1、Chapter 2 Analytic Functions2.1 The concept of the analytic functions1.Derivative of complex functions Def 1.Let where is a domain.Then is said to be differentiable in the complex sense at if (1)exists.:f AAf0zA000()()limzzf zf zzz This limit is denoted by ,or sometimes by .called the derivative of
2、at .Thus is a complex number.0()fz0/()df dz z0()fzf0z0()fz By expressing the variable in definition(1)in terms of the new complex variable we can write that definition as (2)or (3)z0zzz0000()()()limzf zzf zfzz ()()wf zzf z 0()limzdwwfzzz EX.1.Suppose that .At any point ,since is a polynomial in .Hen
3、ce ,or .2()f zzz22000()limlimlim(2)2zzzwzzzzzzzz 2zz z/2dw dzz()2fzz EX.2.Show that is not differentiable on .()f zzSolution：The limit does not exist.0z000000()()()i()i()i()if zf zxxyyxyzzxxyyxy 0 1 0ii01 0 xyxyxyxy 0zxy0Figure.2.1EX.3.Consider not the function .2()|f zz Here We conclude that exists
4、 only an ,its value there being 0.22()()zzzwf zzf zzzz()()zz zzzzzzzzzz/dw dz0z If exists,then is continuous at .0()fzf0ziv.Any polynomial is differentiable on with derivative .01nnaaza z1122nnaa zna z iii.If for all ,then is differentiable at and()0g z zA/fg2()()()()()()()ffz g zg z f zzgg z0z Supp
5、ose that and are differentiable at .Then i.is differentiable at and for any complex numbers and .f0z afbg0z()()()()afbgzafzbg zabgfg()()()()()()fgzfz g zf z g z0zii.is differentiable at and v.(Chain Rule)Suppose that has a derivative at and that has a derivative at the point .Then the function has a
6、 derivative at ,and .0z0zfg0()f z()()F zg f z000()()()F zg f zfz2.The concept of an analytic function A function of the complex variable is analytic in a domain if it has a derivative at each point in .In particular,is analytic at a point if it is analytic throughout some neighborhood of .fzf0z0zDD
7、is analytic at each nonzero point in the finite plane.But is not analytic at any point since its derivative exists only at and not throughout any neighborhood.1()f zz2()|f zz0z If a function fails to be analytic at a point but is analytic at some point in every neighborhood of ,then is called a sing
8、ular point,or singularity,of .f0z0z0zf The point is a singular point of the function .The function ,has no singular points since it is nowhere analytic.0z 1()f zz2()|f zz2.2 A necessary and sufficient condition for differentiability EX1.Consider now the function()2 if zxy Solution:So is not differen
9、tiable in .()()f zzf zz()2()i2 iixxyyxyxy 2iixyxy 1 0,02 0,0yxxy ()f z But their partial derivatives exist and are continuous.(,),(,)2u x yx v x yy,uuvvxyxy Let in a domain ,if is differentiable at ,then()(,)i(,)f zu x yv x yD()f z0zD0000()()lim()zzf zf zfzzz Let us take the special case that .Then
10、As ,the left side of the equation converges to the limit .0izxy0000000(,)i(,)(,)i(,)u x yv x yu xyv xyxx00000000(,)(,)(,)(,)iu x yu xyv x yv xyxxxx0 xx0()fz00()()f zf zzz Thus both real and imaginary parts of the right side must converge to a limit.From the definition of the partial derivatives,this
11、 limit is .Thus .0000(,)(,)ixyxyuvxx00000(,)(,)()ixyxyuvfzxx Next let .Then we similarly have 0izxy000000000()()(,)i(,)(,)i(,)i()f zf zu xyv xyu xyv xyzzyy00000000(,)(,)(,)(,)i()u xyu xyv xyv xyyyyyAs ,we get0yy1iiuvvuyyyy Thus,since exists and has the same value regardless of how approaching ,we ge
12、t By comparing real and imaginary parts of these equations,we drive and called the Cauchy Riemann equations.0()fzz0z0()iiuvvufzxxyyuvxyuvyx Let Here 01200()()()()i()()f zf zzzzfzzz0000000(,)(,)i(,)(,)()()i()u x yu x yv x yv x yf zxxyy0i()(i)(i)uvfzxyxy 12(i)(i)(i)(i)uvxyxyxx 0izzzxy 00(,)(,)uu x yu
