.89 LANG. Introduction to Algebraicand AbelianFunctions. 2nd ed.90 BRONDSTED. An Introduction to Convex Polytopes.91 BEARDON. On the Geometry of Discrete Groups.92 DIESTEL. Sequences and Series in BanachSpaces.93 DUBROVIN!FOMENKOINOVII<oV. ModernGeometry--Methods and Applications. Part I. 2nd ed.94 WARNER. Foundations of Differentiable Manifolds and Lie Groups.95 SHIRYAYEV. Probability, Statistics, and Random Processes.96 CONWAY. A Course in Functional Analysis.

97 KOBLITZ. Introduction to Elliptie Curves and Modular Forms,98 BRÖCXER/I'OM DIECK. Representations of Compact Lie Groups.99 GROVE/BENSON. Finite Refleetion Groups. 2nd ed.

100 BERG/CHRlsTENSEN!REssEL. HarmonieAnalysis on Semigroups: Theory of PositiveDefinite and Related Funetions.

101 EDWARDS. Galois Theory.102 VARDARAJAN. Lie Groups, Lie Algebras and Their Representations.103 LANG. Complex Analysis. 2nd ed.104 DUBROVlN!FOMENKO/NOVIKOv. Modern Geometry--Methods and Applieations. Part 11.105 LANG. SL2(R).106 SILVERMAN. The Arithmetie of Elliptie Curves.107 OLVER. Applieations of Lie Groups to Differential Equations.108 RANGE. Holomorphie Functions and Integral Representations in Several Complex Variables.109 LEHI'O. Univalent Functions and Teichmüller Spaces.110 LANG. Algebraic Number Theory.111 HUSEMÖLLER. Elliptic Curves.112 LANG. Elliptic Functions.113 KARATZAS/SHREVE. Brownian Motion and Stochastic Calculus.2nd ed,114 KOBLITZ. A Course in Number Theory and Cryptography.115 BERGER/GOSTIAUX. Differential Geometry: Manifolds, Curves, and Surfaces.116 KELLEY/SRINIVASAN. Measure and Integral. Vol, 1.117 SERRE. Algebraic Groups and Class Fields.118 PEDERSEN. Analysis Now.119 ROTMAN. An Introduction to Algebraie Topology.120 ZIEMER. Weakly Differentiable Functions: Sobolev Spaces and Functionsof

Bounded Variation.121 LANG. Cyclotomic Fields land 11. Combined2nd ed.122 REMMERT. Theory of Complex Functions.

Readings in Mathemat ics123 EBBINGHAus!HERMES et al. Numbers.

Readings in Mathematics124 DUBROVlN!FOMENKO/NOVIKOV. Modern Geometry--Methods and Applications. Part III.125 BERENSTEIN/GAY. Complex Variables: An Introduction.126 BOREL. Linear Algebraic Groups.127 MAssEY. A Basic Course in Aigebraic Topology.128 RAUCH. Partial Differential Equations.129 FULTON/HARRIS. Representation Theory: A First Course.

Readings in Mathematics130 DODSON/POSTON. Tensor Geometry.131 LAM. A First Course in Noncommutative Rings.132 BEARDON. Iteration of Rational Functions.133 HARRIS. Algebraic Geometry: A First Course.134 ROMAN. Coding and Information Theory.135 ROMAN. Advanced Linear Algebra.136 ADKINS!WEINTRAUB. Algebra: An Approach via Module Theory.137 AxLER/BOURDON!RAMEY. Harmonie Function Theory.