Complex Variables. Chapter 18 Integration in the Complex Plane. March 12, 2013 Lecturer: Shih-Yuan Chen

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1 omplex Varables hapter 8 Integraton n the omplex Plane March, Lecturer: Shh-Yuan hen Except where otherwse noted, content s lcensed under a BY-N-SA. TW Lcense.

2 ontents ontour ntegrals auchy-goursat theorem Independence o path auchy s ntegral ormulas

3 ontour Integrals Recall that the de. o ( x) starts wth a real uncton y (x) that s dened on [a, b] on the x-axs. Ths can be generaled to ntegrals o real unctons o two varables dened on a curve n the artesan plane. b a dx

4 ontour Integrals For the ntegral o a complex x () that s dened along a curve n the complex plane. could be smooth curve, pecewse smooth curve, closed curve, or smple closed curve, and s called a contour or path. s dened n terms o parametrc equatons x x(t), y y(t), a t b, where t s a real parameter. Usng x(t) & y(t) as real & magnary parts, we can descrbe n the complex plane by means o a complex-valued uncton o a real varable t: (t) x(t) y(t), a t b. 4

5 ontour Integrals An ntegral o () on s denoted by or ( ) d s closed. It s reerred to as a contour ntegral or a complex lne ntegral. ( ) d 5

6 ontour Integrals () u(x, y) v(x, y) Let be dened at all ponts on a smooth curve dened by x x(t), y y(t), a t b. Dvde nto n subarcs accordng to the partton a t < t < < t n b. orrespondng ponts on : x y x(t ) y(t ), x y x(t ) y(t ),, n x n y n x(t n ) y(t n ). Let k k k, k,,, n & let P be the norm o the partton. hoose a pont on each subarc. Form the sum * * k xk n ( ) k y k * k * 6 k

7 ontour Integrals De. Let be dened at ponts o a smooth curve dened by x x(t), y y(t), a t b. The contour ntegral o along s ( ) ( * d lm ) ( ) P k n The lmt exsts s contnuous at all ponts on & s ether smooth or pecewse smooth. k k 7

8 ontour Integrals Method o evaluaton Wrte () n the abbrevated orm ( ) d lm ( u v)( x y) lm { ( u x v y) ( v x u y) } u dx v dy Thus, a contour ntegral s a combnaton o two real-lne ntegrals. v dx u dy ( ) 8

9 ontour Integrals Snce x x(t), y y(t), a t b, the RHS o () s b [ u ( x( t), y( t) ) x ( t) v( x( t), y( t) ) y ( t) ] a b a I we use (t) x(t) y(t) to descrbe, the last result s the same as b ( ( t) ) ( t) separated nto two ntegrals. a dt dt [ v( x( t), y( t) ) x ( t) u( x( t) y( t) ) y ( t) ], when dt 9

10 ontour Integrals Thm. I s contnuous on a smooth curve gven by (t) x(t) y(t), a t b, then ( ) d ( ( t) ) ( t) dt ( ) b a I s expressed n terms o, then to evaluate () we smply replace by (t). I not, we replace x & y wherever they appear by x(t) & y(t), respectvely.

11 ontour Integrals Ex. Evaluate d, where s gven by x t, (sol) y t, t 4. ( ) t t t ( t) ( ( t) ) d t 4 4 t t t ( t t )( t) dt t ( ) 4 9t t dt t dt 95 65

12 ontour Integrals Ex. Evaluate d, where s the crcle x cos t, y sn t, t π. (sol) ( ) t t cost sn t e ( t) ( ( t) ) d π e t e ( ) π t t e e dt dt π t

13 ontour Integrals Propertes o contour ntegrals Suppose & g are contnuous n a doman D & s a smooth curve lyng entrely n D. ( ) k ( ) d k ( ) ( ) [ ( ) g( ) ] d ( ) d g( ) ( ) ( ) d ( ) d ( ) ( v) ( ) d ( ) d, d k a constant. d, d

14 ontour Integrals ( x y ) Ex. Evaluate, where. d ( ) ( ) ( x y d x y d x y ) s dened by y x t, t. ( ) ( x y d t t )( ) s dened by x, y t, t. ( ) ( ) ( ) x y d t dt t dt dt t dt dt d 7

15 ( ) ML d ontour Integrals Thm. I s contnuous on a smooth curve & () M or all on, then, where L s the length o. (proo) From (5) o h.7, we can wrte k : length o the chord jonng the ponts k & k. ( ) ( ) n k k n k k k n k k k M * * ( ) ML M n k k n k k k * 5

16 ontour Integrals e Ex. Fnd an upper bound or d, where s the crcle 4. The length o the crcle s 8π. From the nequalty o h.7, e e e e x ( cos y sn y) e x e 4 e d 8πe 4 6

17 ontour Integrals rculaton & net lux T & N denote the unt tangent vector & unt normal vector to a postvely orented smple closed contour. Interpret () u(x,y) v(x,y) as a vector, the lne ntegrals T ds u dx v dy ( 4) N ds u dy v dx ( 5) The lne ntegral n (4) s called crculaton around & measures the tendency o the low to rotate.

