MA 201-COMPLEX ANALYSIS

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1 MA 201-COMPLEX ANALYSIS Autum, Istructor: Devidas Pai Departmet of Mathematics Idia Istitute of Techology Gadhiagar Idia Name: Tutorial Batch:

2 MA 201: Complex Aalysis ( ) (Course cotets) Defiitio ad properties of aalytic fuctios. Cauchy-Riema equatios, harmoic fuctios. Power series ad their properties. Elemetary fuctios. Cauchy s theorem ad its applicatios. Taylor series ad Lauret expasios. Residues ad the Cauchy residue formula. Evaluatio of improper itegrals. Coformal mappigs. Iversio of Laplace trasforms. Texts/Refereces 1. E. Kreyszig, Advaced Egieerig Mathematics (8th Editio), Joh Wiley(1999). 2. R.V. Churchill ad J.W. Brow, Complex variables ad applicatios (7th Editio), McGraw-Hill(2003). 3. J.M. Howie, Complex aalysis, Spriger-Verlag (2004). 4. M.J. Ablowitz ad A.S. Fokas, Complex Variables-Itroductio ad Applicatios, Cambridge Uiversity Press, 1998 (Idia Editio). 5. E.B. Saff ad A.D. Sider, Fudametals of Complex Aalysis with Applicatios to Egieerig ad Sciece(3rd Editio),Pearso Educatio,2003(Idia Reprit 2009).

3 Lectures ad Tutorials: Every week there will be three lectures of oe hour each. There will also be oe tutorial of oe hour. For tutorials, the class will be divided ito two batches ad each batch will be assiged a Course-Associate. Tutorial classes must be treated as part of the course. Certai cocepts may be covered oly i the tutorial classes. These are also meat for a closer iteractio with you i clearig your doubts, if ay, ad also to give you practise i problem solvig. Based o the topics covered, a tutorial sheet will be give to you each week. You are expected to try the problems before had for each tutorial class. You may feel free to uhesitatigly approach your Course Associate i case you have ay doubts. I additio, the Istructor ad the Course Associates will assig a Office Hour each week to eable you to meet them i case you have ay course related difficulties. Policy for Attedace: Attedace i lectures ad tutorials is compulsory. Less tha 80% attedace either i the lectures or tutorials will attract XX grade. I case you miss lectures/tutorials for valid ( medical or other reasos ), you may obtai a medical certificate from the Medical Officer of the Istitute. Make sure that you produce a xerox copy of it to the Istructor i case you fall short of the attedace. Evaluatio Pla: There will be oe Class Test of weightage 15% of oe hour s duratio. The date ad the timig of this test will be aouced subsequetly i the class. The fial examiatio will carry 35% weightage. This will be held durig the Mid-Semester Examiatio week as prescribed by the Academic Caleder.

4 Course Pla I the followig, [K] refers to the text book by E. Kreyzig, Advaced Egieerig Mathematices, 8th Editio, Joh Wiley ad Sos (1999). The sectios of [K] metioed below are oly idicative of the pla of course coverage. Sr. No. Topic Sectio from [K] No. of Lectures 1. Review of complex umbers, complex plae, 12.1, polar form, powers ad roots. 2. Regios i the complex plae, topological cocepts Fuctios of a complex variable, their mappig properties. 3. Limits ad cotiuity, derivatives, aalytic fuctios. 12.4,15.2(part) 2.5 Riema sphere ad stereographic projectio. 4. Elemetary fuctios: expoetial fuctio, logarithmic fuctio, geeral power, trigoometric ad hyperbolic fuctios. 5. Mappigs by special fuctios, coformal mappigs Liear fractioal trasformatios Complex itegratio, lie itegrals,methods Cauchy s itegral theorem, 13.2, Cauchy s itegral formula. 9. Derivatives of aalytic fuctios, Morera s theorem, Liouville s theorem Power series. 14.2, Taylor series, Taylor s theorem Lauret series Sigularities ad zeros, behaviour at ifiity Residues ad residue theorem Evaluatio of real itegrals, improper itegrals

