The Riemann-Stieltjes integral. and some applications in complex analysis and probability theory. Klara Leffler

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1 The Riemnn-Stieltjes integrl nd some pplictions in complex nlysis nd probbility theory Klr Leffler VT 2014 Exmensrbete, 15hp Kndidtexmen i mtemtik, 180hp Institutionen för mtemtik och mtemtisk sttistik

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3 Abstrct The purpose of this essy is to prove the existence of the Riemnn-Stieltjes integrl. After doing so, we present some pplictions in complex nlysis, where we define the complex curve integrl s specil cse of the Riemnn- Stieltjes integrl, nd then focus on Cuchy s celebrted integrl theorem. To show the verstility of the Riemnn-Stieltjes integrl, we lso present some pplictions in probbility theory, where the integrl genertes generl formul for the expecttion, regrdless of its underlying distribution. Smmnfttning Syftet med denn uppsts är tt bevis existensen v Riemnn-Stieltjes integrlen. Därefter ges tillämpningr inom komplex nlys, där vi definierr den komplex kurvintegrlen som ett specilfll v Riemnn-Stieltjes integrlen och sedn koncentrerr oss på Cuchys hyllde integrlsts. För tt vis på mångsidigheten hos Riemnn-Stieltjes integrlen beskrivs även tillämpningr inom snnolikhetsteori, där integrlen ger en generell formel för väntevärdet, oberoende v dess underliggnde fördelning.

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5 Contents 1. Introduction 1 2. Preliminries Functions of bounded vrition Bsic topology Bsic complex nlysis 9 3. The existence of the Riemnn-Stieltjes integrl Some pplictions in complex nlysis Some pplictions in probbility theory Acknowledgements 34 References 35

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7 1. Introduction A bsic tool in nturl science is the integrl. It hs countless pplictions rnging from the most simple in clculus such s computing re nd volume to more dvnced topics such s electrodynmics, where integrls re used to describe interctions between electric chrges. There re severl different types of integrls (see e.g. [11]). In this essy we hve chosen to focus on the so clled Riemnn-Stieltjes integrl. After proving its existence, we give some pplictions within complex nlysis nd probbility theory. However, let us first introduce some bsic concepts nd nottions. Let A be non-empty open subset of K {R, C}, nd let [, b] be n intervl. Furthermore, let f : A K be continuous function nd g : [, b] K function of bounded vrition. A prtition π = {t k } of the intervl [, b] is n ordered set of points stisfying = t 0 < t 1 <... < t n = b. Furthermore, we introduce the following nottion for the norm of prtition π = mx k (t k t k 1 ). Now for ech subintervl [t k 1, t k ] [, b], k = 1, 2,..., n, let σ = {s k } be n ordered set of points s k [t k 1, t k ]. We then define the Riemnn-Stieltjes sum corresponding to π, σ, f, nd g s n S π,σ (f, g) = f(s k ) ( g(t k ) g(t k 1 ) ). k=1 Furthermore, for the continuous function f : A K we introduce the modulus of continuity µ(δ, f) defined for δ > 0 s µ(δ, f) = mx { f(t 1 ) f(t 2 ) : t 1, t 2 [, b], t 1 t 2 δ }, nd we denote the totl vrition of function g : [, b] K of bounded vrition s V b (g). The min im of this essy is to prove the following theorem. Theorem 3.2. Let [, b] be n intervl nd K {R, C}, nd let f : [, b] K be continuous function nd g : [, b] K function of bounded vrition. There exists exctly one element J R such tht for ny δ > 0 such tht π δ. J S π,σ (f, g) 2µ(δ, f)v b (g), From Theorem 3.2 we cn mke well-posed definition of the Riemnn- Stieltjes integrl s b f(t)dg(t) = J. The integrl bers resemblnce to the Riemnn integrl encountered in elementry clculus (see e.g. [1]). If we let g(t) = t we get exctly the Riemnn integrl. Hence the nme Riemnn-Stieltjes integrl, since it is generliztion of Riemnn s integrl mde by the Dutch stronomer Thoms Jn Stieltjes. The proof of Theorem 3.2 follows the outline in [8]. 1

8 A useful consequence of the Riemnn-Stieltjes integrl is tht, under the ssumption tht g hs continuous derivtive in [, b], we hve tht b f(t)dg(t) = b f(t)g (t)dt. The left hnd side bove is our Riemnn-Stieltjes integrl, whilst the right hnd side is Riemnn integrl. In section 4 we mke use of this property. We there chnge our nottion from g to z, which is more commonly used for complex vlued functions. Let A C be non-empty open set, nd let f : A C be continuous function. Furthermore, let z : [, b] A be continuous function of bounded vrition. We consider n oriented curve in the complex plne, which hppens to be generted by the prmetriztion z. Let be given by = { z = z(t) : t [, b] }. Now, we cn use Theorem 3.2 to define the complex curve integrl long s b f(z)dz = (f z)(t)dz(t). (1.1) Before discussing ny ssumptions of f nd z we note tht continuous function need not be of bounded vrition, nor does function of bounded vrition need to be continuous. However, we require tht z is both continuous nd of bounded vrition in order for Theorem 3.2 to pply here. We lso point out tht we hve chosen prmetriztion nd tht the curve of interest hppens to be generted by this prmetriztion. This since we then cn void the requirement tht the prmetriztion function hs non zero first order derivtive. Throughout section 4 we therefore work with the prmetriztions themselves nd not with the equivlent clsses of prmetrized curves. For simplicity, we shll lso ssume tht the prmetriztion z tht genertes the curve is continuously differentible so tht eqution (1.1) holds. Our focus for the complex curve integrl is Cuchy s gretly celebrted integrl theorem. Theorem 4.3 (Cuchy s integrl theorem). Let D be simply-connected domin in the complex plne, nd let f : D C be holomorphic function. Then for ny closed, simple curve D generted by continuously differentible prmetriztion f(z)dz = 0. In Theorem 4.3, the ssumption tht the curve D is generted by continuously differentible prmetriztion could be wekened. However, more technicl detils nd pproximtion rguments would be required. Therefore, we shll stick to the stronger ssumption, enbling clerer outline nd greter emphsis to min methods. Due to the importnce of Theorem 4.3, three different proofs re presented in section 4. First, we consider Cuchy s originl complete works from 1825 nd 1847 ([3] nd [4] respectively). His results re presented in Theorem 4.2. Secondly, we present proof of Theorem 4.3 using rel nlytic methods in which the key ingredient is Green s theorem. The min outline here follows 2

