a note on integrals and hybrid contours in the complex plane
TRANSCRIPT
-
8/12/2019 A Note on Integrals and Hybrid Contours in the Complex Plane
1/14
&
2014
: .
(, , .) :
(1) ( , ) z t , z S C [0,1]t .
(2) 1, ,
( ) ( , ) = + kk n k n n
g z z z , , <
1, 2, ,
0 1 < < < < =n n n n
,lim 0k nn
= , 1,2,...,k n= .
1, 1,( ) ( )n nG z g z= , ( ), , 1,( ) ( )k n k n k nG z g G z= ,( ) ( )n n nG z G z=
( ) lim ( )nn
G z G z
= , .
(3) 1, 1, , 1, 0,
( , ),
= + kk n k n k n k n nnz z z z ,
(4) { }, 1( ) ( ) lim ( ) = = =n
n k n nk nz .
-
8/12/2019 A Note on Integrals and Hybrid Contours in the Complex Plane
2/14
-
8/12/2019 A Note on Integrals and Hybrid Contours in the Complex Plane
3/14
:
(3)
= =
=
(9)
=
= + =
.
-
8/12/2019 A Note on Integrals and Hybrid Contours in the Complex Plane
4/14
= +
+ +, = = + .
+
= , ,
(10)
= =
.
-
8/12/2019 A Note on Integrals and Hybrid Contours in the Complex Plane
5/14
: = + , = + + , =
A :
= = , = , = +
-
8/12/2019 A Note on Integrals and Hybrid Contours in the Complex Plane
6/14
: = + + , + , = +
+
-
8/12/2019 A Note on Integrals and Hybrid Contours in the Complex Plane
7/14
: A = . ,
:
: , 1, 2, 1,1 1 1 1( ) ( ) ( ) ( ) = + + + + +n n n n n nz z z zn n n n
, ,
( ) [ ] ( ) ( )1
1,
0
, , 0,1 and , lim , with , defined
mk mnmk
t t z t dt n
:
1
0
1 1 1 2 1 3 1( ) , , , , ( , ) = + + + +
nnG t dt
n n n n n n n n
1
0
( , ) ( ) = t dt G .
=
( , ) ( ( )) , (0) = =t z t z . ,2
( , ) 2 ( ( )) 2 ( , )
= = =tz t zt z t te t
1
0
( ) ( ( )) ( ) = = z t dt G , , ,1
2 *
0
( ( )) ( ) = z t dt G .
, ( , ) t ( ( )) z t
1
0
( ) ( , ) ( ) = = t dt G .
1
2 *
0
( ( )) ( ) = z t dt G ( )z t .
-
8/12/2019 A Note on Integrals and Hybrid Contours in the Complex Plane
8/14
: = + , 2.5 2 = i
1
0
( ) ( , ) ( ) .1789 1.7624 = = z t dt G i,
1
2 *
0
( ( )) ( ) 1.8830 .4950 = + z t dt G i . ( ( , ) z t )
:
,
= =
= ,
= =
= =
= = .
-
8/12/2019 A Note on Integrals and Hybrid Contours in the Complex Plane
9/14
= .
=
=
.
: = + = + ,
= +F .
+
+ + =
-
8/12/2019 A Note on Integrals and Hybrid Contours in the Complex Plane
10/14
: ( ) = .
= ,
= = .
= +
: = + + = , .
=
+ = + = + =
+ + .
+ + + = + +
+ + F ,
A ( )
= +
-
8/12/2019 A Note on Integrals and Hybrid Contours in the Complex Plane
11/14
-
8/12/2019 A Note on Integrals and Hybrid Contours in the Complex Plane
12/14
-
8/12/2019 A Note on Integrals and Hybrid Contours in the Complex Plane
13/14
: : = + = +
: = + = , = .
= +
-
8/12/2019 A Note on Integrals and Hybrid Contours in the Complex Plane
14/14
=
= ,
( ) ( )
=
,
:
= +
:
= +
=
. !