5 EXERCISE BANK Exercises I 1. Usig z 2 = zz, show that: (a) z 1 + z 2 z 1 + z 2, (b) z 1 w 1 + z 2 w 2 z z 2 2 w w 2 2, (c) z 1 + z z 1 z 2 2 = 2( z z 2 2 ). Note that (b) is a special case of Cauchy-Schwarz iequality(see 5(a) below,) ad that (c) is the usual parallelogram law. 2. By meas of a example, show that, i geeral, Argz 1 +Argz 2 Arg(z 1 z 2 ) where Argz deotes the pricipal vaue of the argumet of z. Show, however, that if Rz 1 > 0 ad Rz 2 > 0, the the above equality holds. 3. If z 1 z 2 0, the show that R(z 1 z 2 ) = z 1 z 2 if ad oly if argz 1 argz 2 = 2π for some iteger. I this case show further that (i) z 1 + z 2 = z 1 + z 2, ad (ii) z 1 z 2 = z 1 z If z 1, z 2,..., z are complex umbers, the show that z 1 + z z z 1 + z z. Moreover, show that equality holds i this iequality if ad oly if for every pair z i, z j such that z i 0, z j 0, i, j = 1,...,, zi z j is real ad > (a) (Cauchy-Schwarz iequality) If z i, w i IC, i = 1, 2,...,, the show that 2 ( ) ( ) z i w i z i 2 w i 2. (b) (Lagrage s Idetity) If z i, w i IC, i = 1, 2,...,, the show that 2 ( ) ( ) z i w i = z i 2 w i 2 i=1 6. Show that if w < 1, the z w wz 1 < 1 if z < 1, ad i=1 i=1 7. Show that there are complex umbers z satisfyig if ad oly if z 0 z 1. i=1 i=1 z w wz 1 = 1 if z = 1 i=1 z z 0 + z + z 0 = 2 z 1 1 i<j z i w j z j w i Show that three distict poits z 1, z 2, z 3 i the complex plae IC form the vertices of a equilateral triagle if ad oly if z z z 2 3 = z 1 z 2 + z 2 z 3 + z 3 z 1. Deduce that if w 1, w 2, w 3 are poits dividig the three sides of the triagle (z 1, z 2, z 3 ) i the same ratio, the the triagle (w 1, w 2, w 3 ) is equilateral if ad oly if the triagle (z 1, z 2, z 3 ) is so.

6 9. If z 1, z 2, z 3 are three distict complex umbers of equal moduli, the show that 2 arg z 2 z 1 z 3 z 1 = arg z 2 z 3. Ca you recall the theorem i school geometry which correspods to this result? 10. For every IN, show that: (a) [1 ( ( 3) + ( 4) ) ] 2 + [ ( ( 1) ( 3) + 5) +...] 2 = 2 ; (b) 1 + cos θ + cos 2θ cos θ = 1 2 (c) si θ + si 2θ si θ = 1 2 cos θ si( 2 )θ + 2 si θ cos( 2 )θ 2 si θ 2 (0 < θ < 2π); (0 < θ < 2π); (d) (1 z 1 )(1 z 2 )... (1 z 1 ) = where z 1, z 2,..., z 1 are the th roots of uity other tha 1; (e) si( π ) si( 2π (Hit: Use (d).) )... si( ( 1)π ) = Show that the sum of the, th roots of every ozero complex umber w is zero. 12. If ω = cis( 2π ), IN, 2, the show that for ay iteger h which is ot a multiple of. 1 + ω h + ω 2h ω ( 1)h = Cosider the equatio p(z) = a 0 z + a 1 z a 1 z + a = 0, where a i IC, i = 0,..., are give complex umbers. Show that if w is a root of this equatio, the w is a root of the equatio a 0 z + a 1 z a 1 z + a = 0. Coclude that if w is a root of the equatio p(z) = 0 with real coefficiets, the so also is w. 14. Show that if z lies o the circle z = 2, the 1 z 4 4z Show that: (i) ( 1 + i) 7 = 8(1 + i), (ii) (1 + 3i) 10 = 2 11 ( 1 + 3i). 16. Let z 1, z 2 be distict complex umbers, ad let k > 0. Show that the set {z IC : z z 1 = k z z 2 } is a circle uless k = 1, i which case the set is a straight lie, the perpedicular bisector of the lie joiig the poits P 1 (z 1 ) ad P 2 (z 2 ). 6

