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1
5=3s
1 ?¨?
?lw
1
2
11
1)( ) ',( ) 2 '
nn
znzn
zz

==?


^??
  b
￡ 1
122 1 1
00
()
()'lim lim( )
nn
nn
n
zz
zz z
zCzz
z

→?→
+
==+?+?=
"
2
2
00
11
111
'lim lim
()
zz
zzz
zzzzz
→?→

+?
==?=?

+?

2/
f
) V?$)3$

1 () yxzf i
2
= 
2
33
() 2 3 if zxy=+

3
() yxxyzf
22
i+=

4 () sin ch icosshf zxyxy= +
3 
1?? 1,0,0,2?=
=
=
=
y
v
x
v
y
u
x
x
u
 z
ü
)) ?? O? O??
2
1
=x
H u,v?
@ C-RHq# () yxvuzf ii?=+= ?
°L
2
1
=x
 V? z
ü
))?3b

2??
2
6
u
x
x
=
 0
u
y
=
 0
v
x
=

2
9
v
y
y
=
 z
ü
)) ?? O? O??
22
23,2 30xy xy= ±='
H u,v ?
@ C-R Hq#
()
33
i2 3if zuvx y=+ = + ?°L 230xy± =
 V? z
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y
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22
i+= ?? 0=z
) V? z
ü
))?3b

4?? cos ch
u
x y
x
=
 sin sh
u
x y
y
=
 sin sh
v
x y
x
=?
 cos ch
v
x y
y
=
 z
ü
)) ?? O??ˉ
ü
u,v?
@ C-RHq# () sin ch icosshf zxyxy= + 
z
ü
)) V? z
ü
))?3b
3·/
f
()f z ￥3? u×i p ?
b
1
5
(1)z?  
2
3
2izz+ 
3
2
1
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4 (,0)
az b
cd
cz d
+
+
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μB??1
3 
1?? ()
4
5( 1)fz z′ =?# ( )zf  z
ü
))3b

2?? () i23
2
+=′ zzf ? ()zf  z
ü
))3b

3?? ()
()
()()
222
2
11
2
1
2
+?
=
=′
zz
z
z
z
zf
? ()zf " ?? 1±=z ?￥ z
ü
)) V?b))3 1±=z
^ ( )zf ￥ ?b
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2

4?? ()
2
()
ad bc
fz
cz d
′ =
+
? ( )zf " ? /( 0)zdcc=?≠?ˉ
ü
))3b
5ˉMf
￥ V??D3?μ
I
1?]$
f
￥3?μ
*tZE$
s
?f
3?1μ
?ZE 1 ?¨3￥?l1
B?ˉMf

0
z
^?3o
1
?

0
z # 
#×
=
^? V? 1
?f
 u× D
=
^?3o1
?
 D
=
^?
V? 2 ?¨3￥ 1Hq''c? 2?￥? ?=b
6
/
5￥?Li  è
a
üb

