THE FUNDAMENTAL THEOREMS OF FUNCTION THEORY

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1 THE FUNDAMENTAL THEOREMS OF FUNCTION THEORY TSOGTGEREL GANTUMUR Contents 1. Contour integrtion 1 2. Gourst s theorem 5 3. Locl integrbility 6 4. Cuchy s theorem for homotopic loops 7 5. Evlution of rel definite integrls 8 6. The Cuchy integrl formul 1 7. The Cuchy-Tylor theorem Morer s theorem The Cuchy estimtes The identity theorem The open mpping theorem Contour integrtion Let Ω C be n open set. A (topologicl) curve in Ω is continuous mp : [, b] Ω, nd it is clled closed curve or loop if () = (b). Loops in Ω cn lso be defined s continuous mps : S 1 Ω. The terms pth, contour nd rc re lso used for curve, sometimes with slight differences in mening. We will not mke ny distinction between ny of these terms. Non-self-intersecting curves re clled simple, nd simple closed curves re clled Jordn curves. If φ : [c, d] [, b] is monotone incresing surjective function, then we sy tht the curve φ : [c, d] Ω is equivlent to the originl : [, b] Ω, nd cll the equivlence clsses of curves under this equivlence reltion oriented curves. Intuitively, given the imge = ([, b]) of the curve, n oriented curve cn be recovered upon identifying the initil nd terminl points, nd specifying how to trverse t self-intersection points. By buse of lnguge we cll the prticulr representtion : [, b] Ω of the underlying oriented curve lso n oriented curve. Note tht one cn tke the intervl [, b] to be, sy, [, 1] t one s convenience. Now, the inverse or the opposite of is defined by reversing the orienttion: 1 (t) = (b + t) for t [, b]. If : [, 1] Ω nd σ : [1, 2] Ω re two curves with (1) = σ(1), then their product or conctention σ : [, 2] Ω is defined s σ(t) = (t) for t [, 1] nd σ(t) = σ(t) for t [1, 2]. When the order of the opertions re not importnt, the bove opertions on curves cn suggestively be written in the dditive nottion s 1 nd + σ σ. The curve : [, b] Ω is clled differentible if C 1 ([, b]), with () = (b) for loops, where the derivtives t nd b re to be understood in the one-sided sense. The curve is clled piecewise differentible in Ω nd written Cpw([, 1 b], Ω) if is the conctention of finitely mny differentible curves. We ssume tht differentible nd piecewise differentible Dte: Mrch 3,

2 2 TSOGTGEREL GANTUMUR curves re oriented, which mounts to sying, e.g., for the cse of differentible curves tht we llow continuously differentible monotone incresing reprmeteriztions of curves. Ω Figure 1. Exmples of oriented curves. Our gol in this section is to define n integrl of function f : Ω C over curve : [, b] Ω. To motivte the definition, we recll here version of the fundmentl theorem of clculus for rel vlued functions. Theorem 1 (Fundmentl theorem of clculus). () If g C 1 ([, b], R) then b (b) If f C ([, b], R) then the function F (x) = g (t) dt = g(b) g(). (1) x stisfies F C 1 ([, b], R) nd F = f on [, b]. Let us record here n immedite corollry tht will be useful. f(t) dt, x b, (2) Corollry 2. Let φ C 1 ([, b], R), nd f C ([c, d], R) with φ([, b]) [c, d]. Then we hve b f(φ(t))φ (t) dt = φ(b) φ() f(x). (3) Proof. By hypothesis, the functions f : [c, d] R nd (f φ)φ : [, b] R re continuous, nd therefore Riemnn intergble. Let F : [, b] R be defined by F (x) = x f(t) dt, x b, (4) which, by the fundmentl theorem of clculus, stisfies F C 1 ([, b]) nd F = f in [, b]. Then we hve φ(b) φ() f(x) = φ(b) φ() On the other hnd, tking onto ccount the fct tht F (x) = F (φ(b)) F (φ()). (5) (F φ) (t) = F (φ(t))φ (t) = f(φ(t))φ (t), t b, (6)

