D, domain or region. Chapter 2. Analytic Functions. Sec. 9. (1) w. function of a complex variable. Note: Domain of f may not be a domain.
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1 Chapter. Analytic Functions Sec. 9. D, domain or region f : D w (1) w f function of a comple variable. Note: Domain of f may not be a domain. (set theory ) (topology ) Let w = u + iv and = + iy u + iv = f iy u, y Re f, y, i.e., v, y Im f, y i.e., uv, Re fy,,im f y, E. f iy y iy u y v y () Real-valued func. of a comple variable, i.e., v E. f : D f y : u y, v (3) Polar coordinates: CH_1
2 i re i uiv f re i, Re f re i, Im f re u r v r E. f u r v r cos sin CH_
3 Sec. 1. Mappings ( cf. Chapter 8 ) D domain or region f : D Question: Geometric behavior of f? E.1. (a) Use Cartesian coordinates: w Let = + iy u y i.e. v y Fi u y & v y Eliminate y from above v 4 y 4 ( u) 4 ( u ) parabola with verte, 1, 1, (1) Assume > : y > v y CH_3
4 upper half line upper branch () : u y, v = range of f (3) Assume < : y > v < upper half line lower branch Appendi. Fig. 3, p. 371 y v C C, u (b) Use polar coordinates: Let i re w i r e w = & E.. 1 f w w Note: symm. under 1 : 1 f f, domain of f \ CH_4
5 (1) () i e, i i we e cos +isin + cos - +isin - =cos - u & v Note: same image i (3) re,, r > 1 i 1 i 1 1 wre e rcos irsin cos i sin r r r 1 1 sin = r cos i r r r u v CH_5
6 u v 1 1 r r r r 1 ellipse 1 1 arg arg arg 1 r > r 1 by symm: f f 1 r > r Appendi, Fig. 17, p. 374 Fig. 18 E i w r e ( r, < ) CH_6
7 Sec. 11. Limits D domain or region f : D, w, D Def. lim f w if >, > < < & D f w < or >, > f N D N w E. Show lim iy 4i i Pf: Let > Find > < i < iy 4 i < <7 < iy i y y y i < i y 4 y 4 < i 4 4 i y i y < need: < 1 Let < 7 Conclusion: < min 1, 7 CH_7
8 Properties: (1) lim & lim f w f w w w 1 1 CH_8
9 Sec. 1. lim f w lim f Re w & lim Im f Im w () Pf: : Re f Re w Re f w f w Im f Im w Im f w f w : Im f w Re f w i f w Re f Re w Im f Im w (3) lim & lim f w g w,, 1 ablim (4) lim af bg aw bw f g w w 1 1 f w (5) lim if w1 g w 1 lim f w & w (6) f( ) < - <, i.e., f(ns ( )) (7) lim f w lim f w Pf: f w f w (8) lim f w lim f w Pf: f w f w CH_9
10 Sec. 13. Limits involving ( in, is one no.; in, ) n bd of : w: w R N R Def. lim f if R >, > f R R f N N or Def. lim f w if >, R > R f w R or f N N w Def. lim f if R >, r > r f R or f Nr NR CH_1
11 Sec. 14. (1) etended comple plane. Note: different from : () Riemann sphere =,, : (3) stereographic projection: p: Riemann sphere & p: Riemann sphere\ 1,, 3 iy,,1 (4),,,,,,,,1 y on a straight line in 1 3 3, y,,,1 t 1,, 3,,1 t1, t,1t 3 1 t 1 y t 1t 1 3 1, 1, t y i stereographic projection:1,, or,,,,1 for CH_11
12 (5) Conversely: ( 1 1 ) ( 1 13y 1 ) i1 1 i Properties: (1) 3 < < 1 3 > > 1 3 = = 1 () circle not passing,,1 circle CH_1
13 circle passing,,1 straight line (3) nbd of,,1, disc centerted at,,1 : R S Def. accumulation pt. of S if R>, N S R Note: accumulation pt. of S S unbdd. E. S n: n,1,, S :Re > S :Im Then accumulation pt. Homework: Sec. 4: 14, 15, 16 Sec. 6: 6, 11, 1, 13, 14 Sec. 8: 4, 7, 1 Etra: (1) Prove that a b < 1 if a < 1 and b < 1. 1 ab CH_13
14 () Prove that a b 1 ab = 1 if a = 1 or b = 1 but not ab 1 (3) Prove that the points a1, a, a 3 are vertices of an equilateral triangle iff a a a a a a a a a (4) Show that and correspond to diametrically opposite poits on the Riemann sphere iff 1 Solu. (3) : a a wa a 3 1 a a w a a a a a aa a a a a = a a a a a a = a 4 a1 w a a1 w a a 1 a a1 w w : a a a a a a Let a1a a3 a1 a a3 CH_14