9 from Greene nd Krntz in [7]. We re well wre tht Green s theorem requires tht the integrnd f hs continuous first order derivtive, nd tht Theorem 4.3 is true even without this requirement. A more generl rel nlysis proof ws presented by Gourst in 1884 (see e.g. [6] nd [18]). However, Gourst s proof is more techniclly complicted which is why we hve chosen the more strightforwrd version with Green s theorem in this essy. We lso present third nd finl proof without the dditionl requirement on f. This proof is inspired by Dixon s proof in [5], which is built round the ide of homotopic curves. To demonstrte the verstility of the Riemnn-Stieltjes integrl, we end this essy by presenting n ppliction of it in probbility theory. In section 5 we therefore consider rndom processes nd prticulrly the expected vlue of rndom vribles. By following the ides of Anevski in [2], we rech the conclusion tht the expected vlue of rndom vribles cn be interpreted s the Riemnn-Stieltjes integrl of the rndom vrible with respect to its distribution function. The results re presented in Theorem 5.4 nd Theorem 5.7 for the one- nd n-dimensionl cses respectively. The min purpose of defining expected vlues with the Riemnn-Stieltjes integrl is tht it provides definition of the expecttion of sum of one continuous nd one discrete rndom vrible s the sum of the two seprte expecttions of the continuous nd discrete vribles. We present this result in Theorem 5.8, which is vlid regrdless of the type of the rndom vribles. Theorem 5.8. If X nd Y re two rndom vribles nd nd b re two rel vlued constnts, then the expecttion of X nd Y stisfies E(X + by ) = E(X) + be(y ). This essy is structured s follows. In section 2, we begin by introducing necessry bckground theory nd nottions. For further detils relted to rel nlysis see e.g. [15], for more on complex nlysis see e.g. [7] or [14], nd for topology see e.g. [17]. In section 3, we then present the min proof of this essy, nmely for the existence of the Riemnn-Stieltjes integrl (Theorem 3.2). Three different proofs of Cuchy s integrl theorem (Theorem 4.3) then follow in section 4. Finlly, in section 5 we present connection between the Riemnn- Stieltjes integrl nd the expecttion of rndom vribles. For further reding on probbility theory nd on the Riemnn-Stieltjes integrl in probbility theory see e.g. [9] or [13]. 3

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11 2. Preliminries In this section we form foundtion for the key concepts of the essy by introducing some bsic theory. Since our gret Riemnn-Stieltjes integrl requires functions of bounded vrition, we begin by discussing wht they re. To prove the existence of the Riemnn-Stieltjes integrl we lso need to tke tour into the world of topology, nd therefore some bsic topology is included. To finish up, we present some bsic concepts of complex nlysis which will ly ground for the pplictions of our Riemnn-Stieltjes integrl in complex nlysis Functions of bounded vrition. The concept of functions of bounded vrition first rose in the lte 19 th century in connection with the lignment of curves when studying convergence of Fourier series. It is precisely with respect to these functions tht we my define the Riemnn-Stieltjes integrl of continuous functions. Definition 2.1. We define prtition π = {t k } of n intervl [, b] s subdivision of [, b] such tht = t 0 < t 1 <... < t n = b, with n N nd t k [, b], k = 0, 1,..., n. A refinement π of prtition π is further subdivision of [, b] such tht every element of π is subset of some element of π. Furthermore, the totl refinement π T of π is the prtition tht corresponds to dding up ll subintervls of [, b]. Let K {R, C} nd g : [, b] K, nd let π be prtition of [, b]. Definition 2.2. A function g : [, b] K is sid to be of bounded vrition on [, b] if there exists n L R such tht n L π (g) = g(t k ) g(t k 1 ) L, k=1 where L π (g) is sum corresponding to prtition π. vrition of function g on the intervl [, b] s V b (g) = sup L π (g). π We define the totl Since every sum L π (g) is bounded, so will the set of ll sums L π (g) be. By our definition of K, Definition 2.2 is vlid for both rel nd complex vlued functions g. In fct, every rel vlued function tht is monotone on given intervl stisfy the conditions of being of bounded vrition. For such function g, the totl vrition of g on [, b] will be V b (g) = g() g(b). The following theorem sums up importnt properties for the totl vrition. Theorem 2.3. Let g : [, b] K be of bounded vrition. I. If [x 1, x 2 ] [, b], g will lso be of bounded vrition on [x 1, x 2 ] nd V x2 x 1 (g) V b (g). Tht is, the totl vrition increses with lrger intervl. 5

12 II. If < c < b, we cn write the totl vrition s V b (g) = V c (g) + V b c (g). Tht is, the totl vrition is dditive on consecutive subintervls. Proof. To prove prt (I) we construct two intervls I = [, b] nd I 1 = [x 1, x 2 ] with totl vritions V = V b (g) nd V 1 = Vx x2 1 (g). Let π 1 be prtition of I 1, nd let π be π 1 with the points nd b djoined. Tht is, π = π 1 {, b}, setting π s prtition of the intervl I. Let L π (g) nd L π1 (g) be sums constructed s in Definition 2.2. We hve L π1 (g) L π (g). nd it follows tht V 1 V. For prt (II) we construct three intervls I = [, b], I 1 = [, c] nd I 2 = [c, b]. Also, set V = V b (g), V 1 = V c (g) nd V 2 = Vc b (g) s the totl vritions on ech intervl. Let π 1 nd π 2 be prtitions of I 1 nd I 2 respectively. Then π = π 1 π 2 will be prtition of I, nd we hve Since V = sup π L π (g) by definition L π (g) = L π1 (g) + L π2 (g). L π1 (g) + L π2 (g) V. Furthermore, V 1 = sup π1 L π1 (g) nd V 2 = sup π2 L π2 (g), so L π1 (g) + L π2 (g) V 1 + V 2 V. (2.1) Conversely, if π is prtition of I, let π be π with the point c djoined. By (I) we then hve L π (g) L π (g) nd furthermore, if π splits into prtitions π 1 of I 1 nd π 2 of I 2 L π (g) L π (g) = L π1 (g) + L π2 (g) V 1 + V 2. Since V = sup π L π (g) we get tht V V 1 + V 2. (2.2) Equtions (2.1) nd (2.2) will leve us with V = V 1 + V 2 which completes the proof Bsic topology. This section is dedicted to n introduction of topologicl spces nd is bsed on Simmons s introductory book on topology nd modern nlysis [17]. A topologicl spce cn be thought of s set which only contins structure tht is relevnt to the continuity of functions defined on the set. A prt of the generl theory behind topologicl spces concerns metric spces in which we del with two elementry concepts: convergent series of rel or complex numbers nd continuous functions of rel or complex vribles. In this essy we will focus on the concept of the difference between two rel or complex numbers or functions, nd prticulrly tht the bsolute vlue of such difference represents the distnce between the two. 6