3 we infer b estblishing the proof. f(φ(t))φ (t) dt = FUNDAMENTAL THEOREMS 3 b (F φ) (t) dt = (F φ)(b) (F φ)() (7) Getting bck to the min gol of this section, we wnt to require complex integrtion to hve the property F (z) dz = F ((b)) F (()), (8) where the left hnd side is the yet-to-be-defined integrl of F : Ω C over the curve : [, b] Ω. This property mimics the first prt of the fundmentl theorem of clculus, nd bsiclly sks complex integrtion to be n opertion tht inverts complex differentition. Note tht this is very strong condition: At the very lest (8) sys tht the integrl of F over depends only on the endpoints () nd (b) of the curve, nd it does not mtter how the curve behves between its endpoints. A clue to how to ensure (8) comes from the fundmentl theorem of clculus itself. If we define g(t) = F ((t)) for t b, then we hve b g (t) dt = g(b) g() = F ((b)) F (()), (9) where g is considered s pir of rel vlued functions defend on the intervl [, b], nd integrtion nd differentition of g re understood componentwise. Now the ide is bsiclly to cll the left hnd side of (9) the integrl of F over. To write g in terms of F nd possibly or, let us ssume tht F is complex differentible, nd tht is differentible. Then by definition, we hve F ((t + h)) F ((t)) = F ((t + h))((t + h) (t)) = F ((t + h)) (t)h, (1) where F is continuous t (t), nd is continuous t t, which yields nd hence, in light of (9), we infer b g (t) = F ((t)) (t), (11) F ((t)) (t) dt = F ((b)) F (()). (12) Our intention is to define the left hnd side to be the integrl of F over. Definition 3. The integrl of f : Ω C over curve C 1 ([, b], Ω) is defined by b f, f(z) dz = f((t)) (t)dt. (13) For piecewise differentible curves the integrl is defined vi linerity : f, n = f, f, n. (14) Remrk 4. () The integrl f, is well-defined, e.g., if f : Ω C is continuous. (b) Let f C (Ω) nd C 1 ([, b], Ω). Let φ C 1 ([c, d]) with φ([c, d]) [, b]. Then by Corollry 2 we hve d c f((φ(t))) (φ(t))φ (t) dt = Putting φ(c) = nd φ(d) = b, we infer φ(d) φ(c) f((t)) (φ(t)) dt. (15) f, φ = f,, (16)

4 4 TSOGTGEREL GANTUMUR which mens tht the integrl f, is invrint under reprmeteriztions of the oriented curve. On the other hnd, if φ(c) = b nd φ(d) =, we hve f, φ = f,, (17) nd so in prticulr f, = f,. (18) (c) If 1 nd 2 re piecewise differentible curves in Ω, nd if f C (Ω), then (d) If f C (Ω) nd C 1 ([, b], Ω) then f, = f, 1 + f, 2. (19) f, mx t b f((t)) b (t) dt. (2) Exmple 5. () Consider f(z) = z nd (t) = re it for t 2π, where r >. By the chin rule (11), which we rewrite here s d dg(z) g(α(t)) = dα(t), dt dz z=α(t) dt (21) we hve (t) = d dt (reit ) = d dz (rez ) d(it) = rie it. z=it dt (22) Then noting tht f((t)) = f(re it ) = re it, we infer 2π 2π z dz = re it rie it dt = r 2 i dt = 2πr 2 i. (23) (b) Let f(z) = z, nd let be s in (). Then we hve 2π z dz = re it rie it dt = r 2 i On the other hnd, the chin rule (21) yields 2π e 2it dt. (24) d dt e2it = 2ie 2it, (25) nd therefore (24) cn be continued s z dz = r 2 i 2π 1 2i ( d dt e2it) dt = r2 e 2it 2 (c) Now consider f(z) = 1 z, with s bove. dz 2π z = rie it dt = i reit 2π 2π = r2 (e 4πi 1) 2 =. (26) dt = 2πi. (27) Exercise 6. For ech n Z, compute the integrl of z n over the circle given by (t) = re it, t 2π, where r >. Bsiclly by construction, we get the following result. Theorem 7 (FTC for holomorphic functions). Let F O(Ω) be holomorphic function, nd suppose tht F is continuous in Ω. Then for ny Cpw([, 1 b], Ω) we hve F (z) dz = F ((b)) F (()). (28) Proof. The function g = F stisfies g C 1 ([, b], R 2 ) with g = (F ), nd n ppliction of the fundmentl theorem of clculus (Theorem 1) finishes the job.