7 17. Sketch the followig sets ad determie which are domais: (a) z 2 + i 10, (b) Iz > Rz, (c) 2z + 3 > 2, (d) 0 argz π/4(z 0), (e) z 4 z. 18. I each of the followig cases, sketch the closure of the set. (a) π < argz < π (z 0); (b) Rz < z ; (c) R( 1 z ) 1 2 ; (d) R(z 2 ) > 0; (e) 0 argz < π 4. (z 0). 19. Determie the accumulatio poits of each of the followig sequeces. (a) z = i, IN ; (b) z = i /; (c) z = ( 1) (1 + i)( 1 ). Exercises II 1. Fid the domai ad the rage of each of the followig fuctios ad also write each fuctio i the form w = u(x, y) + iv(x, y) : (a) f(z) = z+5 z 3 for z 3 = 10; (b) f(z) = z + 7 for Rz > 0; (c) f(z) = z 2 for z i the first quadrat, Rz 0, Iz 0; (d) f(z) = 1 z for 0 < z 1; (e) f(z) = 2z 3 for z i the quarter-disk z < 1, 0 < Argz < π (a) Let f : IC \ {1} IC be defied by f(z) = z 1 z 1. Examie whether lim z 1 f(z) exists. (b) Let f(z) be defied equal to i 2 for z = 1 + 2i ad equal to z 2 (4 + 3i)z i, for z 1 + 2i. z 1 2i Usig ɛ δ method discuss the cotiuity of f at the poit 1 + 2i. Is f cotiuous at all other poits of IC? 3. Uder the iversio mappig w = f(z) = 1 z fid the image i the w-plae of each of the followig curves i the z-plae: (a)the circle z = r; (b)the ray Argz = θ 0, π < θ 0 < π; (c)the circle z 1 = Uder the mappig w = f(z) = z 2 fid the image i the w-plae of each of the followig regios i the z-plae: (a)rz > 0, Iz > 0, ad Rz Iz < 1; (b)the first quadrat Rz > 0 ad Iz > 0.

8 5. Uder the mappig w = f(z) = e z fid the image i the w-plae of each of the followig regios i the z-plae: (a)the vertical lie Rz = 2; (b)the horizotal lie Iz = π/3; (c)the rectagular strip α Rz β, γ Iz δ; (d)the ifiite strip 0 Iz π. 6. Show that if Z = (x 1, x 2, x 3 ) is the (stereographic) projectio oto the Riema sphere of the poit z = x+iy i the complex plae IC, the x 1 = 2Rz z 2 + 1, x 2 = 2Iz z 2 + 1, x 3 = z 2 1 z Show that uder the stereographic projectio σ : z Z all lies ad circles i the z-plae correspod to circles o the Riema shere. 8. Show that the iversio mappig w = 1/z correspods to a rotatio of the Riema sphere by by a agle π about the x 1 -axis. 9. Show that the liear map T : IR 2 IR 2 defied by the matrix: ( ) a b c d is multiplicatio by a complex umber if ad oly if a = d ad c = b. ( Hit:Use the fact that if T is a multiplicatio by a complex umber if ad oly if T is complex liear.) 10. (Geeralized Cauchy-Riema Equatios) If f(z) = u + iv is differetiable at a poit z 0 = x 0 + iy 0 of a domai D, the show that u s = v, u = v s at (x 0, y 0 ) where s ad deote respectively directioal differetiatio i two orthogoal directios s ad at (x 0, y 0 ), such that is obtaied from s by makig a couterclockwise rotatio. 11. Let f = u + iv be aalytic at a poit z 0 ad suppose f (z 0 ) 0. Cosider the curves C 1 : u = R(f(z 0 )) ; C 2 : v = I(f(z 0 )) passig through z 0. Show that C 1 ad C 2 are orthogoal to each other at z Let f(z) = z2 z, z 0 ad f(0) = 0. Show that Cauchy-Riema equatios are satisfied at z = 0, however, f (0) does ot exist. 13. Show that the fuctio f(z) = x 3 + 3xy 2 3x + i(y 3 + 3x 2 y 3y) is differetiable o the coordiate axes; but it is owhere aalytic. 14. Show that the fuctio is etire, ad fid its derivative. f(z) = e x2 y 2 [cos(2xy) + i si(2xy)] 15. If u ad v are expressed iterms of polar coordiates (r, θ), show that the Cauchy-Riema equatios ca be writte i the form u r = 1 r v θ, v r = 1 r u θ. 16. Show that if f is aalytic i a domai D ad either Rf(z) or If(z) or f(z) is costat i D, the f(z) must be costat i D. 8

