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Engineering Mathematics

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This chapter delves into the definitions and fundamental properties of series in the context of engineering mathematics. It covers essential topics such as the convergence of sequences and series, with a focus on Taylor and Laurent series, to approximate functions. The text also explores absolute convergence, power series, and their applications in continuity, differentiation, and integration. By addressing key questions about the convergence of series, this resource aims to enhance understanding of how functions can be represented and approximated through series.

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Engineering Mathematics

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  1. Complex Variables & Applications Chapter 5 Engineering Mathematics • 郑伟诗 • wszheng@ieee.org, • http://sist.sysu.edu.cn/~zhwshi/

  2. Outlilne 1、Definition of Series & Properties 2、Taylor Series 3、Laurent Series 4、Power Series 1/2/2020, Page 2

  3. Series: Definition & Properties What are we caring? (1) Is this sequence converged? When? How? (2) Moreover, are the Sum of these point converged? (3) Can a function f at z be approximated by a series?

  4. Series: Definition & Properties 1/2/2020, Page 4

  5. When does a sequence converge? Series: Definition & Properties 1/2/2020, Page 5

  6. What is a series? Series: Definition & Properties partial sum 1/2/2020, Page 6

  7. When does a series converge? Series: Definition & Properties 1/2/2020, Page 7

  8. Absolute Convergence Series: Definition & Properties Converges!!! 1/2/2020, Page 8

  9. Series and Sequence Series: Definition & Properties 0 1/2/2020, Page 9

  10. When a function at z can be represented by a series? Taylor Series 1/2/2020, Page 10

  11. Taylor's theorem Taylor Series 板书证明 1/2/2020, Page 11

  12. Maclaurin series Taylor Series 1/2/2020, Page 12

  13. Example Taylor Series 1/2/2020, Page 13

  14. Example Taylor Series 1/2/2020, Page 14

  15. Taylor Series Laurent Series If a function f fails to be analytic at a point z0??? 1/2/2020, Page 15

  16. Laurent Series 1/2/2020, Page 16

  17. Another Equivalent Formulation Laurent Series 1/2/2020, Page 17

  18. Absolute Convergence POWER SERIES: Properties 1/2/2020, Page 18

  19. Uniformly Convergent POWER SERIES: Properties 1/2/2020, Page 19

  20. What is uniformly convergent Take for example POWER SERIES: Properties has limit at z POWER SERIES: Properties uniformly convergent 1/2/2020, Page 20

  21. Continuity POWER SERIES: Properties 1/2/2020, Page 21

  22. Integration POWER SERIES: Properties Example 1/2/2020, Page 22

  23. Differentiation POWER SERIES: Properties Example 1/2/2020, Page 23

  24. Uniqueness POWER SERIES: Properties 1/2/2020, Page 24

  25. Uniqueness POWER SERIES: Properties 1/2/2020, Page 25

  26. Multiplication POWER SERIES: Properties Taylor coefficients 前提 1/2/2020, Page 26

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