13 We will begin by defining metric spce nd present its elementry properties, leding up to Cntor s intersection theorem which represents the finl rgument in proving the existence of our Riemnn-Stieltjes integrl. Definition 2.4. Let X be non-empty set. We define metric on X s rel function d : X X R such tht I. d(x, y) 0, nd d(x, y) = 0 x = y II. d(x, y) = d(y, x) III. d(x, y) d(x, w) + d(w, y) for ll elements x, y nd w in X. We refer to the metric d(x, y) s the distnce between x nd y. Definition 2.5. We define metric spce s non-empty set X with metric d on X. Let z = x + iy, x, y R, be complex number. The distnce from z to the origin is given by the bsolute vlue z = x 2 + y 2. If z 1 = x 1 + iy 1, x 1, y 1 R, is nother complex number, the distnce between z nd z 1 is given by d(z, z 1 ) = z z 1. Thus, the complex plne is metric spce with metric d representing distnce nd stisfying I. z 0, nd z = 0 z = 0 II. z = z III. z + z 1 z + z 1. As mentioned, one of the elementry concepts for metric spces is convergent sequences of rel or complex numbers. Therefore, we shll consider the circumstnces under which series converges in metric spce. Definition 2.6. Let X be metric spce with metric d, nd let {x n } = {x 1, x 2,..., x n } be sequence of points in X. If there, for ech ε > 0, exists point x X nd positive integer N such tht n N d(x n, x) < ε 2, then the sequence {x n } is sid to be convergent nd converges to the point x, clled its limit. Equivlently, the sequence {x n } is convergent if there exist point x X such tht d(x n, x) tends to zero with incresing n. Definition 2.7. Let X be metric spce with metric d, nd let {x n } be convergent sequence in X. If, for ech ε > 0, there exists positive integer N such tht m, n N d(x m, x n ) < ε, then {x n } is clled Cuchy sequence. It follows directly from Definition 2.7 tht every convergent sequence is Cuchy sequence since, for some integers m, n N, the convergence of sequence {x n } implies tht m, n N d(x m, x n ) d(x m, x) + d(x, x n ) < ε 2 + ε 2 = ε. However, the converse, tht every Cuchy sequence is convergent, is not necessrily true. For metric spces in which every Cuchy sequence is convergent we rech the following definition. 7

14 Definition 2.8. We define complete metric spce s metric spce in which every Cuchy sequence is convergent nd converges to point in the metric spce. Let X be metric spce with metric d nd let A be subset of X. Then point x is clled limit point of A if ech open sphere centred round x contins t lest one point x 0 A different form x. Furthermore, we sy tht subset A of metric spce X is closed if it contins ll of its limit points. The union of A nd the set of ll its limit points is clled the closure of A, nd is denoted by A. Now, let X be metric spce with metric d, nd let F be non-empty subset of X in which the function d is defined only for points in F. Then F is itself metric spce with the prticulr metric d nd we cll it subspce of X. Theorem 2.9. Let X be complete metric spce, nd let F be subspce of X. F being complete is equivlent to F being closed. Proof. First, we ssume tht the subspce F is complete nd show tht it must be closed. For positive integers n, let B(x, 1 n ) F be sphere of rdius 1 n centred round x contining different point x n. Since d(x n, x) < 1 n tends towrds zero with incresing n, x is limit point of F. The sequence {x n } thereby converges to x in X, nd is therefore Cuchy sequence in X by the rgument following Definition 2.7. Since F is complete, x must be point of F, which implies tht F must be closed. Conversely, we ssume tht F is closed nd show tht F is complete. Let {x n } be Cuchy sequence in F nd thereby lso Cuchy sequence in X. The completeness of X implies tht {x n } converges to point x in X. Since {x n } F, x will be limit point in F. A closed set contins ll its limit points, which implies tht x F. Thus ech Cuchy sequence {x n } in F is convergent nd converges to point in F, nd F is thereby complete by Definition 2.8. Definition Let A be non-empty set of metric spce X with metric d. We define the dimeter of the set A s d(a) = sup { d(x 1, x 2 ) : x 1, x 2 A }. Now, let {F n } n=1 be decresing sequence of X, tht is F 1 F 2 F 3... Then the sequence {d(f n )} n=1 will lso be decresing. Conditions under which n intersection of such sequence will be non-empty is provided in the following theorem by Cntor. Theorem 2.11 (Cntor s intersection theorem). Let X be complete metric spce, nd let {F n } be decresing sequence of non-empty closed subsets of X such tht d(f n ) 0 s n. Then F = n=1 F n contins exctly one point. Proof. The ssumption tht d(f n ) 0 implies tht F cnnot contin more thn one point. Therefore, in order to prove the theorem it is sufficient to prove tht F is non-empty. Ech subset F n is closed nd thereby lso complete by Theorem 2.9. Let x n be point in F n, thus forming sequence {x n } in X. We hve ssumed tht d(f n ) 0 s n. Therefore, for ε > 0, there exists positive integer N 8

15 such tht n N d(f n ) < ε. {F n } is furthermore decresing sequence. Therefore, given ny n, m N, we hve tht x n, x m F n nd d(x n, x m ) < ε. By Definition 2.7 {x n } is thereby Cuchy sequence. Since every F n is closed, F = n=1 F n will be closed nd thereby lso complete. At length the Cuchy sequence will be in F, implying tht lim n x n = x F Bsic complex nlysis. Holomorphic functions nd their properties re of gret importnce for the theory behind complex integrtion, nd in this essy they re the foundtion on which the entire section on pplictions of the Riemnn-Stieltjes integrl in complex nlysis is built. Another importnt spect concerns the curves long which integrtion is performed. In this section we therefore lso consider curves tht cn be continuously deformed into other more conveniently shped curves. This opens up to new pproch to treting complex integrls. Definition Let f be complex vlued function defined in neighbourhood of point z 0 C. Then f is complex differentible t z 0 if the limit f(z 0 + h) f(z 0 ) lim h 0 h exists, nd we define the complex derivtive s this limit, denoted f (z 0 ). Since h C, it cn pproch zero in numerous wys. However, the limit must tend towrds unique vlue f (z 0 ) independent of how h pproches zero. Definition Let A C be non-empty open set. A function f : A C is holomorphic on A if f is complex differentible t every point of A. A necessry condition for function to be complex differentible t point z 0 is tht the Cuchy-Riemnn equtions hold t z 0. Tht is, if the function f(z) = u(x, y) + iv(x, y) is differentible t z 0 = x 0 + iy 0 then u x = v y, u y = v x (2.3) must hold t z 0 = x 0 + iy 0. Equtions (2.3) re clled the Cuchy-Riemnn equtions. Consequently, for function to be holomorphic in n open set A C, Cuchy-Riemnns equtions must hold t every point of A. Theorem If f is holomorphic function in non-empty open set A C, then the Cuchy-Riemnn equtions hold t every point in A. For further detils nd proof of Theorem 2.14 we refer to e.g. pges in [16]. We point out tht the reverse impliction does not pply without some dditionl requirements. Tht is, tht the Cuchy-Riemnn equtions hold is not lone sufficient to ensure holomorphicity. For exmple, we need existence of the first order prtil derivtives of u nd v in the given domin, s well s their continuity. For holomorphic function f : A C in non-empty open set A C, defined by f(z) = u(x, y) + iv(x, y), differentition of f with respect to z gives f z = 1 2 ( u x + v y ) + i 2 9 ( v x u y ).