9 17. Suppose that f(z) is aalytic i a domai D. I additio, if either f(z) or f(z) is aalytic i D, the show that f(z) is costat i D. 18. Show that the followig fuctios are harmoic ad fid a harmoic cojugate for each of them: (i) u 1 (x, y) = 2x(1 y); (ii) u 2 (x, y) = sih x si y. 19. If f(z) is aalytic i a domai D, show that f(z) 2 is ot harmoic uless f(z) is a costat. 20. Fid a fuctio φ(x, y) that is harmoic i the regio of the first quadrat betwee the curves xy = 2 ad xy = 4 ad takes the value 1 o the lower edge ad the value 3 o the upper edge. Exercises III 1. Let p(z) be a polyomial of positive degree. Show that for every M > 0 there exists R > 0 such that for all z IC with z > R we have p(z) > M. ( Put differetly, this meas p(z) as z.) 2. Fid the behaviour of e z as z alog arg z = 0, π/2, π. I particular, verify that e z does ot have the property metioed i the previous exercise. 3. Fid all values of z such that: (a) e z = 2; (b)e z = 1 + 3i; (c)e 2z 1 = Show that: (i) e (2z+i) + e (iz2) e 2x + e 2xy ; (ii) e z2 e z 2. Also, show that (iii) e iz = e iz if ad oly if z = π( IZ {0}). 5. Show that cos z sih y, where z = x + iy. 6. Fid all values of z for which the fuctios cos z, si z are real. 7. Show that all solutios of the equatios (a)cos z = 0 (b)si z = 0 are real. 8. Fid the followig limits. Justify your aswer. si z (a) lim z 0 z ; (b)lim cos z 1 z 0 z. 9. (a) Show that the mappig w = si z is oe-to-oe i the semi-ifiite strip S 1 = {x + iy : π < x < π, y > 0}. (b)uder the same mappig as i (a), what is the image of the smaller semi-ifiite strip 10. Fid all roots of the equatio l z = iπ/2. S 2 = {x + iy : π/2 < x < π/2, y > 0}? 11. Show that: (a)the fuctio Log(z i) is aalytic everywhere except o the half lie y = 1(x 0). Log(z + 4) (b)the fuctio z 2 is aalytic except at the poits (1 i)/ 2, (1 i)/ 2 ad o the portio x 4 + i of the real axis. 12. (i)fid the domais i which the fuctios Argz, Log z are respectively harmoic. (ii)determie the domai of aalyticity for the fuctio f(z) = Log(3z i). Also compute f (z). 13. Determie a brach of f(z) = log(z 3 2) that is aalytic at z = 0, ad also fid f(0) ad f (0).