5 FUNDAMENTAL THEOREMS 5 Remrk 8. The continuity hypothesis on F is in fct superfluous, since it will turn out tht holomorphic functions re infinitely often differentible. However, the bove form (with the continuity hypothesis) will be used to prove tht fct. Corollry 9. In the setting of the preceding theorem, if is closed curve, then we hve F, =. (29) In prticulr, for ny polynomil p nd ny C 1 pw(s 1, C), we hve p, =. (3) Proof. For the first ssertion, Theorem 7 nd the condition () = (b) give F (z) dz = F ((b)) F (()) =. (31) For the second ssertion, since p is polynomil, it is integrble in C, mening tht there is F O(C) such tht F = p in C. Then we pply the first ssertion to finish the proof. 2. Gourst s theorem For set U C tht is not open, the nottion f O(U) mens tht f is holomorphic in n open neighbourhood of U. Let us denote by [, b] the (oriented) line segment with the initil point C nd the terminl point b C. Given three points, b, c C, the tringulr loop [, b, c] is defined to be the oriented loop [, b] + [b, c] + [c, ]. The set of points tht re strictly inside the loop [, b, c] forms n open set τ C, which we cll n open tringle. In this setting, the loop [, b, c] is clled the boundry of τ, nd written τ = [, b, c]. Note tht there is n mbiguity in the orienttion of τ, since the loop [, c, b], whose orienttion is the opposite of tht of [, b, c], gives rise to the sme open tringle τ. The defult convention is to orient τ in such wy tht τ lies on the left of τ, but it is lwys good ide to explicitly mention the chosen orienttion to void confusion. Finlly, the closure of τ, denoted by τ, is the union of τ nd τ, the ltter tken s set. The following theorem ws proved by Édourd Gourst ( ) in This is n improvement over Cuchy s theorem, in which Cuchy ssumed tht not only f is holomorphic, but lso the derivtive f is continuous. While the originl formultion by Gourst employs rectngles, the following tringulr version is due to Alfred Pringsheim ( ). Theorem 1. Let τ C be n open tringle, nd let f O(τ). Then f, τ =. Proof. Let us subdivide τ into 4 congruent tringles τ 1, τ 2, τ 3, τ 4 by connecting the midpoints of the edges of τ. All lengths of the smller tringles re mesured s hlf the corresponding length of the originl tringle τ. Moreover we hve f, τ = f, τ j. (32) 1 j 4 Let τ m be tringle mong the 4 tringles tht gives the lrgest contribution to the sum, nd cll it τ (1), tht is, τ m (with some m between 1 nd 4) stisfies f, τ m f, τ j for ny 1 j 4. Then we hve f, τ 4 f, τ (1). (33) Now subdividing τ (1) into 4 still smller tringles, nd repeting this procedure, we get f, τ 4 n f, τ (n), (34)

6 6 TSOGTGEREL GANTUMUR with ny length of τ (n) being 2 n prt of the corresponding length of τ. In prticulr, if c n is point 1 in τ (n), then the sequence {c n } is Cuchy, so c n c for some c τ. Since f is holomorphic in neighbourhood of τ, by definition we hve f(z) = f(c) + λ(z c) + o(2 n ), z τ (n), (35) with some constnt λ C. We clculte the integrl of f over the boundry of τ (n) to be f, τ (n) = f(c) + λ(z c), τ (n) + o(2 n 2 n ) = o(4 n ), (36) where the integrl vnish by Corollry 9, nd we hve tken into ccount tht the perimeter of τ (n) is of the order O(2 n ). Substituting this into (34) estblishes the proof. It is possible to slightly relx the hypothesis of Gourst s theorem, so tht only holomorphy in the interior nd continuity up to the boundry re ssumed. The rgument is continuity rgument tht cn be used to strengthen mny of the theorems tht follow. Corollry 11. Let τ C be n open tringle, nd let f O(τ) C(τ). Then f, τ =. Proof. Let, b, c be the vertices of τ, nd let { n }, {b n }, {c n } be sequences of points in τ such tht n, b n b, nd c n c s n. As the closure of the tringle τ n defined by [ n, b n, c n ] is entirely in τ, Gourst s theorem pplies to τ n, mening tht f, τ n =. By uniform continuity, f, τ n tends to f, τ, hence f, τ =. 3. Locl integrbility In wht follows, by defult Ω will lwys denote n open subset of C. Definition 12. A continuous function f C(Ω) is clled integrble in Ω if there is F O(Ω) such tht F = f in Ω. It is clled loclly integrble in Ω if for ny z Ω there exists n open neighbourhood U of z such tht f is integrble in U. In combintion with Gourst s theorem, the theorem below implies tht holomorphic functions re loclly integrble. By closed tringle we men set of the form τ, where τ C is n open tringle. Theorem 13. Let D = D r (c) be n open disk, nd let f C (D) stisfy f, τ = for ny closed tringle τ Ω. Then f is integrble in D. Proof. Define F (z) = f, [c, z] for z D. We would like to show tht F = f on D, or equivlently tht F (w) = F (z) + f(z)(w z) + o( w z ). (37) From the definition of F we hve F (w) F (z) = f, [z, w], nd tking into ccount tht w z = 1, [z, w], we infer F (w) F (z) f(z)(w z) = f, [z, w] f(z) 1, [z, w]. (38) Now f = f(z) + o(1) on [z, w], so the right hnd side is of order o( w z ). In the subsequent sections, by sequence of severl theorems, we will prove tht loclly integrble functions re nlytic, therefore lso holomorphic. Hence locl integrbility is equivlent to holomorphy. As simple ppliction of the theorem, we get Cuchy s theorem for disks. Corollry 14. Let f O(D), where D = D r (c) is n open disk. Then f, = for ny piecewise differentible loop C 1 pw(s 1, D) lying in D. 1 For concreteness, e.g., we my tke cn to be the brycenter of τ (n).

