1
p269
绗叚绔犱範棰?/p>
(
涓�
) [ 7, 8, 9, 10, 11, 12, 13, 14 ]
7.
浠?/p>
飪?/p>
C
e
i
z
/
鈭?/p>
z
dz
鍑哄彂锛屽叾涓?/p>
C
鏄鍥炬墍绀轰箣鍛ㄧ嚎
(
鈭?/p>
z
娌挎瀹炶酱鍙栨鍊?/p>
)
锛岃瘉鏄庯細
飪?/p>
(0, +
飩?/p>
)
cos
x
/
鈭?/p>
x
dx
=
飪?/p>
(0, +
飩?/p>
)
sin
x
/
鈭?/p>
x
dx
=
鈭?/p>
(
飦?/p>
/2)
锛?/p>
銆愯В銆?/p>
|
飪?/p>
C
(
R
)
e
i
z
/
鈭?/p>
z
dz
|
飩?/p>
飪?/p>
C
(
R
)
| e
i
z
|/
R
1/2
ds
=
飪?/p>
[0,
飦?/p>
/2]
| e
i
飦?/p>
(cos
飦?/p>
+
i
sin
飦?/p>
)
|/
R
1/2
路
R d
飦?/p>
=
飪?/p>
[0,
飦?/p>
/2]
| e
飥?/p>
R
sin
飦?/p>
|
R
1/2
d
飦?/p>
飩?/p>
R
1/2
飪?/p>
[0,
飦?/p>
/2]
e
飥?/p>
R
sin
飦?/p>
d
飦?/p>
锛?/p>
鐢?/p>
sin
飦?/p>
飩?/p>
2
飦?/p>
/
飦?/p>
(
飦?/p>
飪?/p>
[0,
飦?/p>
/2] )
锛屾晠
R
1/2
飪?/p>
[0,
飦?/p>
/2]
e
飥?/p>
R
sin
飦?/p>
d
飦?/p>
飩?/p>
R
1/2
飪?/p>
[0,
飦?/p>
/2]
e
飥?/p>
(2
R
/
飦?/p>
)
飦?/p>
d
飦?/p>
= (
飦?/p>
/(2
R
1/2
))(1
鈥?/p>
e
飥?/p>
R
)
飩?/p>
飦?/p>
/(2
R
1/2
)
锛?/p>
鎵�浠ワ紝
|
飪?/p>
C
(
R
)
e
i
z
/
鈭?/p>
z
dz
|
飩?/p>
0 (as
R
飩?/p>
+
飩?/p>
)
锛?/p>
鑰岀敱
|
飪?/p>
C
(
r
)
e
i
z
/
鈭?/p>
z
dz
|
飩?/p>
(
飦?/p>
/(2
r
1/2
))(1
鈥?/p>
e
飥?/p>
r
)
鐭?/p>
|
飪?/p>
C
(
r
)
e
i
z
/
鈭?/p>
z
dz
|
飩?/p>
0 (as
r
飩?/p>
0
+
)
锛?/p>
褰?/p>
r
飩?/p>
0
+
锛?/p>
R
飩?/p>
+
飩?/p>
鏃讹紝
飪?/p>
[
r
,
R
]
e
i
z
/
鈭?/p>
z
dz
=
飪?/p>
[
r
,
R
]
e
i
x
/
鈭?/p>
x
dx
=
飪?/p>
[
r
,
R
]
(cos
x
+
i
sin
x
)/
鈭?/p>
x
dx
飩?/p>
飪?/p>
(0, +
飩?/p>
)
cos
x
/
鈭?/p>
x
dx
+
i
飪?/p>
(0, +
飩?/p>
)
sin
x
/
鈭?/p>
x
dx
锛?/p>
飪?/p>
[
r
i
,
R
i
]
e
i
z
/
鈭?/p>
z
dz
=
飪?/p>
[
r
,
R
]
e
i
(
i
y
)
/
鈭?/p>
(
i
y
)
i
dy
=
飪?/p>
[
r
,
R
]
e
飥?/p>
y
e
i
飦?/p>
/4
/
鈭?/p>
y
dy
锛?/p>
= (1 +
i
)/
鈭?/p>
2 路
飪?/p>
[
r
,
R
]
e
飥?/p>
y
/
鈭?/p>
y
dy
= 2(1 +
i
)/
鈭?/p>
2 路
飪?/p>
[
鈭?/p>
r
,
鈭?/p>
R
]
e
飥?/p>
u
^2
du
飩?/p>
(1 +
i
)
鈭?/p>
2 路
飪?/p>
(0, +
飩?/p>
)
e
飥?/p>
u
^2
du
= (1 +
i
)
鈭?/p>
2 路
鈭?/p>
飦?/p>
/2 = (1 +
i
)
鈭?/p>
(
飦?/p>
/2)
锛?/p>
鐢?/p>
Cauchy
绉垎瀹氱悊锛?/p>
飪?/p>
C
e
i
z
/
鈭?/p>
z
dz
= 0
锛屾晠鍏舵瀬闄愪篃涓?/p>
0
锛?/p>
鎵�浠ワ紝
飪?/p>
(0, +
飩?/p>
)
cos
x
/
鈭?/p>
x
dx
+
i
飪?/p>
(0, +
飩?/p>
)
sin
x
/
鈭?/p>
x
dx
= (1 +
i
)
鈭?/p>
(
飦?/p>
/2)
锛?/p>
鍗?/p>
飪?/p>
(0, +
飩?/p>
)
cos
x
/
鈭?/p>
x
dx
=
飪?/p>
(0, +
飩?/p>
)
sin
x
/
鈭?/p>
x
dx
=
鈭?/p>
(
飦?/p>
/2)
锛?/p>
8.