10 14. Fid the pricipal value of (i)(1 + i) i ; (ii)3 3 i. 15. Show that: (i)si 1 z = i l(iz ± 1 z 2 ); (ii)cos 1 z = i l(z ± z 2 1); (iii)ta 1 z = i + z 2 l (i i z ); (iv) sih 1 z = l(z ± z 2 + 1); (v) cosh 1 z = l(z ± z 2 1); (vi) tah 1 z = 1 + z 2 l (1 1 z ). Exercises IV 1. Let f : D IC be a fuctio where D is a domai. Let z 0 D ad assume the regularity coditiod: f f x ad y are cotiuous at z 0. If f is agle preservig at z 0 the show that f (z 0 ) must exist. (Hit: Show that uder the stated coditios, the Cauchy- Riema equatios are satisfied.) 2. Determie the agle through which the tagets to all the curves passig through the poit 1 + i are rotated uder the trasformatio w = z Fid ad sketch the images of the agular regio 0 arg z π/8 i the case of each of the followig mappigs: (i) w = 1/z; (ii) w = i/z; (iii) w = z 2 ; (iv) w = iz Fid ad sketch the images of the regio y > 1 uder the followig trasformatios: (i) w = iz; (ii)w = (1 + i)z; (iii) w = (1 i)z + 2i; (iv) w = iz Fid a aalytic fuctio w = u + iv = f(z) that maps the regio 0 < arg z < π/5 oto the regio u < Fid a fuctio w = f(z) = u+iv that maps the agular regio π/8 arg z π/8 oto the disk w Fid a liear trasformatio mappig the circle z = 1 oto the circle w 3 = 5 ad takig the poit z = i to w = Show that every Möbius trasformatio T : IC IC which fixes three distict poits of IC is ecessarily the idetity trasformatio. Coclude that two Möbius trasformatios which agree o a set of three distict poits of IC coicide. ( z z 3 ) ( ) z 9. Let z 2, z 3, z 3 IC be distict poits ad let M(z) = 2 z 4 z z 4 z 2 z 3 deote the uique Möbius trasformatio which carries z 2, z 3, z 4 oto 1, 0, respectively. (We adopt the covetio that if oe of z 2, z 3, z 4 equals, the the quotiet cotaiig that term is to be take equal to 1.) If z 1, z 2, z 3, z 4 IC ad z 2, z 3, z 4 are distict, the the cross ratio (z 1, z 2, z 3, z 4 ) is defied to be equal to M(z 1 ), i.e.: ( ) ( ) z1 z 3 z2 z 4 z 1 z 4 z 2 z 3 with the covetio as above. Show that give two sets {z 1, z 2, z 3 }, {w 1, w 2, w 3 } of distict poits i IC, there is a uique Möbius trasformatio T such that T z i = w i, i = 1, 2, 3, which is give by (w, w 1, w 2, w 3 ) = (z, z 1, z 2, z 3 ). 10. If z 1, z 2, z 3, z 4 are distict poits i IC ad T is ay Möbius trasformatio, the show that: (T z 1, T z 2, T z 3, T z 4 ) = (z 1, z 2, z 3, z 4 ). Put differetly, this says that the cross ratio is ivariat uder a Möbius trasformatio.

11 11. Fid the Möbius trasformatio that maps three give poits oto three give poits i the respective order: (i) 1, i, 1 oto i, 1, 1; (ii) 1, 0, 1 oto 1, i, 1; (iii)0, 1, oto 1,, 0; (iv)0, 1, oto 1, i, Show that ay Möbius trasformatio w = T z which maps the upper half plae Iz > 0 oto the uit disk w < 1 is ecessarily of the type ( ) z λ T z = e iθ z λ for some λ IC such that Iλ > Show that the Möbius trasformatio maps the uit disk z 1 oto the uit disk w 1. w = z z 0 z 0 z 1, z 0 < Show that the Möbius trasformatio w = z 1 z+1 maps: (i) the half-plae Iz > 0 oto the half-plae Iw > 0, ad (ii) the half-plae Rz > 0 oto the uit disk w < 1. Does this give you a clue to fid a Möbius trasformatio that maps the uit disk z < 1 oto the right half-plae Rw > 0? 15. Fid a ratioal fuctio w = f(z) that maps the agular regio π/6 arg z π/6 oto the uit disk w Fid a fuctio w = f(z) that maps the upper half-plae Iz > 0 oto the strip 0 < Iw < π. 17. Fid a fuctio w = f(z) that maps respectively: (i) the upper half-disk z < 1, ad Iz > 0 oto the first quadrat Rw > 0 ad Iw > 0; (ii) the quarter-disk z < 1, ad Iz > 0, Rz > 0 oto the upper half-plae Iw > 0. (Hit: Make use of the previous questio.) 18. Show that uder the mappig w = 1/z the half-plae Rz α(α > 0) is mapped oto the disk ( u 1 ) 2 ( ) v 2. 2α 2α 11