18 ontour Integrals The net lux across s the derence between the rate at whch lud enters & the rate at whch lud leaves the regon bounded by. It s gven by the lne ntegral n (5). Note that ( ) ( ) T ds N ds ( u v)( dx dy) ( ) ( ( ) ) rculaton Re d ( 6) ( ( ) ) Net lux Im d ( 7) d 8

19 ontour Integrals Ex. Gven the low () ( ), compute the crculaton around & the net lux across the crcle :. ( ) ( ) ( t) e t, ( ) d ( ) π t e t e dt ( ) dt π ( ) π π t 9

20 auchy-goursat Theorem Focus on contour ntegrals where s a postve smple closed curve Smply & multply connected domans Smply connected every smple closed lyng entrely n D can be shrunk to a pont wthout leavng D. Ex. The entre complex plane. Multply connected a doman that s not smply connected.

21 auchy-goursat Theorem auchy s Theorem Suppose that s analytc n a smply connected doman D & that s contnuous n D. For every smple closed contour n D, ( ) d (proo) Snce s contnuous throughout D, the real & magnary parts o () u v & ther st partal dervatves are contnuous n D. From (), ( ) d u( x, y) dx v( x, y) dy v( x, y) dx u( x, y) v x u y da D D u x v y da dy

22 auchy-goursat Theorem auchy-goursat Theorem Suppose s analytc n a smply connected doman D. Then or every smple closed contour n D, ( ) d ( 8) In other words, s analytc at all ponts wthn & on a smple closed contour, then e d Ex. Evaluate. ( ) d Snce e s entre & s a smple closed contour, t ollows rom the auchy- Goursat theorem that e d

23 auchy-goursat Theorem d ( x ) Ex. Evaluate, where : () / s analytc everywhere except at. But s not nteror to or on. ( ) cos Ex. Gven the low, compute the crculaton around & net lux across, where s the square wth vertces ± & ±. rculaton & net lux are both ero. ( ) d cos d ( y 5) 4 d

24 auchy-goursat Theorem For multply connected domans I s analytc n a multply connected doman D, then we cannot conclude that ( ) d or every smple closed n D. Suppose D s doubly connected and & are smple closed contours such that surrounds the hole n D & s nteror to. Also, suppose s analytc on each contour & at each pont nteror to but exteror to.

25 auchy-goursat Theorem As we ntroduce the cut AB, the regon bounded by the curves s smply connected. ( ) d ( ) d ( ) d ( ) ( ) d ( ) d ( 9) AB (9) s called the prncple o deormaton o contours. Thus, one can evaluate an ntegral over a complcated smple closed contour by replacng that contour wth one that s more convenent. BA d 5

26 auchy-goursat Theorem Ex. Evaluate shown n the gure. From (9), we choose. Takng the radus o the crcle to be r, les wthn. : Let e t, t π e d t d d, where s the outer contour π e e t t dt π

27 auchy-goursat Theorem I s any constant complex number nteror to any smple closed contour, then d π, n ( ) ( ) n, n For n s ero or negatve, the ntegral s ero. For n s postve nteger derent rom one, t re t :, π and t d d π re dt r n n nt r e ( ) n ( ) n re e t ( n ) t ( n) π 7

28 auchy-goursat Theorem Ex. Evaluate wth :. d 7 5 ( )( ) d d d π 6π

29 auchy-goursat Theorem I,, are smple closed contours & s analytc on them & at ponts nteror to but exteror to both &, then by ntroducng two cuts, ( ) d ( ) d ( ) d

30 auchy-goursat Theorem G Thm or Multply onnected Domans Suppose,,, n are postve smple closed contours such that,,, n are nteror to but ther nteror regons have no ponts n common. I s analytc on each contour & at each pont nteror to but exteror to all k, k n, n ( ) d ( ) d ( ) k k