1 ?T ()zf 
0
z ? ??
*
1 ( )
0
zf′ ib

2 ?T ()
0
zf′ i
*
1 ()zf 
0
z ?3b

3 ?T
0
z
^ ()zf ￥ ?
*
1 ( )zf 
0
z ? V?b

4 ?T
0
z
^ ()zf ()gz￥B? ?
*
1
0
z 9
^ ( ) ()f zgz+ ( )/()f zgz￥ ?b

5 ?T (,)uxy (,)vxy V?
·
ê?
i 
*
1 () if zuv= + g V?b

6
! () if zuv=+ u×
=
^3￥b ?T u
^
Lè

*
1 ()f z ?? D
=
^è
 ?
T v
^
Lè

*
1 ()f z ?? D
=
^è

3

1
5Lb ?f
()
222
|| yxzzf +==  z
ü
)) ??"
? z=0?))? V?b

2
5L ?f
()
2
|| zzf = ? z=0) V? ?? z=0)?3b

3
5L ?T
00
() ()fz z z fz ??35 ?1 ￥ ?b ?
 èb

4
5L ? () sin ch,() icos shf zxygz xy== (/2,0)z π= 1
ì￥ ???
^ () ()f zgz+ ￥ ?b

5
5Lb ?f
() xyxzzzf iRe
2
+== ?? z=0)
@ C-RHq# ()zf ?? z=0
) V?b

6
5?b? u
^
Lè
? C-RZ?? v9
^
Lè
# ()f z ?? D
=
^è

a
] ? V¤b
7 ?T () vuzf i+=
^ z￥3f
￡
ü
() () ()
2
2
2
|'||||| zfzf
y
zf
x
=
+?
￡ ()
22
|| vuzf += ?
^
()f z  D
u×
=3
()f z 
0
z 3
()f z  D
= V?
()f z 
0
z V?
()f z 
0
z ??
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3
()
22
||
vu
x
v
v
x
u
u
zf
x
+
+
=
 ()
22
||
vu
y
v
v
y
u
u
zf
y
+
+
=
?? () vuzf i+= 13f
#
y
v
x
u
=

x
v
y
u
=

V7
() ()
+?
+
=
+?
2
2
2
2
22
2
2
1
||||
x
v
u
x
u
u
vu
zf
y
zf
x
+
+?
+?
+
x
u
x
v
uv
x
v
x
u
uv
x
v
v
x
u
v 22
2
2
2
2
()() ()
2222
22
22
2
22
2
22
||||
1
1
zfzfvu
vu
x
v
x
u
v
x
v
x
u
u
vu
=+
+
=
+?
+
+?
+
=
9￡
ü O - ó
￡Z?￥US?
T
^
θ?
=
v
rr
u 1

θ?
=
u
rr
v 1
￡
7 θθ sin,cos ryrx ==  ?¨ˉf
p?E5 vu,
@ C-RHq¤
θθ sincos
y
u
x
u
r
u
+
=
()
r
u
rr
x
u
r
y
u
r
y
v
r
x
vv
=
+
=
+?
=
θθθθ
θ
cossincossin
'
θ?
=
v
rr
u 1
b?
() θθ
θ
cossin r
y
u
r
x
uu
+?
=
θθθθ sincossincos
x
u
y
u
y
v
x
v
r
v
+
=
+
=
θ
θθ
=
=
u
r
r
x
u
r
y
u
r
1
sincos
1
9-μ
θ?
=
v
rr
u 1

θ?
=
u
rr
v 1
b
10￡
ü ?Tf
() ivuzf +=  u× D
=3i
@/
Hq-B
*
1 ()zf
^è
b

1 ()zf |
L′b

2 ()zf  D
=3b

3 ()|| zf  D
=
^B?è
b

4 ()zfarg  D
=
^B?è
b

5 cbvau =+  ? aabD c1? ?1
,￥
Lè
b
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4
3 
1 ? ()zf |
L′5 0=v ?? ( )zf  u× D
=3? C-RHq? ??
^
0=
=
y
v
x
u
 0=
=
x
v
y
u
# u u× D
=1Bè
,( )
Lè
Cu = 5 ( ) Civuzf =+= 1Bè
b

2 ? () ivuivuzf?=+=  u× D
=35
( )
y
v
y
v
x
u
=

=

( )
x
u
x
v
y
u
=

=

1
? () ivuzf +=  u× D
=35
y
v
x
u
=

x
v
y
u
=

2
2
1 a
2

Tμ
0=
=
=
=
vy
v
x
v
y
u
x
u

# vu, D
= (1è
sY:-1
( )1
Lè
212211
,,CCCuCu == 
5
( ) CiCCivuzf =+=+=
21
1Bˉè
b

3 ? ()|| zf  D
=1Bè
:1
1
C 5
2
1
22
Cvu =+ 
HsY? x y p
ê?¤
=
+
=
+
022
022
y
v
v
y
u
u
x
v
v
x
u
u
?? ()zf  D
=3
@ C-RHq
x
v
y
u
y
v
x
u
=
=
,} ?

