复变函数总课件第二章

§3

初等函数

1.

指数函数

2.

三角函数和双曲函数

3.

对数函数

4.

乘幂与幂函数

5.

反三角函数与反双曲函数1.

指数函数=

e

x

ezArge

z

=

y

+

2kp它与实变指数函数有类似的性质：(1)"

z

ez

„

0(事实上,

ez

=

e

x

„

0)(2)当z为实数

x时,

f

(

z)

=

e

z

=

e

x(

y

=

0)(3)

f

(

z

)

=

e

z

在复平面上处处解析且

(e

z

)'

=

e

z(见§2的例1(2))定义对z

=x

+iy,定义复变数z的指数函数如下:f

(z)

=

e

z

=

e

x

+iy

=

e

x

(cos

y

+

i

sin

y)

(1)1

21

22121

2

12211z

+zx

+

xx1

+

x2x

x=

e

[cos(

y1

+

y2

)

+

i

sin(

y1

+

y2

)]

=

e=右边+

i(sin

y1

cos

y2

+cos

y1

sin

y2

)]-

sin

y

sin

y=

e

[cos

y

cos

y+

i

sin

y

)+

i

sin

y

)

e

(cos

y=

e

(cos

y事实上,设zj

=

x

j

+

iy

j

(

j

=

1,2)左边=ez1

ez2=

ez1

+z2(4)加法定理:

ez1

ez2由加法定理可推得f

(z)=e

z的周期性:f

(

z

+

T

)

=

f

(

z

),T

=

2

kpi

,

k

˛

Zk为"整数.=

f

(

z

)\

T

=

2kpi=

e

z(cos

2kp

+

i

sin

2kp

)

=

e

z=

e

ze

2

kpi事实上,f

(z

+2kpi

)=e

z

+2

kpiez\e-z

=

1

这个性质是实变指数函数所没有的。又eze-z

=

ex-x

(cos(y

-

y)

+

i

sin(

y

-

y))

=

e0

1

=

12ez2ez1z1

-z

=

eex

(cos

y

+i

sin

y)，\没有幂的意义(2)特别当z的实部x

=

0时,就得Euler公式:

e

yi

=

cos

y

+

i

sin

y

(1)ez仅仅是个符号,它的定义为22i"

y

˛

R

(2)cos

y

=sin

y

=eiy

+

e-iyeiy

-

e-iyeiy

=

cos

y

+

i

sin

ye-iy

=

cos

y

-

i

sin

y从而得到:当x

=0时,由指数函数的定义:2.

三角函数和双曲函数推广到复变数情形--称为z的正弦与余弦函数e

zi

+

e-

zie

zi

-

e-

zi(3)2i

2sin

z

=

cos

z

=定义22

2=

=

cos

z)=eiz

+

e-iz(cos(z

+

2p

)

=3)在z平面上处处解析的

,

且(sin

z

)'

=

cos

z

(cos

z

)'

=

-

sin

z(事实上,

(sin

z)'

=

1

(eiz

-

e-iz

)'

=

1

(eiz

+

e-iz

)

=

cos

z)2i

2正弦与余弦函数的性质当z为实数时,显然这与(2)式完全一致sin

z及cos

z是T

=

2p周期函数ei

(

z

+2p

)

+

e-i

(

z+2p

)

eize2pi

+

e-ize-2pi4)sin

z是奇函数,cos

z是偶函数.2i(sin(

-z)

=

=

-

sin

z;e-iz

-

eize

iz=

cos

z

+

i

sin

z同理cos(-z)=cos

z)5)

由(3)式,Euler

公式对一切z成立(6)

由正弦和余弦函数定义及指数函数的加法定理可推知一些三角公式221

112

1

2+

cos

z

=

1sin

z+

cos

z

sin

zsin(

z

+

z

)

=

sin

z

cos

zcos(

z1

+

z

2

)

=

cos

z1

cos

z

2

-

sin

z1

sin

z

2sin(

x

+

iy)

=

sin

x

cos

iy

+

cos

x

sin

iycos(

x

+

iy)

=

cos

x

cos

iy

-

sin

x

sin

iy(4

)2

isin

iy

=

=

ishy=

chycos

iy

=2e

-

y

-

e

ye

-

y

+

e

y由正弦和余弦函数的定义得\

sin(

x

+

iy)

=

sin

xchy

+

i

cos

xshycos(

x

+

iy)

=

cos

xchy

-

i

sin

xshysinz1cosz1cscz

=tanz

=

sinz

cotz

=

cosz

secz

=cosz

sinz其它三角函数的定义(详见P51)=

shy

fi

¥e

-

y

-

e

y2i(8)

由(4)式知当y

fi

¥

sin

iy

=cos

iy

=

chy

fi

¥(7)sin

z的零点,即方程sin

z

=0的根为z

=kp

(k

˛

Z

)2cos

z的零点为z

=

p

+

kp

k

˛

Z\

在复数范围内

cos

z

£

1,

sin

z

£

1不再成立

.2

2ez

+

e-

zez

-

e-

zshz

=

chz

=定义—称为双曲正弦和双曲余弦函数(thz

=

shz

cthz

=

1

)chz

thz双曲正弦和双曲余弦函数的性质shz、chz都是以2pi为周期的函数chz

--偶函数,shz

--奇函数(chz

)'

=

shz

(

shz

)'

=

chzshz和chz在整个复平面内处处解析三角函数,双曲函数均是由复指数函数定义的,且是周期函数，故它的反函数一定是多值函数.4)由定义shiy

=

i

sin

y

chiy

=

cos

ych(

x

+

iy)

=

chxcos

y

+

ishxsin

y3.