16 Applying Cuchy-Riemnns equtions (2.3) to this expression yields the following result for ll holomorphic functions. Proposition If f : A C is holomorphic function in non-empty open set A C, then f z = f x = i f y. Now, we hve sorted out some bsic theory for holomorphic functions nd turn to the concept of continuously homotopic curves nd the domins in which these so clled deformble curves exist. Definition Let D C be n open, connected set. A closed, smooth curve 0 D is continuously homotopic to the closed, smooth curve 1 D if there exists continuous function z : [0, 1] [0, 1] D such tht I. For every fixed s [0, 1], z s (t) = z(s, t) is prmetriztion of closed, piecewise smooth curve in D. II. The function z 0 (t) = z(0, t) is prmetriztion of 0. III. The function z 1 (t) = z(1, t) is prmetriztion of 1. A closed, smooth curve 0 is lso continuously homotopic to curve 1 in given domin if, in the domin, 0 is continuously homotopic to single point nd 1 is lso continuously homotopic to the sme point. Definition Let D C be n open, connected domin. If every closed, piecewise smooth curve in D is continuously homotopic to point, then D is sid to be simply-connected. In prctice, the concept of continuously homotopic curves is of gret importnce to the theory of holomorphic functions since it lys ground for complex integrtion nd pth independence in simply-connected domins. We will return to this in detil when considering complex curve integrls in section 4. 10

17 3. The existence of the Riemnn-Stieltjes integrl The purpose of this section is to show the existence of the Riemnn-Stieltjes integrl. We follow the method from Hille in [8], using Riemnn-Stieltjes sums to show existence of n element J which cn be pproximted to the Riemnn- Stieltjes integrl. This s opposed to the more common method of using upper nd lower Riemnn-Stieltjes sums nd defining the Riemnn-Stieltjes integrl s the vlue where the infimum of the upper sum nd the supremum of the lower sum coincide. Hille s method proves much techniclly pretty nd provides clerer overview of the proof, nd we therefore consider it fvourble. Before we begin, we need to introduce some required conditions for the existence of the Riemnn-Stieltjes integrl, such s the domin nd functions involved, s well s define the Riemnn-Stieltjes sums with which we re to pproximte our integrl. We hve chosen to show the existence of the Riemnn- Stieltjes integrl for functions defined on non-empty open subset of K {R, C}. This since existence my then be proved by the sme method for both rel nd complex vlued functions. Consider function g : [, b] K of bounded vrition. For every prtition π = {t k } s defined in Definition 2.1, let L π (g) be sum corresponding to π nd g such tht n L π (g) = g(t k ) g(t k 1 ). k=1 since g is function of bounded vrition, there is constnt L such tht L π (g) L. Since ech sum L π (g) is bounded, the set of sums {L π (g)} will lso be bounded, nd Definition 2.2 gives the totl vrition of g on [, b] s V b (g) = sup L π (g), where g is of bounded vrition. For point c [, b] such tht < c < b, Theorem 2.3 gives the following property for the totl vrition of g π V b (g) = V c (g) + V b c (g). For the continuous function f : [, b] K, the nottion of modulus of continuity pplies µ(δ, f) = mx { f(t 1 ) f(t 2 ) : t 1, t 2 [, b], t 1 t 2 δ }. (3.1) For δ > 0 we hve tht µ(δ, f) is monotoniclly incresing function stisfying lim µ(δ, f) = 0 δ 0 + with the exception when f is constnt nd µ(δ, f) = 0. This follows from the evlution of the limit where we hve f(t 1 ) f(t 2 ) with t 1 t 2 tending towrds zero. For prtition π = {t k }, consider point s k in every subintervl [t k 1, t k ] of [, b]. Let σ = {s k } be the set of ll points s k [t k 1, t k ], k = 1, 2,..., n. Definition 3.1. Let π be prtition of n intervl [, b], nd let σ = {s k } be point set where s k [t k 1, t k ] [, b], k = 1, 2,..., n. We define the 11

18 Riemnn-Stieltjes sum corresponding to π, σ, f nd g s n S π,σ (f, g) = f(s k ) ( g(t k ) g(t k 1 ) ), k=1 where f : [, b] K is continuous function nd g : [, b] K is function of bounded vrition. The norm of the prtition π is given by π = mx k (t k t k 1 ). Now, we wnt to prove the following theorem for Riemnn-Stieltjes sums. Theorem 3.2. Let [, b] be n intervl nd K {R, C}, nd let f : [, b] K be continuous function nd g : [, b] K function of bounded vrition. There exists exctly one element J R such tht J S π,σ (f, g) 2µ(δ, f)v b (g), (3.2) for ny δ > 0 such tht π δ. Here, S π,σ (f, g) is Riemnn-Stieltjes sum, (g) is the totl vrition of g, nd µ(δ, f) is the modulus of continuity. V b Proof. This proof is bsed on evluting the distnce between two Riemnn- Stieltjes sums. The vlue on the right hnd side in eqution (3.2) is reched by lternting the prtition corresponding to Riemnn-Stieltjes sum. We begin with fixed prtition, then refine it, nd then crete reltion between the prtition nd its refinement. The finl step of defining the element J is mde by turning to the world of topology nd Cntor s intersection theorem (Theorem 2.11). First, consider the left hnd side of inequlity (3.2) for two rbitrry Riemnn- Stieltjes sums with fixed prtition on the given intervl. Let π = {t k } n k=1 be prtition of the intervl [, b], nd let π δ, for δ > 0. For every subintervl [t k 1, t k ] of [, b], consider two rbitrry points s k,1 nd s k,2 nd form the sets σ 1 = {s k,1 } nd σ 2 = {s k,2 }, k = 1, 2,..., n. Now we cn estimte the distnce between these two sums n ( S π,σ1 (f, g) S π,σ2 (f, g) = f(sk,1 ) f(s k,2 ) )( g(t k ) g(t k 1 ) ) k=1 n f(s k,1 ) f(s k,2 ) g(t k ) g(t k 1 ). k=1 Since π δ is given, we hve tht s k,1 s k,2 δ for every k = 1, 2,..., n nd therefore the inequlity f(s k,1 ) f(s k,2 ) µ(δ, f) holds. For k = 1, 2,..., n we thus hve n S π,σ1 (f, g) S π,σ2 (f, g) µ(δ, f) g(t k ) g(t k 1 ) (3.3) k=1 µ(δ, f)v b (g). Secondly, consider two prtitions π 1 nd π 2 of [, b] such tht π 1 δ nd π 2 is set s the totl refinement of π 1. Let σ 1 = {s k,1 } be point set to the Riemnn-Stieltjes sum S π1,σ 1 (f, g) constructed s before with the corresponding 12