7 FUNDAMENTAL THEOREMS 7 Proof. By the preceding theorem (in combintion with Gourst s theorem) there is F O(Ω) such tht F = f on D. Then the fundmentl theorem of clculus for holomorphic functions (Theorem 7) sttes tht the integrl of f over ny piecewise differentible closed curve must be zero. We cn slightly extend the rgument in the proof of Theorem 13 to get criterion on (globl) integrbility. Theorem 15. A continuous function f C (Ω) is integrble in Ω if nd only if f, = for ny C 1 pw(s 1, Ω). Proof. One direction is immedite from the fundmentl theorem of clculus. For the other direction, ssume tht Ω is connected (otherwise we work in connected components of Ω one by one). Let c Ω, nd for z Ω define F (z) = f, with piecewise differentible curve connecting 2 c nd z. The vlue F (z) does not depend on the prticulr curve, since if σ is nother curve connecting c nd z, then σ is piecewise differentible loop in Ω, so tht f, = f, σ by hypothesis. Now noting tht F (w) F (z) = f, [z, w], the proof proceeds in exctly the sme wy s in the proof of Theorem Cuchy s theorem for homotopic loops Definition 16. Loops, 1 C (S 1, Ω) re clled (freely) homotopic to ech other, nd written 1, if there exists continuous mp Γ : S 1 [, 1] Ω such tht Γ(t, ) = (t) nd Γ(t, 1) = 1 (t) for t S 1. Free homotopy is n equivlence reltion in the spce of loops, nd so this spce is prtitioned into (free) homotopy clsses. The following theorem shows tht t lest in the piecewise differentible cse, the integrl of given holomorphic function over loop depends only on the homotopy clss the loop represents. Theorem 17. For f O(Ω) nd for piecewise differentible loops, 1 C 1 pw(s 1, Ω) with 1, we hve f, = f, 1. (39) Proof. Let us prmetrize the circle S 1 by the intervl [, 1], so the curves will be mps defined on [, 1]. Let Γ : [, 1] 2 Ω be homotopy between nd 1. Since [, 1] 2 is compct, Γ is uniformly continuous, nd the imge Γ = {Γ(t, s) : (t, s) [, 1] 2 } is compct subset of Ω. Fix ε > such tht ε < dist( Γ, C \ Ω). Obviously, f is integrble in ny disk D ε (z) with z Γ. For lrge integer n, let z j,k = Γ( j n, k n ) for j =,..., n, nd k =,..., n. Let Q j,k be the closed qudrilterl with the vertices z j,k, z j+1,k, z j+1,k+1, nd z j,k+1. Then for k =, we modify Q j,k so tht the stright edge [z j,k, z j+1,k ] is replced by the piece of tht lies between z j,k nd z j+1,k. Similrly, for k = n 1, we modify Q j,k so tht the stright edge [z j+1,k+1, z j,k+1 ] is replced by the piece of 1 tht lies between z j+1,k+1 nd z j,k+1. Thus in generl Q j,k with k = or k = n 1 is going to be qudrilterl with curved edge. We choose n to be so lrge tht Q j,k D ε (z j,k ) for ll j nd k. Then note tht f, f, 1 = j,k f, Q j,k, (4) where the contribution from ny edge of Q j,k tht does not coincide with n edge of either or 1 is cnceled due to the opposite orienttions tht common edge inherits from neighbouring polygons. Moreover, ech integrl f, Q j,k is zero becuse f is integrble on D ε (z j,k ) Ω nd Q j,k is polygonl loop in D ε (z j,k ). The theorem is proven. 2 Any two points in connected open plnr set cn be connected by piecewise liner curve.