13、xy 00(,)(,)vv x yv xy Expanding the right side of the equation and using the C.-R equations,and by comparing real and imaginary parts of it,we derive thatSince and we derive that and are differentiable at .12uuuxyxyxy 21vvvxyxyxy 100lim0,xy 200lim0,xy (,)u x y(,)v x y00(,)xyNext we will prove that t
14、hey are also sufficient.Let and are differentiable at .We havehere (,)u x y(,)v x y00(,)xy12uuuxyxyxy 34vvvxyxyxy 0000lim0lim(i)(1,2,3,4)kkxxyyxyk So that()()wf zzf z iuv (i)(i)uvuvxyxxyy 1234(i)(i)xy i.e.So we have 1324i(i)(i)wuvxyzxxzz|1,|1xyzz00()limizwuvfzzxx Theorem 2.2.1 Let on a domain ,is di
15、fferentiable at Bath and are differentiable at and satisfy that and at()(,)i(,)f zu x yv x yD()f z000(,)zxyD(,)u x y(,)v x y00(,)xyuvxyuvyx 00(,)xy Theorem 2.2.2 Let is analytic on Both and are differentiable on and satisfy that and on .()(,)i(,)f zu x yv x yD(,)u x y(,)v x yDuvxyuvyx D Thus,if ,and
16、 exist,are continuous on ,and satisfy the C.-R equations,then is analytic on .uxuyvxvyDfD Corollary:If does exist,then 0()fz0()iiuvvufzxxyyiiuuvvxyyx EX.1.Show that satisfies the C.-R equations but is not differentiable at .()|f zxy0z Solution:Then So satisfies the C.-R equations.But(,)|,|u x yx y(,
17、)0v x y(,)(0,0)0|00|lim0 x yxxuxx (,)(0,0)00|0|lim0 x yyyuyy ()f z00 0 0|0lim01 i 0(1 i)zyxxyyyzxyy EX.2.Determine whether is analytic on .()Ref zzz Solution:So They are continuous on but they satisfy the C.-R equations only at .So that is differentiable at ,is not analytic on .2(,),(,)u x yxv x yxy
18、2,0,uuvvxyxxyxy0z()f z0z We can also express the C.-R equations in terms of polar coordinates,but care must be exercised because the change of coordinates defined by and is differentiable change only if is restricted to the open interval or any other open interval of length and if the origin is omit
19、ted.Without such a restriction is discontinuous because it jumps by on crossing the x-axis.22rxyarg(i)xy(,)2(0)r 2Solution 1cossin,cossinuuuvvvrxyryxsincos,(cossin)uuuvvvrrrrxyrxy EX.3 Using ,we easily see that the Cauchy-Riemann equations are equivalent to saying that cosxrsinyr11,.uvvurrrr cos,sin
20、.xryr EX.4.Define the symbol .Show that the C.-R equations are equivalent to (It is sometimes said,because of this result,that analytic functions are not functions of but of alone.This statement should be taken only as a rough guide.Since is not really a derivative of with to but merely a shorthand
21、notation for ).called complex derivative.12()2ifffzxy0fzzzfzfz()/2iffxy1()2ifffzxySolution:1()2ifffzxy1(i)2iuvuvxxyy.condition0CR EX.5.Let is analytic on the domain ,and ,Then is a constant on .()if zuvD2vu()f zDProof:So that both and are constant .2vu.condition202220CRxyxxyyxyuuuvuuvuuuuu2(1 4)0000
22、,0yyxxyu uuuvvuv()constantf z2.4 Elementary Functions The trigonometric functions sine and cosine,as well as the exponential function and the logarithmic function,are covered in elementary calculus.Let us recall that the trigonometric functions are definded in terms of the ratios of sides of a right
23、-angled triangle.The definition of“angle may be extended to include any real value,and thus and become real-valued functions of the real variable .It is a basic mathematical fact that and are differentiable with derivatives given by and .Alternatively,and can be defined by their power series:cossinc