19 prtition π 1. Tke subintervl [t 0, t 1 ] of [, b] where t 0, t 1 π 1. The first term of S π1,σ 1 (f, g) will be T 1 = f(s 1 ) ( g(t 1 ) g(t 0 ) ). Now, prtition [t 0, t 1 ] into m subintervls T 1 cn then be written s t 0 = t 0 < t 1 <... < t m = t 1. T 1 = m f(s 1 ) ( g(t j) g(t j 1) ). j=1 We cn choose the sme points t j, j = 1, 2,..., m, in π 2 nd write m T 2 = f(s j) ( g(t j) g(t j 1) ) j=1 for the prt of S π2,σ 2 (f, g) over [t 0, t 1 ]. Here, the s j [t j 1, t j ] re rbitrry. With this we cn write m ( T 1 T 2 = f(s1 ) f(s j) )( g(t j) g(t j 1) ) j=1 m f(s 1 ) f(s j) g(t j) g(t j 1). j=1 Since π 1 δ, s 1 s j δ will follow nd therefore f(s 1) f(s j ) µ(δ, f) so tht T 1 T 2 µ(δ, f) m j=1 g(t j) g(t j 1) µ(δ, f)v t1 t 0 (g). (3.4) If this is repeted for every subintervl [t k 1, t k ] of [, b], k = 1, 2,..., n, eqution (3.4) yields n S π1,σ 1 (f, g) S π2,σ 2 (f, g) µ(δ, f) V t k t k 1 (g) = µ(δ, f)v b (g). (3.5) Now, consider two rbitrry Riemnn-Stieltjes sums to the prtitions π 1 nd π 2 with the corresponding point sets σ 1 nd σ 2. Let the union of the sets of division points of π 1 nd π 2 define prtition π 3, which will be refinement of both π 1 nd π 2. Independently consider n rbitrry point set σ 3 with one point in between every pir of consecutive points in π 3. From eqution (3.5) we get S π1,σ 1 (f, g) S π2,σ 2 (f, g) = k=1 = S π1,σ 1 (f, g) S π3,σ 3 (f, g) + S π3,σ 3 (f, g) S π2,σ 2 (f, g) S π1,σ 1 (f, g) S π3,σ 3 (f, g) + S π2,σ 2 (f, g) S π3,σ 3 (f, g) µ(δ, f)v b (g) + µ(δ, f)v b (g) = 2µ(δ, f)v b (g). (3.6) Finlly, for δ > 0, we define H(δ) s the closure of the set of our Riemnn- Stieltjes sums S π,σ (f, g) with π δ, i.e. H(δ) = {S π,σ (f, g) : π δ}. 13

20 It then follows from eqution (3.6) tht the distnce between ny two Riemnn- Stieltjes sums in H(δ) is d(h(δ)) 2µ(δ, f)v b (g). Since µ(δ, f) is positive, monotone incresing function for ll non-constnt functions f, which tends to zero with δ, we hve tht H(δ) will be non-empty subset of K nd lim δ 0 + d(h(δ)) = 0. We now consider Cntor s intersection theorem (Theorem 2.11) which sys tht for the complete metric spce X, decresing sequence {F n } of non-empty, closed subsets of X, stisfying the limit lim d(f n) = 0 n we hve tht n=1 F n contins exctly one element. K {R, C} is complete metric spce due to the completeness of both R nd C, nd H(δ) is closed, non-empty subset of K such tht d(h(δ)) 0 s δ tends towrds zero. Set H n = H( 1 n ). We cn then pply Theorem 2.11 which sys tht there exists exctly one element J in n=1 H n tht stisfies the inequlity J S π,σ (f, g) 2µ(δ, f)vb (g). Definition 3.3. Let [, b] be n intervl. Let f : [, b] K be continuous function nd let g : [, b] K be function of bounded vrition. We define the element J R in J S π,σ (f, g) 2µ(δ, f)v b (g), for ny δ > 0 such tht π δ, s the Riemnn-Stieltjes integrl of f with respect to g over the intervl [, b] nd denote it by J = b f(t)dg(t). (3.7) The min properties of the integrl in eqution (3.7) re presented in the following theorem. Theorem 3.4. Let A K be non-empty open set nd let [, b] be n intervl. For continuous function f : A K nd function g : [, b] A of bounded vrition, the following properties for the Riemnn-Stieltjes integrl of f with respect to g pply: I. The integrl is liner both with respect to the integrnd f nd with respect to the integrtor g such tht b b [αf 1 (t) + βf 2 (t)]dg(t) = α f(t)d[αg 1 (t) + βg 2 (t)] = α where α, β K. b b 14 f 1 (t)dg(t) + β f(t)dg 1 (t) + β b b f 2 (t)dg(t), f(t)dg 2 (t),

21 II. The integrl is dditive with respect to the intervl of integrtion, tht is for < c < b b f(t)dg(t) = c f(t)dg(t) + b c f(t)dg(t). III. The Riemnn-Stieltjes integrl cn be estimted by the following inequlity: b f(t)dg(t) mx f(t) V b (g). (3.8) t [,b] IV. If the integrtor g hs continuous derivtive g, the following equlity pplies b f(t)dg(t) = b f(t)g (t)dt. where the integrl on the right hnd side is n ordinry Riemnn integrl. Properties (I)-(III) in Theorem 3.4 follow directly from the definition of the Riemnn-Stieltjes integrl s limit of sums. For further detils on proofs of these properties we refer to Krntz [11], nd for detils on the proof of property (IV) we refer to Wikström [18]. Before turning to some pplictions of the Riemnn-Stieltjes integrl, note tht since Definition 3.3 is vlid for both complex nd rel vlued functions, the properties of Theorem 3.4 will pply to both complex nd rel vlued functions f nd g. 15

22

23 4. Some pplictions in complex nlysis The purpose of this section is to define the curve integrl in the complex plne, nd the min focus will then be Cuchy s gret integrl theorem. We begin by considering the complex curve integrl, which is ctully specil cse of the Riemnn-Stieltjes integrl. Whilst the Riemnn-Stieltjes integrl simply requires tht the prmetriztion function which generte curve for integrtion is of bounded vrition, the complex curve integrl lso requires tht the prmetriztion function is continuous. After defining the complex curve integrl, we turn to Cuchy s originl work on complex integrtion leding up to his complete finl proof of the integrl theorem. We then present generl version of the theorem long with two modern proofs, one with pplied rel nlysis nd one with homotopy pproch. Let A C be non-empty open set. Consider continuous function of bounded vrition z : [, b] A. Let z be prmetriztion which hppens to generte n oriented curve in the complex plne, = {z = z(t) : t [, b]}. If f : A C is continuous function, then (f z) : [, b] A will be continuous on [, b]. The Riemnn-Stieltjes integrl of (f z) with respect to z then exists ccording to Theorem 3.2, nd we define the curve integrl in the complex plne s the Riemnn-Stieltjes integrl. Definition 4.1. Let z : [, b] A be continuous prmetriztion function of bounded vrition, which genertes curve on [, b]. Furthermore, let A C be non-empty open set nd let f : A C be continuous function. Then (f z) : [, b] C is continuous function nd we define the curve integrl in the complex plne s the Riemnn-Stieltjes integrl of (f z) with respect to z b f(z)dz = (f z)(t)dz(t). This integrl will be completely independent of the chosen prmetriztion. According to [8] nd [11], properties for the complex curve integrl follow without chnge from the corresponding properties in Theorem 3.4 for the Riemnn- Stieltjes integrl. We note however, tht n estimtion of the complex curve integrl s in eqution (3.8), the length of the curve l() is used insted of the totl vrition V b (g). Now, we turn bck in time to consider the ctul work of the French mthemticin Augustin-Louis Cuchy who first presented the bsic ide to his integrl theorem in Mémoire sur l théorie des intégrles définies from his 1814 memoires [10]. There, he treted reltions between first order prtil derivtives of rel vlued functions, nd considered the simple cse of complex integrtion long the boundry of rectngle. In Mèmoire sur les intégrles définies, prises entre des limites imginires [3] of his 1825 memoires he continued to develop this work on definite integrl between complex limits X+iY f(z)dz, (4.1) x 0+iy 0 where [x 0, y 0 ] nd [X, Y ] were the boundry points of the rectngle round which integrtion ws considered. He studied the two line integrls obtined by 17