15 ( from a a a aa a a a a ) , w or w If 1, then a1 a a3 w or w (4) : In either case, Say,,, &,, i 1 1 1, 3 i : Check: d, 1 1 If 1, then d, diametrically opposite CH_15
16 E. i 3 lim 1 1 Pf: For R >, find > < 1 < i 3 1 > R i > R < <1 Let < < 1 R 1 lim f w lim f w Note: 1 lim f lim f 1 lim f lim 1 f CH_16
17 Sec. 14. Continuity Def. f conti. at if (1) f defined, () lim f eists, (3) lim f f. >, > f f < < >, > f N N f Def. f conti. on region R if f conti. at each pt. of R Properties: (1) f, g conti. at &, a b af bg conti. at () f, g conti. at & a, b f g conti. at (3) f, g conti. at &, a b & g (4) f poly. f conti. on f conti. at g Pf: By (1), need only check: lim n n n May assume n CH_17
18 < n n n1 n n n1 1 n 1 n < n n 1 < < < also, < Let < < min 1, n 1 1 n (5) f conti. at & g conti. at f g f conti. at (6) f conti. at & f > f on N 1 f > Pf: Let > < < f f 1 f f f < 1 f (7) f conti. at Re f,im f, f & f conti. at (8) f conti. on region R, R closed & bdd R f ma f : R CH_18
19 Pf: f : R is conti. & R compact (7) f assumes its ma. by advanced calculus (9) f conti. on region R, R closed & bdd f unif. conti. on R i.e., >, > 1 <, 1, R f f < 1 Homework: Sec.14: E. 1 (f), (g), 4, 9. Use definition directly to prove (a), (b), (c) E. 1. horiontal translation by a: f a, where a real E.. rotation through angle arg a w.r.t. : f a, where a 1 E. 3. reflection (through -ais): f CH_19
20 Sec. 15. Def. f differ. at if derivative f df f f lim d eits. i.e., w >, > < f < f w < Note: difference for real-variable * N E.1. f at f f = = (1) Assume (real) i.e., Then () Assume iy i.e., iy Then 1 CH_
21 not eist. f at =, f f as Note: cf. real-variable case: f : differ at any Note 1. f differ. at = (in previous eample), but not at any other pt. of nbd of f y. Re f y Im f Re f & Im f have conti. partial derivatives of all orders at f differ at. 3. f differ. at f conti. at. 4. A real-valued func. of a comple variable either has derivative = or the derivative not eist. Pf: D, f : D Assume f real-valued & f eists for some D CH_1
22 Then f f f f lim is real. ( real ) f iy f lim is purely imaginary. iy ( y real ) iy f E. f Re (cf. Sec.16, E. 8 (b), (c)) f Im E.. ( Sec.16, E. 9 ) if f if Determine whether f eists. Solu. f f (*) (1) Let along -ais i.e., (*) = 1 CH_
23 () Let along y-ais i.e., iy (*) = iy iy 1 (3) Let along = y i.e., i i 1i i 1 i 1i (*) = i f not eist. Note: f involving f not differ. Homework: Sec. 16, E. 8 CH_3
24 Sec. 16. Differentiation formulas d (1) d n n n1, ( n integer ) Assume f, g differ. at, then ()af bg af bg (3) f g fg fg f gf fg (4) g g if g (5) ( chain rule ) f differ. at, g differ. at g f g f f f g f differ. at & Homework: Sec.16, E. 3, 4 (L Hospital rule) CH_4
25 Sec. 17. Cauchy-Riemann equ. ( a bridge between comple analysis & calculus ) f u, u, v, v y y Thm. Assume w f,, y, w u iv w, y u, y iv, y & f eists. u u v v eist at, y Then (1),,, y y () f u, y iv, y = v, y iu, y y y (3) u vy, uy v at, y ( Cauchy-Riemann equ.) Pf: (1) & (): f lim f f (i) Consider u, v : Let, y,, y,,,, u y u y v y v y f lim i CH_5
