Complex Integration. Line Integral in Complex Plane A complex definite integrals are called complex...
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Transcript of Complex Integration. Line Integral in Complex Plane A complex definite integrals are called complex...
![Page 1: Complex Integration. Line Integral in Complex Plane A complex definite integrals are called complex line integrals It denotes as Here, the integrand is.](https://reader036.fdocuments.in/reader036/viewer/2022062303/5519d228550346047c8b4c64/html5/thumbnails/1.jpg)
Complex Integration
![Page 2: Complex Integration. Line Integral in Complex Plane A complex definite integrals are called complex line integrals It denotes as Here, the integrand is.](https://reader036.fdocuments.in/reader036/viewer/2022062303/5519d228550346047c8b4c64/html5/thumbnails/2.jpg)
Line Integral in Complex Plane
• A complex definite integrals are called complex line integrals• It denotes as
• Here, the integrand is integrated over a given curve . • The curve is a path of integration with parametric
representation
• Examples
( )f z C
C
![Page 3: Complex Integration. Line Integral in Complex Plane A complex definite integrals are called complex line integrals It denotes as Here, the integrand is.](https://reader036.fdocuments.in/reader036/viewer/2022062303/5519d228550346047c8b4c64/html5/thumbnails/3.jpg)
Line Integral in Complex Plane (cont.)
• We assume the curve is smooth, thus it has continuous and non-zero derivatives
• The definition of the derivatives
![Page 4: Complex Integration. Line Integral in Complex Plane A complex definite integrals are called complex line integrals It denotes as Here, the integrand is.](https://reader036.fdocuments.in/reader036/viewer/2022062303/5519d228550346047c8b4c64/html5/thumbnails/4.jpg)
Definition of Complex Line Integral
• Suppose that the time in parametric representation is divided into
• The is indeed partitioned into
where
C
![Page 5: Complex Integration. Line Integral in Complex Plane A complex definite integrals are called complex line integrals It denotes as Here, the integrand is.](https://reader036.fdocuments.in/reader036/viewer/2022062303/5519d228550346047c8b4c64/html5/thumbnails/5.jpg)
Definition of Complex Line Integral (cont.)
• On each subdivision of we choose any point• The we can form the sum
• The limit of the sum is termed as line integration of over the path of integration of
• Important definition
C i
( )f z
C
![Page 6: Complex Integration. Line Integral in Complex Plane A complex definite integrals are called complex line integrals It denotes as Here, the integrand is.](https://reader036.fdocuments.in/reader036/viewer/2022062303/5519d228550346047c8b4c64/html5/thumbnails/6.jpg)
Basic Properties
• Linearity
• Sense of reversal
• Partitioning path
![Page 7: Complex Integration. Line Integral in Complex Plane A complex definite integrals are called complex line integrals It denotes as Here, the integrand is.](https://reader036.fdocuments.in/reader036/viewer/2022062303/5519d228550346047c8b4c64/html5/thumbnails/7.jpg)
Evaluation of Complex Integral
Proof : We have z x iy
![Page 8: Complex Integration. Line Integral in Complex Plane A complex definite integrals are called complex line integrals It denotes as Here, the integrand is.](https://reader036.fdocuments.in/reader036/viewer/2022062303/5519d228550346047c8b4c64/html5/thumbnails/8.jpg)
Step to Evaluate
![Page 9: Complex Integration. Line Integral in Complex Plane A complex definite integrals are called complex line integrals It denotes as Here, the integrand is.](https://reader036.fdocuments.in/reader036/viewer/2022062303/5519d228550346047c8b4c64/html5/thumbnails/9.jpg)
Examples
![Page 10: Complex Integration. Line Integral in Complex Plane A complex definite integrals are called complex line integrals It denotes as Here, the integrand is.](https://reader036.fdocuments.in/reader036/viewer/2022062303/5519d228550346047c8b4c64/html5/thumbnails/10.jpg)
Examples (cont.)
![Page 11: Complex Integration. Line Integral in Complex Plane A complex definite integrals are called complex line integrals It denotes as Here, the integrand is.](https://reader036.fdocuments.in/reader036/viewer/2022062303/5519d228550346047c8b4c64/html5/thumbnails/11.jpg)
Examples (cont.)
![Page 12: Complex Integration. Line Integral in Complex Plane A complex definite integrals are called complex line integrals It denotes as Here, the integrand is.](https://reader036.fdocuments.in/reader036/viewer/2022062303/5519d228550346047c8b4c64/html5/thumbnails/12.jpg)
Dependence on Path
• If we integrate from to along different paths, then the integral in general will have different values.
• A complex line integral depends not only on the endpoints of the path, but also on the path itself.
( )f z 0z 1z
![Page 13: Complex Integration. Line Integral in Complex Plane A complex definite integrals are called complex line integrals It denotes as Here, the integrand is.](https://reader036.fdocuments.in/reader036/viewer/2022062303/5519d228550346047c8b4c64/html5/thumbnails/13.jpg)
Example
![Page 14: Complex Integration. Line Integral in Complex Plane A complex definite integrals are called complex line integrals It denotes as Here, the integrand is.](https://reader036.fdocuments.in/reader036/viewer/2022062303/5519d228550346047c8b4c64/html5/thumbnails/14.jpg)
Example (cont.)