24 seprting the rel nd imginry prts in his definite integrl nd eventully reched the following expressions X x 0 X x 0 f(x + iy 0 )dx + i Y y 0 f(x + iy)dy Y f(x + iy )dx + i f(x 0 + iy)dy, y 0 which he clled the extreme vlues of the integrl in eqution (4.1). Cuchy stted tht equlity between these extreme vlues would occur s long s f ws finite for ll vlues of x nd y on nd within ny chosen boundries between the points (x 0, y 0 ) nd (X, Y ). If opposite occurred, tht f hd singulrities for some vlue of x nd y on the boundries, the difference between the extreme vlues would be infinite or indefinite. The cse when f ws finite for ll x nd y on the boundries, but might hve singulrities within the boundries, would generte finite difference between the extreme vlues which Cuchy denoted by. To summrize X Y X Y f(x+iy 0 )dx+i f(x+iy)dy = f(x+iy )dx+i f(x 0 +iy)dy+, x 0 y 0 x 0 y 0 where represented the result of possible singulrities of f within the specified boundries. Cuchy s -nottion corresponds to wht we tody cll residues nd denote by 2πi n k=1 Res(z k), z k being singulrities of f, with k = 1, 2,..., n for some n. Cuchy continued by investigting wht hppened when letting f(x+iy) vnish when sending ech of the boundry vlues of the rectngle towrds ±infinity. He finlly reched tht = 0, which implied tht his definite integrl would ultimtely equl zero. However, his method hd certin limittions. Cuchy s ssumption tht the integrls would vnish with their integrnd functions nd tht he could therefore evlute limits under the integrl sign ws considered true in generl, but there existed cses where the opposite occurred, which required certin modifictions to the integrnd functions. So bsiclly, Cuchy s proof in [3] ws insufficient since his results did not pply for ll cses. nous vons supposé que les intégrles comprises dns cette formule s èvnouissint vec les fonctions qu elles renferment. C est ce qui lieu en génerl. Nénmoins le contrire rrive dns un petit nombre de cs prticuliers, et lors les équtions que nous vons étblies dovient être modifiées. However, in Sur les intégrles qui s étendent à tous les points d une courbe fermée [4] from 1847, he proved some remrkble theorems tht provides the necessry conditions for the existence of definite integrls of complex functions long closed curves, llowing him to provide complete proof of his integrl theorem. The results re presented in Theorem 4.2 below. In order to preserve sense of originlity to Cuchy s work, gret effort hs been mde to hold to Cuchy s originl nottions nd peculir lnguge in the trnsltion from French. 18

25 Theorem 4.2. I. The position of vrible point P is determined by the rectngulr or polr coordintes, or coordintes of other nture, x, y, z,... which vry continuously long with the position of the point. Let S be n re mesured in given plne or on given surfce, nd let S be bounded by single closed curve. Let the point P trvel round the curve bounding S in given direction, nd set s s this positively oriented rc. Finlly, let k be function of the vribles x, y, z,... with derivtives reltive to s, nd set (S) s the vlue cquired by the integrl kds when the vrible point P hs trvelled one revolution long the curve s round the re S. If by stright lines or curves on plne or on given surfce, the re S is divided into subres such tht A, B, C,... (A), (B), (C),... re the corresponding (S) to S for ech subre A, or B, or C,..., we get S = A + B + C +... nd furthermore (S) = (A) + (B) + (C) +... in which the function k is defined s finite nd continuous on every point of every contour. II. Let the conditions from prt (I) pply, then set k = XD s x + Y D s y + ZD s z +... where X, Y, Z,... re functions of x, y, z,... chosen such tht the sum Xdx + Y dy + Zdz +... is n exct differentil, nd let the surfce S vry with vrying insensible degrees of the curve which bounds S. Such chnges will not ffect the vlue of the integrl (S) when k is finite nd continuous t ech successive point on the curve. III. Let the conditions from prt (II) pply. Suppose tht the function k ceses to be finite nd continuous t the single points P, P, P,... situted in the interior of S. Let, b, c,... be very smll re elements of S, ech enclosing one of the points of discontinuity. Then the eqution (S) = () + (b) + (c) +... provides n expression for the integrl (S) s sum of single integrls over ech of the re elements, b, c,... In the prticulr cse when k 19

26 is finite nd continuous for ll points in the interior of S, the single integrls will vnish simply leving (S) = 0. The finl result in (III) of Theorem 4.2 corresponds to Cuchy s integrl theorem, nd his originl proof is presented below. Proof of Theorem 4.2 (III). Let S be plnr surfce. Then x nd y re reduced to two rectngulr or polr coordintes, or coordintes of other nture, which determine the position of point in the plne of the surfce S. If we let X nd Y be two continuous functions of x nd y, nd if we ssume k = XD s x + Y D s y, then we hve (S) = ± (D y X D x Y ) dxdy, where the double integrl extends to ll points of the surfce S. Let x nd y be two mesurble lengths from n rbitrry point P which corresponds to the coordintes x nd y, directing the lengths in mnner to obtin positively incresing coordintes. The ± sign is determined by considering if the orienttion of rottion round S genertes n incresing s or not. In the prticulr cse where the sum is n exct differentil, we get Xdx + Y dy D y X = D x Y, nd the eqution which determines the vlue of (S) is reduced to the simple cse of (S) = 0. Cuchy s method includes seprting the rel nd imginry prts of the definite integrl in eqution (4.1), nd considering simple closed curve with positive orienttion s his pth of integrtion. He considers the prticulr cse when k = Xdx + Y dy + Zdz +... is n exct differentil, which mens tht (X, Y, Z,...) represents conservtive vector field with potentil kds, where the integrl is independent of the pth of integrtion. Thus, Cuchy dels with simply-connected domin. No rel rgument is given for why he is ble to rewrite the definite integrl long simple closed curve s double integrl over the re enclosed by the curve. The justifiction however, lies in the fct tht k is mde up of continuous functions in bounded domin. The nottions Cuchy uses to express derivtives, D s x nd D s y, refer to the totl differentils dx dy ds nd ds respectively. He considers totl differentils throughout Theorem 4.2 nd mkes no comment on the difference between totl nd prtil differentils. From now on, we will however distinguish between totl nd prtils differentils such s d ds nd s respectively. Now, let us turn to the generl version of Cuchy s integrl theorem. 20