8 8 TSOGTGEREL GANTUMUR If loop is homotopic to constnt pth, i.e., δ with δ : [, b] Ω such tht δ z for some z Ω, then is sid to be topologiclly trivil or null-homotopic, nd this fct is written s. Corollry 18. If C 1 pw(s 1, Ω) is topologiclly trivil, then f, = for ny f O(Ω). A somewht trivil wy to ensure tht prticulr closed curve in Ω is topologiclly trivil is to simply require tht every closed curve in Ω is topologiclly trivil. Definition 19. A set Ω C is clled simply connected if it is connected nd every closed curve in Ω is topologiclly trivil. Exercise 2. A str-shped sets re chrcterized by the property tht there is c Ω such tht z Ω implies [z, c] Ω. For exmple, convex sets re str-shped. Show tht str-shped sets re simply connected. Corollry 21. If Ω is simply connected then f, = for f O(Ω) nd C 1 pw(s 1, Ω). Definition 22. Curves, 1 C ([, b], Ω) re clled homotopic reltive to their endpoints, nd written {,b} 1, if there exists continuous mp Γ : [, b] [, 1] Ω such tht Γ(t, ) = (t) nd Γ(t, 1) = 1 (t) for ll t [, b], nd Γ(t, s) = (t) for ll s [, 1] nd t {, b}. Note tht {,b} 1 implies in prticulr tht () = 1 () nd (b) = 1 (b). Similrly to the free homotopy cse, the spce of curves tht re fixed t their endpoints is prtitioned into (reltive) homotopy clsses. Corollry 23. For f O(Ω) nd for piecewise differentible curves, 1 C 1 pw([, b], Ω) with {,b} 1, we hve f, = f, 1. Proof. One cn show tht the curve 1 is topologiclly trivil piecewise differentible loop, by constructing homotopy tht, e.g., first follows the homotopy between nd 1 reltive to the endpoints to collpse onto 1, nd then contrcts 1 to point. 5. Evlution of rel definite integrls Recll tht Euler s min motivtion for studying complex functions ws to find new wy to integrte rel functions. With the help of Cuchy s theorem, we cn now mke Euler s procedure precise. In this section, we wnt to look t generl method to tret improper Riemnn integrls. Definition 24. Given f : (, b) R with < b, the improper Riemnn integrl of f over (, b) is defined s b β f(x) = lim lim f(x). (41) α β b α Moreover, for f : (, b) (b, c) R, we define Exmple 25. We hve c 1 f(x) = b f(x) + c b f(x). (42) b x 2 = lim b 1 x 2 = lim ( 1 ) = 1. (43) b x 1

9 FUNDAMENTAL THEOREMS 9 The trivil link between complex nd rel integrtions is the observtion tht complex contour integrl reduces to usul Riemnn integrl if the contour hppens to be rel intervl. Indeed, if (t) = t, with t b, then b f(z) dz = f(t) dt, (44) since (t) = 1. Now we illustrte the method with n exmple. Exmple 26. Let us compute the improper integrl First, we write R 1 + x 2 = lim R R 1 + x 2. (45) 1 + x 2 = 1 R 2 R 1 + x 2. (46) Next, we consider the curve + R (t) = Reit, t π, which is the upper hlf of the circle of rdius R centred t the origin, nd let R be the conctention of the rel intervl [ R, R] nd the semicircle + R. Then we hve R R 1 + x 2 = dz R 1 + z 2 dz z 2. (47) R Now we will evlute the first integrl in the right hnd side. Since the function f(z) = 1 1+z 2 is holomorphic in Ω = C \ {i, i}, nd the loop R is freely homotopic in Ω to the loop ε (t) = i + εe it, t 2π, we infer R dz 1 + z 2 = ε dz 1 + z 2 = 2π iεe it dt 2π εe it (2i + εe it ) = idt. (48) 2i + εeit It is intuitively cler tht the integrnd is pproximtely 1 2 when ε > is smll. To obtin precise bound, note tht 1 2i + εe it 1 = εe it εe it 2i 2i(2i + εe it ) 4 2εe it ε 2, (49) s long s < ε 1, which shows tht 2π idt 2i + εe it R 2π dt 2 since this is true for ny smll ε >, we conclude tht dz 2π 1 + z 2 = idt 2i + εe it = 2π 2π εdt 2 dt 2 = πε. (5) = π. (51) Finlly, for the integrl over the semicircle + R in (47), we hve dz z 2 R 2 πr s R, (52) 1 + R nd therefore the conclusion is 1 + x 2 = 1 2 lim R R R 1 + x 2 = π lim R + R dz 1 + z 2 = π 2. (53)

