Lesson 8
Contour integration
1
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Understanding the Impact of Climate Change on Agriculture, Lecture notes of Mathematics
The effects of climate change on agricultural productivity, focusing on extreme weather events, changing precipitation patterns, and the potential for adaptation strategies. It provides insights into the challenges farmers face in adapting to these changes and the role of scientific research in developing sustainable agricultural practices.
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2020/2021
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Lesson 8
Contour integration
- (^) Last lecture we talked about expansion in Taylor series:
- (^) We also discussed analytic functions
- (^) This lecture we will discuss contour integration
- (^) This will show that analyticity in the unit circle is equivalent to having a Taylor series
- (^) We will also look into Laurent series
f (z) =
k=
fkz
k
f (z) =
k=
fkz
k
- (^) A curve in the complex plane is defined by
a map from an interval I = ( a , b ):
where 0.4 0.5 0.6 0.
= M (I) M : I C
- (^) A curve in the complex plane is defined by
a map from an interval I = ( a , b ):
where
= 0.5^ 1.0^ 1.5^ 2.0^ 2.
- 0.4 0.5 0.6 0.
= M (I) M : I C
- (^) A curve in the complex plane is defined by
a map from an interval I = ( a , b ):
- (^) A Jordon curve is a curve which is
- (^) Closed: it forms a loop M (a) = M (b) where
= 0.5^ 1.0^ 1.5^ 2.0^ 2.
- 0.4 0.5 0.6 0.
= M (I) M : I C
- (^) A curve in the complex plane is defined by
a map from an interval I = ( a , b ):
- (^) A Jordon curve is a curve which is
- (^) Closed: it forms a loop M (a) = M (b)
- (^) Simply connected: it does not intersect
itself (except at the endpoints)
where
= 0.5^ 1.0^ 1.5^ 2.0^ 2.
- 0.4 0.5 0.6 0.
= M (I) M : I C
- (^) A curve in the complex plane is defined by
a map from an interval I = ( a , b ):
- (^) A Jordon curve is a curve which is
- (^) Closed: it forms a loop M (a) = M (b)
- (^) Simply connected: it does not intersect
itself (except at the endpoints)
- (^) Oriented: it has a left and right
- (^) Sufficiently smooth: M is continuously
differentiable
where
= 0.5^ 1.0^ 1.5^ 2.0^ 2.
- 0.4 0.5 0.6 0.
= M (I) M : I C
- (^) A contour in the complex plane is defined
by a finite number of smooth curves
- (^) A Jordon contour is a contour which is
- (^) Closed
- (^) Simply connected
- (^) Oriented
- (^) Sufficiently smooth: the contour is
continuous and piecewise differentiable
Example 0.3 0.5 0.6 0.7 0.8 0.9 1.
0IX M *
1
[I] HI½RIEGYVZI = M (I)
'SRXSYVMRXIKVEXMSRSZIVXLIGYVZI MWHI½RIHF]
f (z) /z =
b a
f (M (x))M
(x) /x
0IX M 1 ,... , M *
1
[I] HI½RIEGSRXSYV
= 1 · · · = M 1 (I) · · · M (I)
'SRXSYVMRXIKVEXMSRSZIVXLIGSRXSYV MWHI½RIHTMIGI[MWI
f (z) /z =
1
f (z) /z + · · · +
f (z) /z
;LIR MWE.SVHERGSRXSYV [IHIRSXI
f (z) /z =
f (z) /z
0IX M *
1
[I] HI½RIEGYVZI = M (I)
'SRXSYVMRXIKVEXMSRSZIVXLIGYVZI MWHI½RIHF]
f (z) /z =
b a
f (M (x))M
(x) /x
0IX M 1 ,... , M *
1
[I] HI½RIEGSRXSYV
= 1 · · · = M 1 (I) · · · M (I)
'SRXSYVMRXIKVEXMSRSZIVXLIGSRXSYV MWHI½RIHTMIGI[MWI
f (z) /z =
1
f (z) /z + · · · +
f (z) /z
;LIR MWE.SVHERGSRXSYV [IHIRSXI
f (z) /z =
f (z) /z
2S[GSRWMHIVXLIGYVZI EWP]MRKMR R
2 PIX = M (T) [LIVI M ( ) = (x( ), y( )) JSV x : T R ERH y : T R WSXLEX M : R 2 6IGEPP E PMRIMRXIKVEP MR R 2 SZIVXLIGYVZI MWHI½RIHF] (u(x, y) /x + v(x, y) /y) =
(u(M ( ))x ( ) + v(M ( ))y ( ) /
9
2S[GSRWMHIVXLIGYVZI EWP]MRKMR R
2 PIX = M (T) [LIVI M ( ) = (x( ), y( )) JSV x : T R ERH y : T R WSXLEX M : R 2 6IGEPP E PMRIMRXIKVEP MR R 2 SZIVXLIGYVZI MWHI½RIHF] (u(x, y) /x + v(x, y) /y) =
(u(M ( ))x ( ) + v(M ( ))y ( ) / ;IGERXLYWVIGEWXGSQTPI\GSRXSYVMRXIKVEXMSREWPMRIMRXIKVEXMSRJSV f (z) = u(z) + Bv(z) f (z) /z =
f (M ( ))M ( ) /
2S[GSRWMHIVXLIGYVZI EWP]MRKMR R
2 PIX = M (T) [LIVI M ( ) = (x( ), y( )) JSV x : T R ERH y : T R WSXLEX M : R 2 6IGEPP E PMRIMRXIKVEP MR R 2 SZIVXLIGYVZI MWHI½RIHF] (u(x, y) /x + v(x, y) /y) =
(u(M ( ))x ( ) + v(M ( ))y ( ) / ;IGERXLYWVIGEWXGSQTPI\GSRXSYVMRXIKVEXMSREWPMRIMRXIKVEXMSRJSV f (z) = u(z) + Bv(z) f (z) /z =
f (M ( ))M ( ) / =
[u(M ( ))x ( ) v(M ( ))y ( ) + B (u(M ( ))y ( ) + v(M ( ))x ( ))] / =
(u /x v /y) + B
(v /x + u /y)
h?2Q`2K +VIIR WXLISVIQ
7YTTSWIXLEX u, v ERHXLIMVTEVXMEPHIVMZEXMZIWEVIGSRXMRYSYWXLVSYKLSYXEWMQTP]
GSRRIGXIHHSQEMR D [LSWIFSYRHEV]MW SVMIRXIHMRXLITSWMXMZI[E] XLIPIJXLERH
WMHISJ MWMRWMHI D 8LIR
(u /x + v /y) =
D