27 Theorem 4.3 (Cuchy s integrl theorem). Let D be simply-connected domin in the complex plne, nd let f : D C be holomorphic function. Then for ny simple, closed curve D generted by continuously differentible prmetriztion we hve tht f(z)dz = 0. (4.2) The rel nlysis pproch to this theorem is bsed on Cuchy s method of rewriting the curve integrl long closed curve into double integrl over the re enclosed by the chosen curve. Therefore, we introduce Green s theorem. This is done in full wreness tht it requires the dditionl condition tht the integrnd function f hs continuous first order derivtive. We re lso wre of the discussions leding up to Gourst s more generl proof, but will stick to the strightforwrd proof tht Green s theorem entils. Let f : D C be holomorphic function nd set f(z) = u(x, y) + iv(x, y), seprting the rel nd imginry prts of f. The definite curve integrl of f long ny closed curve D, generted by continuously differentible prmetriztion cn then be written s f(z)dz = (udx vdy) + i (vdx + udy). Here u, v : D R re two hrmonic functions which stisfy Cuchy-Riemnns equtions, nd D R 2 is the corresponding set of D when C is identified with R 2 chosen so tht (x, y) D if nd only if z = x + iy D. Theorem 4.4 (Green s Theorem). Let D C be simply-connected domin nd let continuously differentible prmetriztion generte positively oriented, simple closed, piecewise smooth curve in D. Let u(x, y) nd v(x, y) be two continuous rel-vlued functions with continuous first order prtil derivtives. Then, ( ( ) v u(x, y)dx + v(x, y)dy = x u ) dxdy, (4.3) y where A is regulr region in D corresponding to the re enclosed by. Proof. Since A is regulr region, it is both x- nd y-simple. Let A be defined by x b nd y 1 (x) y y 2 (x), where y = y 1 (x) is oriented from left to right nd y = y 2 (x) is oriented from right to left. Consider the lst term in the integrl of eqution (4.3). Since u(x, y) is continuous on bounded regulr domin we cn rewrite this term s A u y dxdy = b dx y2(x) y 1(x) u y dy = Also, dx = 0 on the verticl xes of A, so b ( u(x, y)dx = u(x, y1 (x)) u(x, y 2 (x)) ) dx Similrly, v(x, y)dy = A b Combining these two results completes the proof. 21 A ( ) u(x, y 2 (x))+u(x, y 1 (x)) dx. v x dxdy.

28 Now, we re redy to compute rel nlysis proof of Theorem 4.3. Rel nlysis proof of Cuchy s integrl theorem (Theorem 4.3). Let D C be simply-connected domin nd f : D C holomorphic function. Assume tht f hs continuous first order derivtive nd set f(z) = u(x, y) + iv(x, y). By Theorem 4.4 the rel nd imginry prts of the definite curve integrl over the simple closed, continuously differentible curve cn be rewritten s double integrls over the re enclosed by ( ) ( ) f(z)dz = udx vdy + i vdx + udy = A ( v x u ) dxdy + i y A ( u x v ) dxdy. y (4.4) Since f is holomorphic, Cuchy-Riemnns equtions (2.3) pply. Eqution (4.4) thus becomes ( v A x + v ) ( u dxdy + i x A x u ) dxdy = 0. x The third proof of Cuchy s integrl theorem presented in this essy is one bsed on the ide of proof in publiction from 1971 by Dixon [5] together with publiction with comments on Dixon s proof by Loeb [12]. It is proof which combines locl properties for holomorphic functions with integrtion long simple closed curves generted by continuously differentible prmetriztion. We will begin by presenting required conditions for this homotopy proof in three lemms treting pth independence nd conditions of holomorphicity for integrls. We lso introduce Cuchy s integrl formul s min rgument behind the proof. We begin by showing tht the vlue of the integrl in Theorem 4.3 does not depend on the curve tht hppens to be generted by our chosen prmetriztion. Lemm 4.5 (Pth independence). Let D C be n open, connected domin nd let continuously differentible prmetriztion : [, b] D generte smooth curve, lso denoted by. If f nd F re holomorphic functions from D to C such tht F = f, then f(z)dz = F ((b)) F (()). Proof. Let be defined by (t) = 1 (t) + i 2 (t), t [, b]. Since is smooth curve it hs the properties of being continuous on the intervl [, b] s well s hving continuous derivtive on [, b]. Tht is, the boundry limits lim t (t) nd + lim (t) t b both exist. For smll rel vlued constnt ɛ > 0 we cn therefore write (b) () = lim ɛ 0 + ( (b ɛ) (+ɛ) ) = lim ɛ b ɛ +ɛ (t)dt = b (t)dt. (4.5)

29 Let D R 2 be the corresponding set of D when C is identified with R 2. For continuously differentible function g : D R we hve tht nd by eqution (4.5) g((b)) g(()) = (g ) (t) = g x ((t))d 1 dt + g y ((t))d 2 dt b (g ) (t)dt = b ( g x ((t))d 1 dt + g ) y ((t))d 2 dt. dt Now, set f(z) = u(x, y) + iv(x, y), z = x + iy, where u, v : D R. Since f is holomorphic, u nd v will stisfy Cuchy-Riemnns equtions. Thus, F ((b)) F (()) = b ( u = x ((t))d 1 dt + i v x ((t))d 1 dt + u ) y ((t))d 2 dt + i v y ((t))d 2 dt dt b (( u = x ((t))d 1 dt v ) ( x ((t))d 2 v + i dt x ((t))d 1 dt + u )) x ((t))d 2 dt dt b ( u = x + i v )( ) b d1 x dt + id 2 F dt = dt x ((t)) (t)dt. Since F is holomorphic function we hve tht F z Therefore, for the primitive function F to f we get F ((b)) F (()) = b = F x (f )(t) (t)dt. by Proposition From Theorem 3.4 (IV) we hve the following property for the complex curve integrl b f(z)dz = (f )(t) (t)dt. Therefore we finlly get tht f(z)dz = F ((b)) F (()). Since the curve integrl hs the property of being dditive with respect to pth, we cn extend the result to piecewise smooth curves. In this cse, let the curve be piecewise smooth curve, mde up by severl smooth curves = n. We get n n ( F (z)dz = F (z)dz = f((zj )) f((z j 1 )) ) = f((b)) f(()), j j=1 j=1 where z j 1 nd z j re the initil nd terminl points respectively corresponding to j, j = 0, 1,..., n. This implies tht the complex curve integrl is independent of ny chosen piecewise smooth curve between the points nd b. The homotopy proof tht we will present in this essy is bsed on so clled locl Cuchy theory. Tht is, if the integrnd f is holomorphic in specified 23