10 1 TSOGTGEREL GANTUMUR Remrk 27. The illustrted method works for ny integrl of the form p(x) q(x), (54) where p nd q re polynomils stisfying deg(q) deg(p) + 2 nd q(x) for x R. Exercise 28 (Jordn s lemm). With + R s in the preceding exmple, show tht f(z)e iαz dz s R, (55) + R if α >, nd f(z) M(1 + z ) s for z C, with some constnts M nd s >. 6. The Cuchy integrl formul In this section, we will prove the Cuchy integrl formul, which my be considered s the cornerstone of complex nlysis. Before stting the result, let us recll the nottions D r (c) = {z C : z c < r} nd D r (c) = {z C : z c r}. Moreover, given disk D = D r (c), we denote by D the oriented curve given by (t) = c + re it for t 2π. Theorem 29 (Cuchy 1831). Let f O(Ω), nd let D r (c) Ω with r >. Then we hve f(ζ) = 1 f(z)dz for ζ D r (c). (56) 2πi z ζ Proof. The function D r(c) F (z) = f(z), z Ω \ {ζ}, (57) z ζ is holomorphic in Ω \ {ζ}, nd with ε > smll, the loop ε defined by ε (t) = ζ + εe it for t 2π, is homotopic in Ω \ {ζ} to the circle D r (c). Hence we hve f(z)dz D r(c) z ζ = f(z)dz ε z ζ. (58) By complex differentibility, there is function g : Ω C, continuous t ζ, such tht Substituting this into (58), we get f(z)dz z ζ D r(c) ε f(z) = f(ζ) + g(z)(z ζ). (59) = f(ζ) ε dz z ζ + g(z)dz. (6) ε The first integrl in the right hnd side leds to the fmilir computtion dz 2π z ζ = iεe it dt εe it = 2πi. (61) As for the second integrl, let δ > be such tht g(z) g(ζ) < 1 whenever z ζ < δ. Then for < ε < δ, we hve g(z)dz ( g(ζ) + 1) 2πε. ε (62) We conclude tht f(z)dz z ζ 2πif(ζ) ( g(ζ) + 1) 2πε, (63) D r(c) for ny < ε < δ, mening tht the left hnd side is equl to.

11 FUNDAMENTAL THEOREMS 11 Corollry 3 (Men vlue property). In the setting of the theorem, with (t) = c + re it for t 2π, we hve f(c) = 1 f(z)dz 2πi z c = 1 2π f(c + re it )dt, (64) 2π which shows tht the vlue of f t c is the verge of f over the circle D r (c). Corollry 31. Let f O(Ω), ζ Ω, nd let Cpw(S 1 1, Ω \ {ζ}) be loop homotopic to D ε (ζ) in Ω \ {ζ} for some ε >. Then we hve f(ζ) = 1 f(z)dz 2πi z ζ. 7. The Cuchy-Tylor theorem Definition 32. A function f : Ω C is clled (complex) nlytic in Ω, if for ny c Ω, one cn develop f into power series centred t c, with nonzero convergence rdius. The set of ll nlytic functions in Ω is denoted by C ω (Ω). By n n-fold differentition of the power series f(z) = n (z c) n, (65) n= we infer tht the coefficients re given by n = f (n) (c)/n!. In other words, if f C ω (Ω) nd c Ω, then the following Tylor series converges in neighbourhood of c: f (n) (c) f(z) = (z c) n. (66) n! n= Termwise differentition of power series lso implies tht nlytic functions re holomorphic, i.e., C ω (Ω) O(Ω). In fct, the converse O(Ω) C ω (Ω) is lso true. Theorem 33 (Cuchy 1841). Let f C ( D r (c)) with r >, nd ssume tht f(ζ) = 1 f(z) dz for ζ D r (c). (67) 2πi z ζ Then the power series D r(c) f(ζ) = n (ζ c) n, (68) with n = 1 f(z) dz, (69) 2πi D r(c) (z c) n+1 converges in D r (c). In prticulr, we hve O(Ω) C ω (Ω) for open sets Ω C. Proof. Without loss of generlity, let us ssume c =, nd strt with the integrl formul f(ζ) = 1 f(z) dz 2πi D r z ζ, for ζ D r. (7) This cn be rewritten s where we hve used f(ζ) = 1 ( 2πi D r n= 1 z ζ = 1 z 1 1 ζ/z = 1 (1 + ζz ) z +... = ) f(z)ζ n z n+1 dz, (71) n= ζ n. (72) zn+1