31 auchy-goursat Theorem Ex. Evaluate, where :. Surround ± by crcle contours & that le entrely wthn. d d d : and :

32 d d d d d d d π π auchy-goursat Theorem

33 auchy-goursat Theorem It can be shown that the auchy-goursat theorem s vald or any closed n a smply connected doman D. As shown n the gure, s analytc n D, then ( ) d

34 Independence o Path De. Let & be ponts n D. A contour ntegral ( ) d s sad to be ndependent o the path ts value s the same or all contours n D wth an ntal pont & a termnal pont. Suppose & are contours n a smply connected doman D, both rom to. Note that & orm a closed contour. Thus, s analytc n D, ( ) d ( ) d ( ) d ( ) d ( )

35 Independence o Path Analytcty mples path ndependence Thm. I s an analytc uncton n a smply connected doman D, then ( ) d s ndependent o. Ex. Evaluate d where s the contour rom to shown n the gure. Snce () s entre, we can replace by any, say : x, y t, t d d ( t) dt 5

36 Independence o Path ( ) A contour ntegral that s ndependent o rom to s wrtten. De. Suppose s contnuous n D. I there exsts a uncton F such that F () () or n D, then F s called an antdervatve o. The most general antdervatve o () s wrtten ( ) d F( ) Snce F has a dervatve at each pont n D, t s necessarly analytc & hence contnuous n D. d ( ) d 6

37 Independence o Path Fundamental Theorem o ontour Integrals Thm. Suppose s contnuous n D & F s an antdervatve o n D. Then or any n D rom to, (proo) ( ) d F( ) F( ) ( ) As s a smooth curve dened by (t), a t b. Use () & the act that F () () or n D, ( ) d ( ( t) ) ( t) dt F ( ( t) ) ( t) dt F( ( t) ) F b a b d dt ( ( t) ) F( ( b) ) F( ( a) ) F( ) F( ) a b a b a 7 dt

38 Independence o Path Ex. Use () to solve the prevous example. Ex. Evaluate d rom to. d ( ) ( ) cos cos d sn d ( ).4, where s any contour cos d sn sn.489 8

39 Independence o Path I s closed, then & consequently ( ) d ( 4) ( ) d ( ) Snce the value o d &, t s the same or any n D connectng these ponts. In other words, depends on only I s contnuous & has an antdervatve F n D, then s ndependent o. ( ) I s contnuous & s ndependent o n D, then has an antdervatve everywhere n D. d

40 Independence o Path (proo) Assume that s contnuous, ( ) ndependent o n D, & F s a uncton dened by F ( ) ( s) ds s where s s a complex varable, s a xed pont n D, & s any pont n D. F hoose so that s n D and & can be joned by a straght segment n D. d ( ) F( ) ( s) ds ( s) ( s) ds ds 4

41 Independence o Path (proo) As xed, we can wrte ( ) ( ) ds ( ) ( ) ( ) ( ) ( ) ds F F ( ) [ ( s) ( ) ] ds Now s contnuous at the pont. It means that or any ε > there exsts a δ > so that (s) () < ε whenever s < δ. ds 4

42 Independence o Path F (proo) onsequently, we choose so that < δ, F ( ) [ ( s) ( ) ] ds ( ) ( ) lm F ( ) F( ) [ ( s) ( ) ] ds ε ε ( ) or F ( ) ( ) 4

43 Independence o Path It s known that I s analytc n a smply connected doman D, s necessarly contnuous throughout D. Its contour ntegral s ndependent o the path. ombne wth the result just obtaned, Thm. I s analytc n a smply connected doman D, then has an antdervatve F everywhere n D or there exsts a uncton F so that F () () or all n D. 4

44 Independence o Path Recall rom h.7 that / s the dervatve o Ln. It means that under some crcumstances Ln s an antdervatve o /. Suppose D s the entre complex plane wthout the orgn. / s analytc n D. I s any smple closed contour contanng the orgn, In ths case, Ln s NOT an antdervatve o / n D, snce Ln s not analytc n D. (Ln als to be analytc on the nonpostve real axs!) d π 44

45 Independence o Path Ex. Evaluate wth shown n the gure. D s the smply connected doman dened by x d ( ) >, y Im( ) Re > Here, Ln s an antdervatve o /, snce both Ln & / are analytc n D. d Ln Ln ( ) π loge loge Ln 45

46 Independence o Path I & g are analytc n a smply connected doman D contanng rom to, then the ntegraton by parts ormula s vald n D: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) d g g d g d g d g d g d d g d g d d 46