26 Re & Im limits eist, i.e., u, v eist = u, y iv, y (ii) Consider u, v : y y Let y,, y, Then f,,,, u y u y iv y v y lim y y iy ( y) = y y,,,, v y v y u y u y lim i y y y y Re & Im limits eist, i.e.,, u v eist & f v, y iu, y y y y y (3): (i) & (ii) Cauchy-Riemann equ. E. 1. f iy y iy u y v y u vy uy y v E.. f y u y u, uy y v v, vy Cauchy-Riemann iff y,, CH_6
27 f eists iff = u f u iv u v uvy uyv v Note 1. y u v y ( Jacobian determinant of u, v w.r.t., y ) Note. f eists C-R equ. Countereample: Sec.19, E. 6 E. f if if Then f not eist, but u v & u v at y y, ( Sec.19, E. 6 ) CH_7
28 Sec. 18. Thm. f uy, ivy, u u v v eist in a nbd of, y & conti. at, y,,, y y Cauchy-Riemann equ. satisfied at y, f eists. Note 1. proof by MVT for 1-variable func. & Taylor s thm for variables. u u v v eist & conti. in a nbd of, y Note.,,, y y gradient of u, v, i.e., u, v eist at, y difference between comple analysis & calculus: C -R equ. Ref. T. Apostal, Mathematical Analysis, nd ed., p. 357 Pf: :,,,, u y u y u y u y y y (1) y 1,,,, v y v y v y v y y y () y 1 where and as CH_8
29 Reason: u yu yu y u y u u y y,,,, y 1 = u, yy y u, y u, y u, y y y y y = u, yu, y y y u, y u, y y y 1 u y u y u y u y,,,, y y as y y y y f f u iv u iv y y i (1) + () i y y 1 = u iv i y y 1i f f i u iv u iv 1 as f u iv eists. E. 1. f e e cos y isin y u e cos y u vy v e sin y uy v, y & conti. f eists & f f = u iv e cosy ie siny CH_9
30 E.. f u y, v u, u y, v, v & conti. y y Cauchy-Riemann equ. satisfied only at f eists only at. CH_3
31 Sec. 19. Cauchy-Riemann equ. in polar form Assume, rcos irsin f ur, ivr,, i.e.,r, u, v What s the relation between u, u, v, v r eists? if r f r, (1) Find relations between u, uy, v, v y& ur, u, vr, v. u, y, rcos, y rsin u u cos u sin r y u ursin uyrcos Similarly for v: v v cos v sin r y v vrsin vyrcos () Cauchy-Riemann equ. in polar form: u vy uy v 1 ur vycos vsin v r u v rsin v rcos rv y r (3) Derivative of f : ( E.19, 8 & 9 (a)) f re u iv v iu i Moreover, y y CH_31
32 1 e i = ur ivr e cos u ivsinuy ivy i r, 1 re = i r, i v iu = v iu = u iv = i r, u iv e i = cos i sin f ( E. 8 on p. 55 ) Homework: Sec.19, E. 1 (b), (d) E. 3 (b),(d) E. 4 (b),(d) E. 6, E. 9 (4) Thm f ur, ivr,, i re i at re, ur, u, vr, v eist in a nbd of & conti. at Cauchy-Riemann equa. satisfied at f eists (3) f on domain D f = constant on D Pf: f uy, ivy, CH_3
33 f u iv vy iuy = on D u v u v on D y y MVT for -variables 1, D, with 1 D 1 u u u 1 1 = (eistence need u, u, v, v conti.) y y u const. on each line segment in D, D, line segments connecting 1& u = const. on D. 1 Similarly for v. conclusion (6) domain D & f : D analytic f = constant on D ( Sec. 1, E. 7 (a)) on D Pf: ( Proved before ) f f = const. on D CH_33
34 Sec. 1. Reflection principle Thm. f analy on domain D D contain part of -ais & symm. w.r.t. -ais. Then f f D f real for D f f Pf: : Let D, real Then f f f f real : Define F f for D (1) Check: F analytic on D Let f u, y iv, y CH_34
35 ,, F U y iv y F f u, yiv, y U, yu, y& V, y v, y f analytic u v & u v y y U, y u, y,, 1, U V Vy y vy y vy y y, y, 1,, Uy y u y Uy V V y v y C-R equa. satisfied & 1 st partial conti. ( not proved yet in Chap.4 ) ( p.17, Corollary ) F analy. () F f for D infinitely many pts F = f on D ( Note: for conti. func. false: ) Sec. 58 f f D f f D CH_35
36 E. 1. f 1 on Then f real f f = 1 = 1 E.. f i Then f i not real for Also, f i f i CH_36