24、ossincossinddsincosddcossin Convergence must also be proved:such a proof can be found in any calculus text.Alternatively,can be defined as the unique solution to the differential equation ,and satisfying ;and can be defined as the unique solution to .3524sin,cos13!5!2!4!xxxxxxx sinx()f x()()0fxf x(0
25、)0,(0)1ffcosx()()0,(0)1,(0)0fxf xff The exponential function,denoted ,may be defined as the unique solution to the differential equation ,subject to the initial condition that .The exponential function,can also be defined by its power series:xe()()fxf x(0)1f231/2!/3!xexxx In this section these funct
26、ions will be extended to the complex plane.The extension should be natural in that the familiar properties of ,and are retained.sin cosexpLn1.Exponential Function We know from elementary calculus that for real ,can be represented by its Maclaurin series:Thus it would be most natural to define by for
27、 xxe2311!2!3!xxxxe iye2(i)(i)11!2!yyyof course,this definition is not quite legitimate as convergence of series in has not yet been discussed.Chapter 4 will show that this series does indeed represent a well-defined complex number for each given ,but for the moment the series is used informally as t
28、he basis for the definition that follows,which will be precise.A slight rearrangement of the series shows that But we recognize this as being simply .y2435i(1)i()2!4!3!5!yyyyyeycosisinyySo we define icosisinyeyy If ,we define izxy(cosisin).zxeeyy Note that if is real(that is,if ),this definition agr
29、ees with the usual exponential function .Some of the important properties of :z0y xeze 1)for all .Let and .Then,by our definition of ,1212zzzze ee12,z z 111izxy222izxyze12121122(cosisin)(cosisin)zzxxe eeyy eyy121212cos()isin()xxeyyyy12zze 3)is periodic;the period for is .Suppose that for all .Settin
30、g we get .If .Hence any period is of the form ,.Suppose that ,that is .Then ,and so for some integer .zeze2 iz wzeez0z 1we iwst|10swees itti1te cosisin1ttcos1t sin0t 2tkkzez 2)is never zero For any ,we have since we know that the usual exponential satisfies .Thus can never be zero,because if it were
31、,then would be zero,which is not true.1zze ee1e zezzee 4)is analytic on and .By definition,So ,and Thus and ,so the C.-R equations hold,and and are continuous hence is analytic.Also,since zezzdeedz()(cosisin)xf zeyy(,)cosxu x yey(,)sinxv x yeycos,sinxxuueyeyxy sin,cosxxvveyeyxyuvxyuvyx,uuvxyxvyzei(c
32、osisin).xzdfuveyyedzxx 5)Let denote the set of complex numberssuch that ;symbolically,Then maps in a one-to-one manner onto the set .0Ayixy002yyy000i|and2 Ayxy xyyyze0Ay0 In fact,If ,then ,and so for some integer .But because and both lie in where the difference between the imaginary parts of any po
33、ints is less that ,we have .This argument shows that is one-to-one.Let with ,We claim the equation has a solution in .The equation 12zzee121zze122 izzkk1z2z0Ay212zzzew0wzewz0Ay is then equivalent to the two equations and ,.This is merely .Figure.2.2 Particular,is one-to-one manner onto the set .ixye
34、w|xewi|ywewln|xw00,2yyyarg wi,0,zzxyye|Im0zz yx02y0y0uvzwe02.The trigonometric functions Using Euler formulation We get and icosisinyeyyiisin2iyyeeyiicos2yyeey Since is now defined for any ,we are led to formulate the following definition:for any complex number .izeziiiisin,cos2i2zzzzeeeezzzFigure.2
35、.3yx0uv0zwei Again if is real,these definitions agree with the usual definitions of sine and cosine.z By the definitions,we have(1)and are analytic on sinzcosz(sin)cos;(cos)sinzzzz(2)and is periodic;the period is .sinzcosz2(3)is an even function;and is an odd function.coszsinz(4)Their formulas hold