47 auchy s Integral Formulas auchy s ntegral ormula Thm. Let be analytc n a smply connected doman D and be a smple closed contour lyng entrely wthn D. I s any pont wthn, then ( ) ( ) ( 5) d π (proo) Let be a crcle centered at wth radus small enough that t s nteror to. 47

48 auchy s Integral Formulas By the prncple o deormaton o contours, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) d d d d d d π 48

49 auchy s Integral Formulas Snce s contnuous at, or any ε >, there exsts a δ > such that () ( ) < ε whenever < δ. Thus, we choose to be δ/ < δ, then by the ML-nequalty, ( ) ( ) d ε δ δ π πε In other words, the absolute value o the ntegral can be made arbtrarly small by takng the radus o to be sucently small. Ths can happen only the ntegral s ero. (5) can thus be obtaned. 49

50 auchy s Integral Formulas Snce most problems do not have smply connected domans, a more practcal restatement o auchy s ntegral theorem: ( ) I s analytc wthn & on a smple closed contour & s nteror to, ( ) 4 4 d ( ) d π ( ) d π Ex. Evaluate wth :. 4 4 ( 5) ( ) π ( 4 )

51 auchy s Integral Formulas Ex. Evaluate wth : 4. ( ) ( ) d d π π π 6 9 and 9 d 9 5

52 auchy s Integral Formulas ( ) k ( ) Ex., where k a b & are complex numbers, gves rse to a low n the doman. I s a smple closed contour contanng n ts nteror, then a b ( ) d d π ( a b) Thus, the crculaton around s πb, & the net lux across s πa. I were n the exteror o, both the crculaton & the net lux would be ero by auchy s theorem. 5

53 auchy s Integral Formulas When k s real, the crculaton around s ero but the net lux across s πk. The complex number s called a source or the low when k > & a snk when k <. 5

54 auchy s Integral Formulas auchy s ntegral ormula or dervatves Thm. Let be analytc n a smply connected doman D and be a smple closed contour lyng entrely wthn D. I s any pont wthn, then ( n ) n! ( ) ( 6) d n π (proo) ( ) lm lm ( ) ( ) ( ) ( ) π ( ) ( ) d ( ) d

55 auchy s Integral Formulas Snce s contnuous on, there exsts a real number M so that () M or on. Besdes, let L be the length o and δ be the shortest dstance between ponts on &. Thus, or all on, I we choose δ/, then ( ) ( ) ( )( ) d lm π or δ δ δ δ δ and

56 auchy s Integral Formulas ( ) ( )( ) ( ) ( ) ( ) d ( ) ( ) δ ML Snce the last expresson goes to ero as, d ( ) ( ) ( ) π d d 56

57 auchy s Integral Formulas I () u(x,y) v(x,y) s analytc at a pont, then ts dervatves o all orders exst at that pont & are contnuous. onsequently, rom u & v have contnuous partal dervatves o all orders at a pont o analytcty. ( ) ( ) x y u x y v x v x u y u y v x v x u 57

58 auchy s Integral Formulas Ex. Evaluate wth :. The ntegrand s not analytc at & 4, but only les wthn. By (6) we have d 4 4 ( ),, n ( ) ( ) d 4 6! 4 4 π π π 58

59 auchy s Integral Formulas Ex. Evaluate wth shown n the gure. Thnk o as the unon o two smple closed contours &. ( ) d ( ) ( ) ( ) ( ) ( ) I I d d d d d 59

60 auchy s Integral Formulas Use (5) & (6) or I & I, respectvely. ( ) ( ) ( ) ( ) ( ) I I d d I d I π π π π π π π π 4 6 4! 6 6

61 auchy s Integral Formulas Louvlle s Theorem The only bounded entre unctons are constants. Suppose s entre & bounded, that s, () M or all. Then or any pont, ( ) M/r. ( n ) ( ) n! ( ) n! M n! d πr ( ) n n n π r r π By takng r arbtrarly large, we can make ( ) as small as we wsh. ( ) or all ponts n the complex plane. Hence must be a constant. M 6

62 auchy s Integral Formulas Fundamental theorem o algebra I P() s a nonconstant polynomal, then the equaton P() has at least one root. Suppose that P() or all. Ths mples that the recprocal o P, () /P(), s an entre uncton. Snce () as, must be bounded or all nte. It ollows rom Louvlle s thm that s a constant & thereore P s a constant. ontradcton! We conclude that there must exst at least one 6 number or whch P().

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