T? V¤
=
+
=
0
0
y
u
u
x
u
v
y
u
v
x
u
u
3¤ 0=
=
y
v
x
u
b] ? V3¤ 0=
=
vy
v
x
v
# vu, (1è
sY:1
21
,CvCu == 5
() CiCCivuzf =+=+=
21
1Bˉè
b

4 ? zarg  D
=
^B?è
1
C 5 ( ) 0≠zf V7 ( ) 0≠+= ivuzf  O
()
<<?
><+
>
=
0,0,arctan
0,0,arctan
0,arctan
arg
vu
u
v
vu
u
v
u
u
v
zf
π
π
<<?
><+
>
=
0,0
0,0
1
1
1
1
vuC
vuC
uC
π
π
9- ()zfarg sY1? x y p
ê?¤
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5
0
1
1
222
2
=
+
=
+
vu
x
u
v
x
v
u
u
v
x
u
v
x
v
u
u
0
1
1
222
2
=
+
=
+
vu
y
u
v
y
v
u
u
v
y
u
v
y
v
u
u
e

Ti ?¨ ()zf 3 
La′?
@ C-RHq¤
=
=
0
0
y
u
v
x
u
u
y
u
u
x
u
v
3¤ 0=
=
y
u
x
u
] ?9 V p¤ 0=
=
y
v
x
u
' u v (1
Lè
sY:1
2
C
3
C V7
() CiCCivuzf =+=+=
32
1Bˉè
b

5 ? cbvau =+  ? aab c1? ?1
,￥
Lè
 ? ú a b? ?1 0 ' 0
22
≠+ba 
?5N
H aab c ?1
,bZ? cbvau =+ sY? x y p
ê?¤
=
+
=
+
0
0
y
v
b
y
u
a
x
v
b
x
u
a
 ?¨3f
() ivuzf += ￥
La′? u v
@ C-RHq¤
=
+
=
0
0
y
u
a
x
u
b
y
u
b
x
u
a
3¤ 0=
=
y
u
x
u
] ?9 V p¤ 0=
=
y
v
x
v
?f
( )zf 1Bè
b
11./
1"
^?? ?$

1
zz
ee =  
2 zz coscos =  
3 zz sinsin =
3 
1
zyxxxz
eeyyeyyee ==?=+=
i
)sini(cos)sini(cos

2 ( ) ()zeeee
ee
z
zzzz
zz
cos
2
1
2
1
2
cos
iiii
ii
=+=+=
+
=

b

3 ()() )(
i2
1
i2
1
i2
1
sin
iiiiii zzzzzz
eeeeeez?
=?=?=

= ( ) zee
zz
sin
i2
1
ii
=?
b
12s/
Z?￥ ??3b

3 01 =+
z
e  
4 0cossin =+ zz 
3 
3eZ??N? 1?=
z
e ?
^
￥31
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6
( ) ( )[ ] ( )kkz 21i21argi|1|ln1Ln +=+?+?=?= ππ ",2,1,0 ±±=k

4??
()
ii
ii
1
sin cos,
2i 2
zz
zz
ee
zz ee
=? =? + #
( )1i1
i2i2
+?=?
zz
ee
i1
i1
i2
+
=
z
e
() ()[]πkz 2iargi|i|ln
i2
1
iLn
i2
1
i1
i1
Ln
i2
1
+?+?=?=?
+
=
",2,1,0,
4
1
2
2i2
i
±±=?
=?
+?= kkk ππ
π
13￡
ü

1 ()
212121
sinsincoscoscos zzzzzz?=+ ;
()
212121
sincoscossinsin zzzzzz?=+ ;

2 1cossin
22
=+ zz  
3 zzz cossin22sin =  
4
z
z
z
2
tan1
tan2
2tan
= 

5 zz cos
2
sin =?
π
 () zz coscos?=+π 

6
22 2 22 2
| cos | cos sh,| sin | sin shzxyzxy=+ =+
￡ 
1P ()
() ()
[]
2121
2
1
cos
21
zzizzi
eezz
+?+
+=+=
· =
2121
sinsincoscos zzzz?
i2i222
22112211
iiiiiiii zzzzzzzz
eeeeeeee


