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1, f(z)ex:i) f(z);ii) f '(z)=f(z)iii) Im(z)=0, f(z)=ex, x=Re(z)

1, f(z)=ex(cos y+i sin y), f '(z)=f(z), y=0, f(z)=ex. f(z). exp z=ex(cos y+isin y).(2.3.1): |exp z|=ex,Arg(exp z)=y+2kp(2.3.2)

(2.3.2)exp z0.ex, exp z:exp z1exp z2 = exp(z1+z2)(2.3.3)

, z1=x1+iy1, z2=x2+iy2,

exp ziii), , , ezexp z. , ez, exp z, ez=ex(cos y+isin y)(2.3.4), x=0, eiy=cos y+isin y(2.3.5)

, exp z, 2kpi, ez+2kpi=eze2kpi=ezk.

2. . ew=z(z0)w=f(z). w=u+iv, z=reiq, eu+iv=reiq,u=ln r, v=q.w=ln|z|+iArg z

Arg z, w=f(z), 2pi,Ln z=ln|z|+iArg z(2.3.6)

Ln z=ln|z|+iArg z(2.3.6)Arg zarg z, Ln z, ln z, Ln z, ln z = ln|z|+iarg z(2.3.7)Ln z=ln z+2kpi(k=1,2,...)(2.3.8). k, (2.3.8), Ln z., z=x>0, Ln zln z=ln x, .

1 Ln 2, Ln(-1).[] Ln 2=ln 2+2kpi, ln2. Ln(-1)=ln 1+iArg(-1)=(2k+1)pi(k), ln(-1)=pi., , . .

, . :

. ln z, ln|z|, arg z. z=x+iy, x

, ln z. (2.3.8), Ln z, .

Ln z, .

3. ab , a, b, abab=eblna, . a0, b, abebLna, ab=ebLn a(2.3.9)Ln a=ln|a|+i(arg a+2kp), ab. b, ab=ebLna=eb[ln|a|+i(arg a+2kp)]=eb(ln|a|+iarg a)+2kbpi=eblna,ab.

b=p/q(pq, q>0), abq, k=0,1,...,(q-1)., ab.

zn, (zn)'=nzn-1.

4. (2.3.5)eiy=cos y+isin ye-iy=cos y-isin y, , z, (2.3.12).

ez2pi, cos zsin z2p, cos(z+2p)=cos z,sin(z+2p)=sin z.cos z:cos(-z)=cos zsin z:sin(-z)=-sin z(cos z)'=-sin z, (sin z)'=cos z(2.3.13), eiz=cos z+isin z(2.3.14), , .

cos(x+iy)=cosxcosiy-sinxsiniy, sin(x+iy)=sinxcosiy+cosxsiniy.ziy,

cos zsin z.y, |siniy||cosiy|, , |sinz|1|cosz|1. :

, ,.chzshz2pi, chz, shz, , :(chz)'=shz,(shz)'=chz(2.3.18) chiy=cosy, shiy=isiny(2.3.19)

5. , z=cos w,wz, w=Arccos z.

, , :

. , :.

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