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