36、on 22sincos1,zzsin()cos2zz121221sin()sincossincoszzzzzz(5)As As (),sin0zkkz1(),2zkcos0z (6)and do not hold for some .EX.|sin|1x|cos|1x zcosi22yyyyeeey Note that in exponential form,the polar representation of a complex number becomes ,which is sometimes abbreviated to .iarg|zzz eizre EX.1.Find the r
37、eal and imaginary parts of expze Solution:Let ;then .Thus .Hence and .izxycosisinzxxeeyeycosexp()cos(sin)isin(sin)xzeyxxeeeyeycosReexp()()cos(sin)xzeyxeeeycosImexp()sin(sin)xzeyxeeey3.The logarithmic function The logarithmic function is the inverse function of the exponential function.satisfies ,is
38、called the logarithmic of ,noted .Because is periodic,must be multiple valued.i,0zxyz wwzewzLnwzzeLnwzLet So We derive that and i,andizrewuviiuvereuer2vkkSo i.e.ln(2)iwrkLnzln|iArgwzz Where is the argument of .is its principal value.Arg2zkz argzLnzln|iargi2wzzkk If we fix a ,we will get a branch.0k
39、We chose the principal branch as followlnln|iargwzzz is analytic on lnz:|0,and0Dz xy111(ln)()wwzeez and1 212LnLnLnz zzz4.Power function Let and ,we define“raised to the power ”.has distinct values,these values differ by factors of the form .iabi,0zxyzzLnzze(lni2)zkelni2zkeezi2kxeSo if ,we haveIt sho
40、ws has infinitely values.0b i22i2kkbkaeee wz If .0,baln|i(arg2)(arg2)i|azzkaaazkwzeeze|ww wz 1)As is single valued.annwzz 2)As is in its lowest terms As it has distinct values,then it has exactly distinct values.paq(arg2)i|zkpqwzze0,1,1kqq 3)As is irrational.If (is irrational).It has infinity values
41、.ai2i2i2()1()naman m aeeea nmmn a If is an integer we know that is entire(with derivative )namely is analytic on .z1zz But in general,is analytic only on the domain of .Namely.zlnzlnln1()()(ln)zzzeezz112eechi 22 i2iee22(cos1 isin1)(cos1 isin1)2ieech2sin1 ish2cos1 EX.1.cosisin(12i)Find all their valu
42、es.EX.2.Ln(1)ln(1)lniln(34i)EX.3.21Lni(21)ikln1i2ikln|i|iargii2ki22kii24ln5iarctani23kcos(2 2)isin(2 2)kk2Ln12ikeeiLn(1)i(21)ikee(21)ke(2)2kei(i 2i)iLni2keeii1 i2i(1)(1 i)(ln2 2i)ke(ln2 2)(ln2 2)kike22cos(ln2)isin(ln2)kek(1 i)Ln2e EX4.Differentiate the following functions,giving the appropriate doma
43、in on which the functions are analytic:a.,b.,c.,d.,e.,f.,g.zeesin()ze2/(3)zez 1ze cosz1/(1)ze ln(1)ze Solution:a.on()zzeezeeeb.on(sin()cos()zzzeee c.on .2222(3)2()3(3)zzzee zzezz3i d.|1zze Choose the branch of the function that is analytic on .Then we must choose the domain .Such thatif ,then is not
44、 both real and .Notice that is real iff for some integer .When is even,is positive;when is odd,is negative.Here ,where and iff .So if we define ,running through positive and negative integers and zero .Then is real and iff .Since is entire,it is certainly analytic on .wwi|0,0 xy yxA1ze Imiyzkkkzekze
45、|zxeeRexz1xe 0 x|i|0,(21)Axy xykk1ze 0zA1ze A(1)21zzzeeezAze1 e.So and .Thus ,.If were analytic,which would occur iff (that is,if ,or if ).Thus there is not an(nonempty)open set on which is analytic.|coszzcoscos(i)cos cos(i)sin sin(i)zxyxyxycos coshisin sinh.xyxy(,)cos cosu x yxhy(,)sin sinv x yxhys
46、in cosuxhyx sin cosvxhyycoszuvxysin0 x 0 x,1,2,xkk A|coszz f.we conclude that is analytic on the set on which ;Namely,the set ,.|1/1zze|1/1zze10ze 2iAzk21()1(1)zzzeee z g.Since(the principal branch of)the is defined and analytic on the same domain as the square root,namely,we can use the result of d).|ln(1)zzei|0,0Axy yxln(1)1zzzeeeln