12 12 TSOGTGEREL GANTUMUR Ech term in the series under integrl in (71) cn be estimted s f(z)ζ n z n+1 f ( ) D r ζ n, where f Dr = sup f(z), (73) r r z D r so s function of z, the series converges uniformly on D r. Therefore we cn interchnge the integrl with the sum, resulting in f(ζ) = 1 ζ n f(z)dz 2πi D r z n+1. n= Now the individul term of the series stisfies ( ) f(z)dz ζn ζ n D r z n+1 2π f D r, r implying tht the series converges loclly normlly in D r. 8. Morer s theorem The following result ws proved by Gicinto Morer ( ) in Theorem 34. A function f C (Ω) is holomorphic if ny of the following conditions holds. () f is loclly integrble. (b) f, τ = for ny closed tringle τ Ω. Proof. Condition () follows from (b) by Theorem 13. Now suppose tht () holds. Then by definition, ech point in Ω hs neighbourhood U nd F O(U) such tht F = f on U. The Cuchy-Tylor theorem gurntees tht F C ω (U), nd by termwise differentiting we infer f C ω (U). This mens tht f C ω (Ω), or in other words f O(Ω). As n ppliction, one cn prove tht the loclly uniform limit of holomorphic functions is holomorphic. This is to be contrsted with the sitution in the rel differentible cse where the uniform limit of smooth functions is not smooth in generl. The following theorem is often clled the Weierstrss convergence theorem. Theorem 35 (Weierstrss 1841). Let {f k } O(Ω) be sequence such tht f k f loclly uniformly in Ω for some function f : Ω C. Then f O(Ω) nd f (n) k f (n) loclly uniformly in Ω, for ech n N. Proof. First of ll we hve f C (Ω), without using complex nlysis. Now let τ Ω be closed tringle. Then since τ is compct, f n converges uniformly to f on τ, nd so we hve f, τ = lim k f k, τ =, implying tht f O(Ω) by Morer s theorem. For the second prt of the clim we employ the Cuchy estimtes. Let D 2δ () Ω for some δ >. Then since f k f O(Ω), for k N the Cuchy estimte gives completing the proof. f (n) k f (n) Dδ () n! δ n f k f D2δ (), We close this section with theorem tht offers mny different chrcteriztions of holomorphic functions. This is n indiction tht the initil phse in the development of the theory is now complete. If we were building rocket, t this point we hve ssembled it nd re redy to strt testing. Theorem 36. Let Ω C be open, nd let f C (Ω). Then the following re equivlent.

13 FUNDAMENTAL THEOREMS 13 () f is holomorphic in Ω, i.e., f O(Ω). (b) For ll closed tringles τ Ω, the integrl of f over the boundry of τ is zero. (c) f is loclly integrble in Ω. (d) For ll open disks D with D Ω nd for D, one hs f() = 1 f(z)dz 2πi D z. (e) f is nlytic in Ω, i.e., f C ω (Ω). Proof. The impliction (e) () follows from termwise differentition of power series. The impliction () (b) is Gourst s theorem in 2, nd (b) (c) is the integrbility theorem (Theorem 13) in 3. Then () (d) is the rgument leding to the Cuchy integrl formul in 4 nd 6, where we hve lso used the impliction () (c), nd (d) (e) is the Cuchy- Tylor theorem in 7. Using ll of these, (c) () is proven s Morer s theorem in The Cuchy estimtes We hve the following fundmentl estimtes on power series coefficients, which roughly sys tht the lrgest term in the series determines the mximum bsolute vlue the power series cn hve in given region. Theorem 37 (Cuchy 1835). Let f(z) = n (z c) n be convergent in n open neighbourhood of D ρ (c). Then we hve or equivlently, n ρ n mx f(z) for n =, 1,..., (74) z c =ρ f (n) (c) n! ρ n mx f(z) for n =, 1,.... (75) z c =ρ Proof. It is immedite from the Cuchy-Tylor theorem (Theorem 33) tht n = 1 f(z) dz 1 2π (z c) n+1 2π 2πρ 1, (76) ρn+1 which is the desired estimte. D ρ(c) Functions nlytic in the entire complex plne C re clled entire functions. In 1844, Cuchy proved tht bounded entire functions re constnt, but this theorem is now known s Liouville s theorem. The reson for this is ttributed to Crl Borchrdt ( ), who lerned the theorem from Joseph Liouville ( ) in 1847, nd then published it under the nme Liouville s theorem in Corollry 38 (Liouville s theorem). If f O(C), nd if there exists constnt M > such tht f(z) M for z C, then f must be constnt function. Proof. By the Cuchy-Tylor theorem, the Tylor series of f centred t the origin converges uniformly on ny closed disk. Applying the Cuchy estimtes to this series on D ρ with ρ >, we get the following bound on the n-th coefficient n ρ n mx z =ρ f(z) ρ n M. Since ρ cn be rbitrrily lrge, this estimte shows tht n = for ll n = 1, 2,.... Liouville s theorem cn be used to prove the fundmentl theorem of lgebr. Corollry 39. Any nonconstnt polynomil hs t lest one root in C. Proof. Suppose tht polynomil p hs no root. Then f = 1 p O(C). If p(z) = + 1 z n z n with n, then p(z) n z n for lrge z, mening tht f is bounded. By Liouville s theorem f must be constnt, contrdicting the hypothesis.

