Chapter2:Analytic FunctionsshaUni.ofSci&TechFCV&ITNovember5.20193/45angWan
Chapter 2: Analytic Functions Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 3 / 45
s2.1 The concept of analytic functionshaUni.ofSci&TechFCV&ITNovember5.20194/45angWan
§2.1 The concept of analytic function Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 4 / 45
DerivatrveofcexfunctionsDerivative of complex functionsDefinition(2.1.1)Let f be a function whose dmain of definition contains a neighborhood ofa point zo. The derivative of f at zo, written f'(zo), is defined by theequationf(z) -f(zo)f'(z0) = limZ-202→20dfprovided this limit exists.This limit is denoted sometimes by(zo). Thedzfunction f is said to be differentiable at zo when its derivative at zo existsFCV&IThaUni.of Sci&Tech)November5,20195/45angWanTch
Derivative of complex functions Derivative of complex functions Definition (2.1.1) Let f be a function whose dmain of definition contains a neighborhood of a point z0. The derivative of f at z0, written f 0 (z0), is defined by the equation f 0 (z0) = limz→z0 f(z) − f(z0) z − z0 provided this limit exists. This limit is denoted sometimes by df dz (z0). The function f is said to be differentiable at z0 when its derivative at z0 exists. Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 5 / 45
Derivative of complex functions(z) = - 20thenf(20 +z) -f(z0)f'(zo) = lim△zA-→0or△w = f(z+z) - f(2)wdwlimf'(z) :dzA2-0zExample(2.1.1)Supposethatf(z)=z2.AtanypointzE CAw(z + z)2 - z2limlimlim(2z+△z)=2zAz42→0△z△2→0A2-Hencedw/dz=2zorf(z)=2zFCV&ITNovember 5, 2019FangWan(Chanegsha Uni. of Sci & Tech)6/45
Derivative of complex functions ∆(z) = z − z0 then f 0 (z0) = lim ∆z→0 f(z0 + ∆z) − f(z0) ∆z or ∆w = f(z + ∆z) − f(z) f 0 (z) = dw dz = lim ∆z→0 ∆w ∆z Example (2.1.1) Suppose that f(z) = z 2 . At any point z ∈ C, lim ∆z→0 ∆w ∆z = lim ∆z→0 (z + ∆z) 2 − z 2 ∆z = lim ∆z→0 (2z + ∆z) = 2z. Hence dw/dz = 2z or f 0 (z) = 2z. Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 6 / 45
DerivatrveofcomplexfunctionsExample(2.1.2)Showthat f(z)=z is not differentiableonCSolution.See the figure, Vzo E C,f(z) - f(zo)( - ro) - i(y - yo)i(r -to) +i(y -yo)iZ-201Ar→0,Ay = 0-1 4r=0.4y→0tyZoaoFCV&ITFangWanshaUni.of Sci&Tech)November 5.20197/45
Derivative of complex functions Example (2.1.2) Show that f(z) = ¯z is not differentiable on C. Solution.See the figure, ∀z0 ∈ C, f(z) − f(z0) z − z0 = (x − x0) − i(y − y0)i (x − x0) + i(y − y0)i = ( 1 ∆x → 0, ∆y = 0 − 1 ∆x = 0, ∆y → 0 O x y z0 Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 7 / 45
DerivatrveofconnplexfunctionExample(2.1.3)Considerthefunctionf(z)=z/2,whereAwf(z+z) -f(z)△zAz△z()()△(z+△z)+ (z+z)三艺十zAzWe conclude that dw/dz exists only at z=O, its valuetherebeing 0.Itis, however, true that if f'(zo)exists, then f is continuous at zo.FCV&ITha Uni. of Sci &Tech)November5.20198/45angWangCha
Derivative of complex functions Example (2.1.3) Consider the function f(z) = |z| 2 , where ∆w ∆z = f(z + ∆z) − f(z) ∆z = |z + ∆z| 2 − |z| 2 ∆z = (z + ∆z)(¯z + ∆z) − zz¯ ∆z = ¯z + z · (z + ∆z) ∆z + (z + ∆z). We conclude that dw/dz exists only at z = 0, its value there being 0. It is, however, true that if f 0 (z0) exists, then f is continuous at z0. Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 8 / 45