++
=
() () () ( ) ( ) ( ) ( )()
4
2121212121212121
iiiiiiii zzzzzzzzzzzzzzzz
eeeeeeee
++++
+++++
=
=
() ()
2
2121
ii zzzz
ee
+?+
+
VnP =·' ( )
212121
sinsincoscoscos zzzzzz?=+ 
P = ()
() ()
[ ]
2121
ii
21
i2
1
sin
zzzz
eezz
+?+
=+
·
2121
sincoscossin zzzz +=
()( ) ()( )
22112211
iiiiiiii
i2
1
2
1
2
1
i2
1
zzzzzzzz
eeeeeeee

++=
() () () ()
[]
21212121
iiii
i4
1
zzzzzzzz
eeee
++
+=
()() () ()
[ ]
21212121
iiii
i4
1
zzzzzzzz
eeee
++
+?+
=
() ()
[]
2121
ii
22
i4
1
zzzz
ee
+?+
i2
1
=
() ( )
[ ]
2121
ii zzzz
ee
+?+
VnP =·' ()
212121
sincoscossinsin zzzzzz +=+

2
2
ii
2
ii
22
i2i2
cossin
+
+

=+
zzzz
eeee
zz
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7
()( ) 12
4
1
2
4
1
i2i2i2i2
=++++=
zzzz
eeee

3P = ( )
zz
eez
2i2i
i2
1
2sin
=
· = ()()
zzzz
eeeezz
iiii
2
1
i2
1
2cossin2

+?=
()( )
zzzz
eeee
2i2i2i2i
i2
1
11
i2
1

=+=
VnP =·' zzz sincos22sin = b

4
z
z
z
z
z
z
zz
zz
z
z
z
2
2
22
tan1
tan2
cos
sin
1
cos
sin
2
sincos
cossin2
2cos
2sin
2tan
=
=
==

5?
1?
() () ()zzzz?+?=
+=?
sin
2
coscos
2
sin
2
sin
2
sin
ππππ
()
() ()
()
izizzizi
eeeez

+=+=?=
2
1
2
1
cos
zcos=
?
1¤ () zzzz cossinsincoscoscos?=?=+ πππ

6P"
2222
|cos | |cos ch isin sh | cos ch sin shzxyxy xyxy=? = +
"
222222
cos (1 sh ) sin sh cos shx yxyxy++ =+
P" | sin | | sin ch i cos sh | sin ch cos shzxyxy xyxy=+ = +
"
22 2222
sin (1 sh ) cos sh sin shx yxyxy++ =+b
14
a
ü 1? y →∞
H |sin( i )|x y+ |cos( i )|x y+ t?í kv
2? t1ˉ
H |sin | 1t ≤ |cos | 1t ≤ ?? ?b
3 1
i-i
||
|sin | | |
2i 2
zz yy
ee e e
z

=≥  |cos |z ] ?b
2
! i,tyyR=∈5
||
|sin |
2
yy
ee
t
= 5? y →∞
HA ?5
!?? ?b
15 p ()iLn?  ( )i43Ln +?
ì￥?′b
3 () ()?
+?=+?+?=? π
π
π kk 2
2
i2iargi|i|LniLn
",2,1,0,
2
1
2i ±±=?
= kkπ
() ()
2
i
iargi|i|lniln
π
=?+?=?
() ( )[]πk2i43argi|i43|lni43Ln ++?++?=+?
+?
+= ππ k2
3
4
arctani5ln
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8
()",2,1,0,12
3
4
arctani5ln ±±=
+= kk π
() ()?
4
+=+?++?=+?
3
arctani5lni43argi|i43|lni43ln π b
16￡
ü
￥/
?é 1
12 1 2
Ln( ) Ln Lnzz z z= +  2
12 1 2
Ln( / ) Ln Lnzz z z=?b
￡
ü 1
12 12 12 1 2 1 2
Ln( ) ln(| |) i Arg ln ln i Arg +i Argzz zz zz z z z z= +=++
12
Ln Lnzz=+
2
12 12 12 1 2 1 2
Ln( / ) ln(| / |) iArg / ln ln iArg -iArgzz zz zz z z z z=+=?+
12
Ln Lnzz=?b
17
a
ü/
?
T
^?? ? 1
2
Ln 2 Lnzz=  2
1
Ln Ln
2
zz= b
3