对数函数(k

=

0,–1,)w

=

Lnz

=

ln

r

+

i(q

+

2pk

)(k

=

0,–1,–2,)\或

Lnz

=

ln

z

+

iArgz

=

ln

z

+

i(arg

z

+

2kp

)(1)对数的定义定义指数函数的反函数称为对数函数。即，把满足ew

=

z(z

„

0)的函数w

=

f

(z)称为对数函数,记作w

=Lnz令w

=

u

+

iv z

=

reiq

那么eu+iv

=

reiq

u

=

ln

r,

v

=

q

+

2kp

(k

˛

Z

)这说明一个复数z(z

„0)的对数仍为复数,它的实部是z的模的实自然对数；它的虚部是z的幅角的一般值,即虚部无穷多,其中"两个相异值相差2p的一个整数倍.记作lnz+

i

arg

z

=

ln

z即,

w

=

Lnz是z的无穷多值函数当k

=

0时,

Lnz

=

(2)为Lnz的一单值函数,称为Lnz的主值(主值支

)(k

˛

Z

)Lnz

=

ln

z

+

i2kp故在复数域中,正实数对数有无穷多个值,负数也有对数.例如

当z

=

a

>

0

Lnz

的主值

ln

z

=

ln

aLnz

=

ln

a

+

2pik k

˛

ZLnz

=

ln

a

+

(2k

+

1)pia

=

1 ln(

-

1

)

=

ln

1

+

pi

=

piLn

(

-

1

)

=

(

2

k

+

1

)piLnz的主值ln

z

=ln

a

+pi当z

=-a(a

>0)特别(2)

对数函数的性质（见P47)(3)

解析性主值:ln

z

=ln

z

+i

arg

z,其中

ln

z

除原点外在其它点均连

续;而arg

z在原点与负实轴上都不连续.见§1-6例1\除原点及负实轴外,ln

z在复平面内处处连续.z

=ewzdzdz1

=

1

=

1ewdw(ew

)'=ew

„0

\

dw

=

(ln

z)'=z即

(ln

z)'

=

1\w

=ln

z除原点及负实轴外是解析的.且(Lnz

)'=1Lnz的每个分支除了原点和负实轴外均是解析的，z4.

乘幂与幂函数b(a

)b(

z

)=

e

bLna

.乘幂ab定义设a,b为复数,且a

„0,定义乘幂

a

b实变数情形,a

>0,b为实数.

Lna

=

ln

a

+

i2kp—多值—一般为多值\

ab

=

ebLna

=

eb(ln

a

+i

2kp

)①当b为整数ab

=

ebLna

=

eb(lna+i

2kp

)

=

eb

ln

aebi

2kp=

eb

ln

a

(cos

2kpb

+

i

sin

2kpb)

=

eb

ln

a\b为整数时,它是单值函数.②当b

=p

(p,q为互质的整数,且q

>0)ln

a

pq

pq

pqabi

(arga+2kp

)(ln

a

+i

arga+2kpi

)q=

e

=

e

epq

qp(arg

a

+

2kp

)

+

i

sin

(arg

a

+

2kp

)]

p

ln

a=

e

q

[cos(k

=0,1,2,3,q

-1)—q支③一般而论,ab具有无穷多支.b(2)当b=1/n(n正整数)时，乘幂a

与a

的n次根意义一致。

(1)当b=n(正整数)时，乘幂ab与a的n次幂意义一致。=

e

Lnae

Lna

e

Lnaa

n

=

e

nLna

=

e

Lna

+

Lna

+

+

Lnan个=

aaaaLnaarg

a

1n

n

1n

1n

1nln

a

i+

2

kp=

e

ea

=

e

=

e(ln

a

+

i

arg

a

+

2

kpi

)n

n=

n(k

=

0,1,2n

-

1)a

(cos

arg

a

+

2kp

+

i

sin

arg

a

+

2kp

)

=

n

a2i22Ln1

=

e2

(ln

1

+2kpi

)

=

e2kp)p2p2-(

2

kp

++

2

kpi

)i

(ln

i

+

i=

e

=

ei

i

=

e

iLni2

3

222

Lnii

32

(ln

i

+ip

+2kpi

)

i

2

(

p

+2kp

)=

cos(p

+4kp

)

+

i

sin(p

+4kp

)3

3(k

=

0,1,2)=

e

3

=

e

3

=

e(

k