30 domin D nd continuously differentible prmetriztion ζ genertes closed curve D, then the integrl f(ζ) f(z) dζ ζ z is holomorphic function of z on the whole of D. We specify these conditions in Lemm 4.5. Lemm 4.6. Let D C be simply-connected domin nd let f : D C be holomorphic function. Furthermore, let ϕ be the mpping D D C defined by ϕ(ζ, z) = f (z) f(ζ) f(z) ζ z if ζ z if ζ = z (4.6) such tht ϕ(, z) is holomorphic on D for ech point z D. Set g(z) = ϕ(ζ, z)dζ. Then g is holomorphic on D. Proof. Since f is holomorphic in D the function ϕ(, z) will, for ech z D, be continuous t z nd holomorphic everywhere else in D. If is continuous, piecewise smooth, closed curve in D nd z / then g(z) = ϕ(ζ, z)dζ = f(ζ) dζ f(z) ζ z 1 ζ z dζ. It follows from the properties of ϕ tht g is holomorphic everywhere in D except on. Let z be fixed point on. By Lemm 4.5 we cn replce with nother continuous, piecewise smooth, closed curve Υ such tht z / Υ without chnging the vlues of g in surrounding to z. Then g will be holomorphic in the surrounding to z nd thus on the whole of D. Integrls of the bove form re the bckbone of one of the min rguments of the homotopy proof of Theorem 4.3. Therefore, we tke closer look t these integrls. Lemm 4.7. Let D C be simply-connected domin. Furthermore, let closed curve C D be generted by the continuously differentible prmetriztion ζ(θ) = z + re iθ, with r > 0 nd θ [0, 2π]. Then for n Z { (ζ z) n 2πi when n = 1 dζ = 0 when n 1. C Proof. The closed curve C will be the boundry of disc centred bout the point z, of rdius r > 0. Now, we use the fct tht ζ(θ) is differentible function nd pply property (IV) of the Riemnn-Stieltjes integrl of Theorem 3.4 to the integrl 2π 2π (ζ z) n dζ = (z + re iθ z) n ire iθ dθ = r n+1 ie i(n+1)θ dθ. C The fct tht n+1 ei(n+1)θ is primitive function to ie 1(n+1)θ whrn n 1 completes the proof. 24

31 1 Lemm 4.7 estblishes result for integrls of ζ z tken over the boundry of disc centred bout the point z. However, we lso need to consider the cse when evluting this sme integrl over ny simple, closed, smooth curve such tht z cn be point either in the interior or exterior of. Lemm 4.8. Let D C be simply-connected domin nd let f : D C be holomorphic function. Let D be simple closed, smooth curve generted by continuously differentible prmetriztion. Furthermore, let i denote the interior of nd e denote its exterior. Then 1 ζ z dζ = { 2πi when z i 0 when z e. (4.7) Proof. By Lemm 4.5 this integrl is pth independent nd therefore the vlue of the integrl will be the sme whether tken over or over the boundry of smll circle C D, positively oriented, with center t c, nd with one point in common with. The integrl cn thus be expressed s 1 ζ z dζ, C where C is closed curve prmetrized by ζ(θ) = c + re iθ, with r > 0 nd θ [0, 2π]. Just s for eqution (4.7), the vlue of this integrl will depend on if z lies in the interior or the exterior of C, denoted C i nd C e respectively. We evlute the frction term s geometric series for z c. For ll z C i we will hve z c < ζ c nd 1 ζ z = 1 ζ c ( n=0 z c ζ c ) n. Similrly, for ll z C e we will hve z c > ζ c which gives ( ) n 1 ζ z = 1 ζ c. z c z c For fixed c, r nd z / C these series re normlly convergent in the vrible ζ. If z C i we hve tht ( ) n 1 ζ z dζ = 1 z c dζ = (z c) n 1 dζ ζ c ζ c (ζ c) n+1 C C n=0 where ll the integrls on the right hnd side will vnish by Lemm 4.7, except for n = 0 when the vlue will be 2πi. If z C e C 1 ζ z dζ = C 1 z c ( n=0 ζ c z c 0 ) n dζ = n=0 n=0 C 1 (z c) n+1 (ζ c) n dζ. C Here, the integrls on the right hnd side will vnish without exception by Lemm 4.7. Now, we hve reched the min rgument for the homotopy proof of Theorem

32 Theorem 4.9 (Cuchy s integrl formul). Let D C be simply-connected domin, nd let f : D C be holomorphic function. Furthermore, let D be simple, closed, positively oriented curve generted by continuously differentible prmetriztion. Let i denote the interior of. Then for ll z i f(z) = 1 2πi f(ζ) dζ. (4.8) ζ z Proof. The integrnd is holomorphic function of ζ on, except t ζ = z. Let the point z i be given nd fixed. Consider the function ϕ(ζ, z) defined s in eqution (4.6) for ζ D \ {z}. Furthermore, let Φ be holomorphic function such tht Φ = ϕ. Since is closed curve, we will hve () = (b) on some intervl [, b] D. Thus, by Lemm 4.5 we hve which gives 0 = Φ((b)) Φ(()) = f(z) ϕ(ζ, z)dζ = 1 ζ z dζ = f(ζ) f(z) dζ, ζ z f(ζ) dζ. (4.9) ζ z Since z i, the vlue of the integrl on the left hnd side in eqution (4.9) is 2πi by Lemm 4.8. Thus, f(z) = 1 f(ζ) 2πi ζ z dζ. It cn be show inductively tht repeted differentitions with respect to z under the integrl sign in eqution (4.8) yields formuls for successive derivtives of f. Under the sme conditions s in Theorem 4.9, this gives generlized Cuchy integrl formul f (n) (z) = n! 2πi f(ζ) dζ (4.10) (ζ z) n+1 with n = 1, 2, 3,... For detiled proof we refer to Theorem in [7]. As finl step before we present the homotopy proof of Theorem 4.3, we introduce two theorems contining importnt restrictions on the behviour of holomorphic functions. Theorem 4.10 (Cuchy estimtes). Let f be holomorphic inside nd on the positively oriented curve C given by ζ z 0 = r, generted by continuously differentible prmetriztion. If f(ζ) M for ll ζ C, then the derivtives of f t z 0 stisfy f (n) (z 0 ) n!m r n for n = 1, 2, 3,... Proof. By Cuchy s generlized integrl formul, eqution (4.10), we hve f (n) (z 0 ) = n! f(ζ) dζ. 2πi (ζ z 0 ) n+1 C 26

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