14 14 TSOGTGEREL GANTUMUR Exercise 4. If f O(C), nd if there exist M > nd s such tht f(z) M(1 + z s ) for z C, then f must be polynomil of degree t most s. 1. The identity theorem Recll tht n ccumultion point of set D C is point z C such tht ny neighbourhood of z contins point w z from D. We sy tht z D is n isolted point if it is not n ccumultion point of D. If ll points of D re isolted D is clled discrete. Theorem 41 (Identity theorem). Let f C ω (Ω) with Ω connected open set, nd let one or both of the following conditions hold. () The zero set of f hs n ccumultion point in Ω. (b) There is c Ω such tht f (n) (c) = for ll n. Then f in Ω. Proof. By definition, connectedness of Ω mens tht if Ω = A B for some open nd disjoint sets A nd B, then it is necessrily either A = Ω or B = Ω. Our strtegy is to show tht both A = {z Ω : f (n) (z) = n} nd its complement B = Ω \ A re open, nd tht A is nonempty. This would estblish tht A = Ω, nd hence f in Ω. It is esy to see tht A is open, becuse c A implies tht f in smll disk centred t c by Tylor series rgument. To see tht B is open, we write it s B = n B n with B n = {z Ω : f (n) (z) }. Since B n is the preimge of the open set C \ {} under the continuous mpping f (n) : Ω C, we infer tht B n is open, nd thus B is open. Prt (b) of the theorem is estblished, since c A (nd so A ) by hypothesis. For prt (), it remins to prove tht A is nonempty. Let c Ω be n ccumultion point of {z Ω : f(z) = }, nd suppose tht c A. Let n be the smllest integer such tht f (n) (c). Then we hve f(z) = (z c) n g(z) for some continuous function g with g(c). This implies the existence of smll open disk D ε (c) in which f(z) = hs only one solution z = c, contrdicting tht c is n ccumultion point of the zero set of f. The following corollry records the fct tht n nlytic function is completely determined by its restriction to ny non-discrete subset of its domin of definition. In other words, if it is t ll possible to extend n nlytic function (defined on non-discrete set) to bigger domin, then there is only one wy to do the extension. Corollry 42 (Uniqueness of nlytic continution). Let u, v C ω (Ω) with Ω connected open set, nd let u v in non-discrete set D Ω. Then u v in Ω. 11. The open mpping theorem The Cuchy estimtes (Theorem 37) cn lso be used to prove the open mpping theorem. Theorem 43. Let Ω be connected open set, nd suppose tht f O(Ω) is not constnt function. Then f : Ω C is n open mpping, i.e., it sends open sets to open sets. Proof. Without loss of generlity let us ssume tht Ω nd tht f() =. We will prove tht smll disk centred t the origin will be mpped by f to neighbourhood of the origin. 1 Let D r Ω with r >, nd let w f(d r ). Then the function φ(z) = f(z) w is nlytic in D r. Choose < ρ < r so smll tht f(z) = hs no solution with z = ρ, so tht δ = inf f(z) >. This is possible by the identity theorem since f is not constnt nd Ω is z =ρ connected. Since ρ < r, the Tylor series of φ bout converges uniformly in the closed disk D ρ. Now we pply the Cuchy estimte to φ nd get ( ) 1 φ() sup φ(z) = inf f(z) w, z =ρ z =ρ

15 which, tking into ccount tht φ() = w 1, is equivlent to FUNDAMENTAL THEOREMS 15 inf f(z) w w. z =ρ We hve f(z) w f(z) w δ w for z = ρ, therefore the bove estimte gives w δ/2. It follows tht D δ/2 f(d r ). Thus for exmple, one cnnot get (nonzero-length) curve s the imge of n open set under holomorphic mpping. In prticulr, the only rel-vlued holomorphic functions defined on n open set in C re loclly constnt functions. The open mpping theorem cn be used to obtin proof of mximum principles. Corollry 44 (Mximum principle). Let f O(Ω) with Ω n open subset of C. () If Ω is connected nd f(z) = sup f t some z Ω, then f is constnt. Ω (b) If Ω is bounded nd f C ( Ω), then we hve sup Ω f mx f. Ω Proof. The hypothesis in () sys tht f(z) is boundry point of the imge f(ω), since otherwise there would hve to be point in f(ω) with bsolute vlue strictly greter thn f(z). If f is not constnt, by the open mpping theorem f(ω) cnnot include ny of its boundry points, leding to contrdiction. For prt (b), there is z Ω with f(z) = sup Ω f since Ω is bounded nd f is continuous on Ω. If z Ω then we re done; otherwise pplying prt () to the connected component of Ω tht contins z concludes the proof. We end this section with two simple corollries of the open mpping theorem. Corollry 45 (Preservtion of domins). If Ω C is connected open set nd f O(Ω) nonconstnt, then f(ω) is lso connected open set. Exercise 46. Let f O(C), nd suppose tht f(z) s z. Then f(c) = C.