Supposethef andgaredifferentiableatzoEC.Then(0) af +bg is differentiable at zo and(af+bg)(z)=af'(z)+bg(z)for any complex numbers a and b.(ii) fg is diffentiableat zo and(fg)(z)=f'(z)g(z)+f(z)g(z)(ii)Ifg(z)0forall zEACC,thenf/gisdifferentiableat zoand() = I(2)g() -g(2)f(2)g2(z)(iv)Anypolynomial ao+aiz+..+anzn is differentiableon Cwithderivativeai+2a2z+.+nanzn-1(v) Suppose that f has a derivative at zo and that g hs derivative at thepoint f(zo).Then the function F(z)=g[f(z)) has a derivativeatz0,andF'(zo) = g'[f(zo)]f'(zo)FCV&ITNovember 5, 20199/45aUni.of Sci&Tech)
Derivative of complex functions Suppose the f and g are differentiable at z0 ∈ C. Then (i) af + bg is differentiable at z0 and (af + bg) 0 (z) = af0 (z) + bg0 (z) for any complex numbers a and b. (ii) fg is diffentiable at z0 and (fg) 0 (z) = f 0 (z)g(z) + f(z)g 0 (z) (iii) If g(z) 6= 0 for all z ∈ A ⊂ C, thenf /g is differentiable at z0 and f g 0 (z) = f 0 (z)g(z) − g 0 (z)f(z) g 2(z) (iv) Any polynomial a0 + a1z + · · · + anz n is differentiable on C with derivative a1 + 2a2z + · · · + nanz n−1 . (v) Suppose that f has a derivative at z0 and that g hs derivative at the point f(z0). Then the function F(z) = g[f(z)] has a derivative at z0, and F 0 (z0) = g 0 [f(z0)]f 0 (z0) Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 9 / 45
DerivatrveofcomplexfunctionsTheconcept of an analytic functionDefinition(analyticfunction)A function f of the complex variable zis analytic in an open set if it has aderivative at each point in that set. In particular, f is analytic at a pointzo if it is analytic throughout some neighborhood of zoExample. f(2) = ↓ is analytic at each nonzero point in the finite plane.f(z)=z/2is notanalytic atanypoint since itsderivativeexistsonlyatz=O and notthroughoutanyneighborhoodFCV&ITNovember 5, 201910/45sha Uni. of Sci & Tech)angWa
Derivative of complex functions The concept of an analytic function Definition (analytic function) A function f of the complex variable z is analytic in an open set if it has a derivative at each point in that set. In particular, f is analytic at a point z0 if it is analytic throughout some neighborhood of z0. Example f(z) = 1 z is analytic at each nonzero point in the finite plane. f(z) = |z| 2 is not analytic at any point since its derivative exists only at z = 0 and not throughout any neighborhood. Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 10 / 45
DerivatrveofcoXfunetionDefinition(singularpoint)If a function f fails to beanalytic at apoint zo but is analyticat somepoint in every neighborhood of zo,then zo is called a singular point, orsingularity of f.Example。 The point 2 = 0 is a singular point of the function f(2) = 1The points z=i, z=-i are the singular points of1+22111Thepointsz=O,±,allaresingularpointsof元—2元n1sin1/zThefunction f(z)=zj2has no singularpoints sinceit is nonwhereanalytic.FCV&ITNovember 5, 2019sha Uni. of Sci & Tech)11/45angWanslCha
Derivative of complex functions Definition (singular point) If a function f fails to be analytic at a point z0 but is analytic at some point in every neighborhood of z0, then z0 is called a singular point, or singularity of f. Example The point z = 0 is a singular point of the function f(z) = 1 z . The points z = i, z = −i are the singular points of 1 1 + z 2 . The points z = 0, ± 1 π , ± 1 2π , · · · , ± 1 nπ , · · · , all are singular points of 1 sin 1/z . The function f(z) = |z| 2 has no singular points since it is nonwhere analytic. Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 11 / 45
Derivathveofcomnlexfunctiong2.2 Necessary and sufficient conditions of analyticfunctions12/45haUni.ofSci&TechFCV&ITNovember5,2019angWa
Derivative of complex functions §2.2 Necessary and sufficient conditions of analytic functions Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 12 / 45