浠?/p>
飪?/p>
C
鈭?/p>
z
ln
z
/(1 +
z
)
2
dz
鍑哄彂锛屽叾涓?/p>
C
鏄鍥炬墍绀轰箣鍛ㄧ嚎锛岃瘉鏄庯細
飪?/p>
(0, +
飩?/p>
)
鈭?/p>
x
ln
x
/(1 +
x
)
2
dx
=
飦?/p>
锛?/p>
飪?/p>
(0, +
飩?/p>
)
鈭?/p>
x
/(1 +
x
)
2
dx
=
飦?/p>
/2
锛?/p>
銆愯В銆戝湪鍓插幓鍘熺偣鍙婃瀹炶酱鐨?/p>
z
骞抽潰涓婏紝
鈭?/p>
z
锛?/p>
ln
z
閮?/p>
鑳藉垎鍑哄崟鍊艰В鏋愬垎鏀紝
鈭?/p>
z
鍙栧湪姝ｅ疄杞寸殑涓婂哺鍙栨鍊?
鐨勯偅涓垎鏀紝
ln
z
鍙栧湪姝ｅ疄杞寸殑涓婂哺鍙栧疄鏁板�肩殑閭ｄ釜
鍒嗘敮锛庤
f
(
z
) =
鈭?/p>
z
ln
z
/(1 +
z
)
2
dz
锛?/p>
f
(
z
)
鐨勬湁闄愬鐐瑰彧
鏈?/p>
飥?/p>
1
锛屼笖
飥?/p>
1
鏄?/p>
f
(
z
)
鐨?/p>
2
闃舵瀬鐐癸紟
Res[
鈭?/p>
z
ln
z
/(1 +
z
)
2
;
飥?/p>
1] = lim
z
飩?/p>
飥?/p>
1
((1 +
z
)
2
路
f
(
z
))
鈥?/p>
= lim
z
飩?/p>
飥?/p>
1
(
鈭?/p>
z
ln
z
)
鈥?/p>
= lim
z
飩?/p>
飥?/p>
1
(((1/2) ln
z
+ 1
)
鈭?/p>
z
/
z
)
= ((1/2) ln (
飥?/p>
1) + 1
)
鈭?/p>
(
飥?/p>
1)/(
飥?/p>
1)
=
飥?/p>
((1/2)
飦?/p>
i
+ 1
)
i
= (1/2)
飦?/p>
飥?/p>
i
锛?/p>
褰?/p>
r
< 1 <
R
鏃讹紝
飪?/p>
C
鈭?/p>
z
ln
z
/(1 +
z
)
2
dz
=
飪?/p>
C
(
r
)
+
飪?/p>
C
(
R
)
+
飪?/p>
L
(1)
+
飪?/p>
L
(2)
= 2
飦?/p>
i
Res[
鈭?/p>
z
ln
z
/(1 +
z
)
2
;
飥?/p>
1] = 2
飦?/p>
+
飦?/p>
2
i
锛?/p>
飪?/p>
L
(1)
鈭?/p>
z
ln
z
/(1 +
z
)
2
dz
=
飪?/p>
(
r
,
R
)
鈭?/p>
x
ln
x
/(1 +
x
)
2
dx
飩?/p>
飪?/p>
(0, +
飩?/p>
)
鈭?/p>
x
ln
x
/(1 +
x
)
2
dx
(
褰?/p>
r
飩?/p>
0
+
锛?/p>
R
飩?/p>
+
飩?/p>
鏃?/p>
)
飪?/p>
L
(2)
鈭?/p>
z
ln
z
/(1 +
z
)
2
dz
=
飪?/p>
(
R
,
r
)
(
飥?/p>
鈭?/p>
x
)(ln
x
+ 2
飦?/p>
i
)/(1 +
x
)
2
dx
=
飪?/p>
(
r
,
R
)
(
鈭?/p>
x
ln
x
)/(1 +
x
)
2
dx
+ 2
飦?/p>
i
飪?/p>
(
r
,
R
)
鈭?/p>
x
/(1 +
x
)
2
dx
飩?/p>
飪?/p>
(0, +
飩?/p>
)
鈭?/p>
x
ln
x
/(1 +
x
)
2
dx
+ 2
飦?/p>
i
飪?/p>
(0, +
飩?/p>
)
鈭?/p>
x
/(1 +
x
)
2
dx
(
褰?/p>
r
飩?/p>
0
+
锛?/p>
R
飩?/p>
+
飩?/p>
鏃?/p>
)
锛?/p>
鍥犱负
z
路
鈭?/p>
z
ln
z
/(1 +
z
)
2
飩?/p>
0 (
褰?/p>
|
z
|
飩?/p>
+
飩?/p>
鏃?/p>
)
锛?/p>
C
R
C
r
R
r
Ri
ri
C
R
C
r
L
1
L
2