T (?? ?b 1
2
Ln 2ln| | iArg(2 ),2Ln 2ln| | 2iArg( )zz zzz z=+ +7" 
2
111i
Ln ln | | i Arg( ),Ln ln | | Arg( )
2222
zz z zz z=+ +7" b
18 p
2
i1
π
e 
1i
exp
4
π





i
3 ( )
i
i1+ ￥′b
3
eeeee i
2
sini
2
cos
2
i
2
i1
=?
==

ππ
ππ
()
11 1
i
44 4 4
1i 2
exp cos isin 1 i
444
ee e e
π
πππ
== =




( )iln3iarg3 2iiLn3 2iln3
3
k
k
ee ee
π
π
++

== = ( )",2,1,0,3lnsini3lncos
2
±±=+=
ke
kπ
()
() [ ] ( )( )
i iln|1 i| iarg1 i 2
iLn 1 i
1i
k
ee
π++ +++
+= =
+==
+
+?
2
2ln
sini
2
2ln
cos
2
4
1
2
42
2ln
i kk
ee
ππ
π
 ",2,1,0 ±±=k
19￡
ü
1
()'
aa
zaz
=  ? a1
L
b
￡
ü
ln 2 ln 2 1
1
()'( )' (ln)'
a azki azki a a
ze aze azz
z
ππ++?
== ==b
20￡
ü 1
22
ch sh 1zz? =  2
22
sh ch ch 2zz z+=
3 ()
12 1 2 1 2
sh sh ch ch shzz z z z z+= + 
( )
12 1 2 1 2
ch ch ch sh shzz z z z z+= + b
￡
ü 1
22 2 2
ch sh ( ) ( ) 1
22
zz zz
ee ee
zz

+?
=? =
2
22
22 2 2
sh ch ( ) ( ) 1
222
zz zz z z
ee ee e e
zz

++
+= + = =
3
112 2 112 2 1212
12 12
()( )()( )
sh ch ch sh
44
zzzz zzzz zzzz
eeee eeee e e
zz zz
+?
+ +
+= + =
()
12
sh zz=+b
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Z? 1 sh 0z =  2 ch 0z =  3 sh iz = b
3 1? sh 0z = ¤
2
1
z
e = 
1
Ln1 i,0,1,2,
2
zkkπ===±±"b
2? ch 0z = ¤
2
1
z
e =? 
1(21)
Ln(1) i,0,1,2,
22
k
π
+
=?= =±±"b
3? sh iz = ¤ i
z
e = 
1
Ln i i(2 ),0,1,2,
2
zkkπ==+ =±±"b
23￡
ü,sh z￥Qf
2
Arsh Ln( 1)zzz=++b
￡
! sh wz= 
2
210
2
ww
ww
ee
ze ze
==' 3¤
2
1
w
ezz= ++,
#
2
Arsh Ln( 1)wzzz==++b
24X?
ü
@
?￥ˉ
] ()zf 1

1 ()
2
i+z  
2
3
z  
3
1
1
2
+z

p
@?￥
[#
@L?
]L￥Z?b
3 
1 () () ()()i2i2'?=+== zzzfzV 1
@
?
() ( ) ( )[ ] ( ) ( )12i11ii
2222
+++?=++=+= yxyxyxzzf
?
@L?
]LZ?sY1 ()
1
1 Cyx =+ ( )
2
22
1 Cyx =+? b

2
@
() ()
22
33' zzzfzV === ? ( ) ( ) ( )
22223
3i3 yxyyxxzzf?+?==,
@LZ? ()
1
22
3 Cyyx =?  ?
]LZ? ( )
2
22
3 Cyxx =? b

3
@
() ()
()
2
2
22
1
2
'
1
2
'
1
1
'
+
=?
+
=?
+
==
z
z
z
z
z
zfzV
? ()
()
2222
22
222
41
2i1
2i1
1
1
1
yxyx
xyyx
xyyxz
zf
++?
+?
=
++?
=
+
= 
@LZ?1
()
1
22
2
22
41
C
yxyx
xy
=
++?

?
]LZ?1
()
2
2222
22
41
1
C
yxyx
yx
=
++?
+?
b