37 Sec.. Analytic functions Def. D domain, f : D, D f analytic at if f eists for in a nbd of ( holomorphic ) E. f not analytic at any pt. Def. f analytic in a region R if f analytic at every R ( f analytic on an open set containing R ) E. R: 1 f analytic in R f analytic in some domain D containing R Def. f entire func. if f analytic in Note: poly. are entire functions Def. singular pt. of f if f not analytic at but f analytic at some pt. in every nbd of E.1 f 1 Then singular pt. of f E. f has no singular pt. CH_37
38 Properties: (1) Sum, product, difference, quotient, composite of analytic functions are analytic () f analy. on domain D f conti. on D, f satisfies C-R equation on D CH_38
39 Sec.. Harmonic functions Def. h: R, R region h, yis harmonic in R if h C R & h h on R yy = h ( Laplace equation ) Prop. If f u, y iv, y analytic in R, then u & v harmonic. Pf: f analytic u, v C R. ( proved in Chap. 4, p.17, Corollary ) u v y & u y v u v y & u yy v y u C R vy vy u uyy u harmonic. Similarly, v harmonic Def. u, v harmonic in R & u vy, uy v Then v is harmonic conjugate of u. Note 1: v harmonic conjugate of u u iv analytic in domain D Note : v harmonic conjugate of u u harmonic conjugate of v i.e. u iv analytic v iu analytic E. f iy y iy u y, v y CH_39
40 Then u, u y y v y, v y v u & v u y y Note 3: v harmonic conjugate of u -u harmonic conjugate of v u harmonic conjugate of v. Pf: by definition. i.e., u iv analytic v iu, v iu analytic. iu iv iu iv Note 4: v harmonic conjugate of u & u harmonic conjugate of v on domain D u, v constant on D ( Sec., E.11 ) i.e. u iv & v iu analytic u, v constant Pf: uviv u analytic = u v1 i u v analytic & real-valued derivative = u v constant ( by E. 9 (c)) Similarly, u v constant CH_4
41 u, v constant Note 5: v1, v harmonic conjugate of u v1 v c; or f uiv, f u iv analytic 1 1 f 1 f const. Question: Given u harmonic in R, does v harmonic conjugate of u? does f analytic in R u Re f? Ans: In general, no. Yes, if R simply connected. ( Chap. 9 ) 3 E., 3 u y y y Then u harmonic on ( simply connected ) Find its harmonic conjugate & corresponding analytic func. Solu. u 6y v, y uy 3y 3 v v 6y, y v 3 3y v3y v 3y 3 3y 3 3 c 3 v 3y c ( need: integrations ) CH_41
42 f y 3 3 y i3y 3 c 3 = i c analytic u Re f Homework: Sec. : E.1 (a), (b), E. (a), (b), 4 (a), (b), 7 (b), (c), 9, 1 (b), (c), 11 (a) Etra: Find conditions on a, b, c & d 3 3 a b y cy dy is harmonic. Find the conjugate harmonic function and corresponding analytic function. Ans. u a b y ay by , 3 3 v 3a y3by ay b a uivaib 3 ia Etra: 1. If u is harmonic, so is u. If f is analytic, so is f Polar form of Laplace s equation: ( E.1),, f u r iv r analytic in domain D i re 1 ur v & u r rvr ( Cauchy-Riemann equa.) CH_4
43 1 1 urr v v r r r u rv r v, vrr, v r, vr are conti. v r vr ru r rurr v rv r u rv r rurr u rur E. Solve e 1 Solu. Let iy e cos yisin y1 1cos isin polar coordinate e 1& y n, n, 1,, 1 i n i n, n, 1,, Inverse function: Let w, w Let w i re, r >, < Assume e w rcos isin, where iy CH_43
44 e cos y isin y e r& y n, n, 1, ln r ln r i, R I \ : m, or ln w iargw Given w, solutions of e w: ln w iargw n i: n, 1,,... ln r, ln r, w Argw i.e., w e w ln w iarg w CH_44
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