1. Engineering Mathematics
Complex Variables & Applications
Chapter 4
郑伟诗

2. Outlilne 1、Definition of Integral 2、Condition for Existence of
Integral and Methods of Calculation
3、Properties of Integral
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3. Curve, Contours
arc
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4. Contours Simple arc / Jordan arc Simple closed curve / Jordan curve
The arc C is a simple arc, or a Jordan arc, if it does not cross itself.
Simple closed curve / Jordan curve
When the arc C is simple except for the fact that z(b)=z(a), we say C
is a simple closed curve, or a Jordan curve.
Positively oriented curve
The positive orientation is the counterclockwise direction.
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5. Contours Differentiable arc Length of C Contour Simple closed contour
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6. Contour Integral
Suppose function is defined in domain D, C is a contour in D from point A to point B. Divide curve C into n segmented lines, the points of division are denoted by
Randomly pick a point from each segment
of curve
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7. Contour Integral (
If has an unique limit regardless of the division of C and partition method of ,then we call this limit value as the integral of function on curve C, denoted by
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8. Contour Integral Along a contour C 2019/7/8, Page 8 Wei-Shi Zheng
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9. Contour Integral To compute 2019/7/8, Page 9 Wei-Shi Zheng
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10. Contour Integral About the definition:
then this definition is same to the definition of integral for single real variable function.
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11. Contour Integral *Example1: *Solution: The line equation is 11
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12. Contour Integral
these two integral have nothing do with path-integral C
then regardless of the curve movement
to point
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13. Contour Integral *Example 2: *Solution: (1) The parametric equation is
y=x
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14. Contour Integral (2) parametric equation is y=x 2019/7/8, Page 14
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15. Contour Integral (3) integration path is composed by two line segments
parametric equation of straight-line segment along x-axis is
parametric equation of straight-line segment from point 1 to point 1+i is
y=x
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16. Contour Integral *Example 3: *Solution:
Parametric equation of integration path
(since |z|=2)
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17. Contour Integral *Example 4: *Solution:
Parametric equation of integration
path is:
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18. Contour Integral when n=0 when
Important Conclusion: integral value is independent to the center point and radius of the circle.
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19. ? Contour Integral With Branch Cut 2019/7/8, Page 19 Wei-Shi Zheng
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20. Properties of Integral
Complex integral has similar properties with definite integral of real variable function.
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21. Properties of Integral
板书证明
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22. Anti-Derivatives
板书证明
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23. ? Not D but a curve
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24. Cauchy–Goursat theorem
板书证明
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25. Cauchy–Goursat theorem
Applications:
板书证明
simple closed contour,
closed contours (intersection: finite / infinite)
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26. Cauchy–Goursat theorem
Example:
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27. Cauchy–Goursat theorem
Recall the following theorem
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28. Cauchy–Goursat theorem
板书证明
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29. Cauchy–Goursat theorem
principle of deformation of paths
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30. Cauchy–Goursat theorem
Example: ?
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31. Cauchy Integral Formula
板书证明
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32. Cauchy Integral Formula
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33. Cauchy Integral Formula
Gauss's mean value theorem
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34. Extensions: Analytic
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35. Extensions: Analytic
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36. Extensions: Analytic
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37. Extension: Liouville’s theorem
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38. Extension: Max Modulus
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39. Extension: Max Modulus
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40. Extension: Max Modulus
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