12 Exercises V 1. Suppose f(t) = u(t) + iv(t) is a cotiuous complex-valued fuctio o a iterval [a, b]. Prove that b a f(t)dt b a f(t) dt. 2. Usig the last exercise, prove a stroger versio of the M L iequality, viz. C f(z)dz C f(z) dz. 3. Evaluate C zdz, where C is the right-half z = 3eiθ ( π/2 θ π/2) of the circle z = 3 from z = 3i to z 3i. 4. Fid a upper boud for C e1/z dz, where the itegral is take alog the cotour C, which is the quarter circle z = 1, 0 argz π/2 from the poit 1 to the poit i. 5. Let z 1/2 deote the brach z 1/2 = re iθ/2 (r > 0, π/2 < θ < 3π/2). Without actually fidig the value of the itegral, show that z lim R CR 1/2 z 2 dz = 0, + 1 where C R deotes the semicircular path z = Re iθ (0 θ π). 6. Itegrate z 2 alog (a) the lie segmet from 0 to i, (b) the arc of the parabola y = x 2 from 0 to 2 + 4i, both directly ad by fidig a primitive of the itegrad. 7. Evaluate C z1/2, where C deotes the semicircular path z = 2e iθ (0 θ π) from the poit z = 2 to the poit z = Let z be the pricipal value of the square root of z. Evaluate dz z alog (a) the upper semicircle z = 1, (b) the lower semicircle z = 1. Why do they differ? 9. Evaluate z zdz where C is the closed cotour cosistig of the lie segmet from 1 to 1 ad the semicircle C z = 1, y 0, take i the couterclockwise directio. 10. Suppose f(z) is holomorphic ad f (z) is cotiuous i a domai cotaiig a closed curve C. (The hypothesis about cotiuity of f is redudat but we are ot i a positio to prove this.) Prove that C f(z)f (z)dz is purely imagiary. 11. Assume f(z) is holomorphic (f (z) cotiuous) ad satisfies the iequality f(z) 1 < 1, throughout a domai D. Prove that f (z) C f(z) dz = 0 for every closed curve C i D. 12. Prove that a domai D is simply coected if ad oly if f(z)dz is path idepedet for every fuctio C f(z) which is holomorphic i D. 13. For each of the followig fuctios, examie whether Cauchy s theorem ca be applied to evaluate the itegrals aroud the uit circle take couterclockwise. Hece or otherwise evaluate the itegrals (a) l(z + 2), (b) 1 z, (c) z, (d) e z2, (e) tah z, (f) z, (g) 1 3 z Fid: (a) z 2 z+2 C z 3 2z dz, where C is the boudary of the rectagle with vertices 3 ± i, 1 ± i traversed clockwise. 2 (b) si z C z+3idz, C : z 2 + 3i = 1 (couterclockwise). 15. Evaluate B f(z)dz where f(z) is (a) z+2 si z/2, (b) z 1 e where B is the boudary of the domai betwee z = 4 z ad the square with sides alog x = ±1, y = ±1, orieted i such a way that the domai always lies to its left. 16. C is the uit circle traversed couterclockwise. Itegrate over C, (a) ez 1 z z, (b) 3 cosz si z 2z i, (c) z π (d) 2z Itegrate z 4 1 over (a) z + 1 = 1, (b) z i = 1, each curve beig take couterclockwise. [Hit: Resolve ito partial fractios] 12

Chapter 1. Complex Numbers. Dr. Pulak Sahoo

Chapter 1. Complex Numbers. Dr. Pulak Sahoo Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistat Professor Departmet of Mathematics Uiversity Of Kalyai West Begal, Idia E-mail : sahoopulak1@gmail.com 1 Module-2: Stereographic Projectio 1 Euler