Engineering Mathematics Complex Variables Applications Chapter 5 wszhengieee

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

Outlilne 1、Definition of Series & Properties 2、Taylor Series 3、Laurent Series 4、Power Series

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?

Series: Definition & Properties

Series: Definition & Properties • When does a sequence converge?

Series: Definition & Properties • What is a series? partial sum

Series: Definition & Properties • When does a series converge?

Series: Definition & Properties Absolute Convergence Converges!!!

Series: Definition & Properties • Series and Sequence 0

Taylor Series When a function at z can be represented by a series?

Taylor Series • Taylor's theorem 板书证明

Taylor Series • Maclaurin series

Taylor Series • Example

Taylor Series • Example

Laurent Series • Taylor Series If a function f fails to be analytic at a point z 0? ? ?

Laurent Series

Laurent Series • Another Equivalent Formulation

POWER SERIES: Properties • Absolute Convergence

POWER SERIES: Properties • Uniformly Convergent

POWER SERIES: Properties • What is uniformly convergent Take for example has limit at z POWER SERIES: Properties uniformly convergent

POWER SERIES: Properties • Continuity

POWER SERIES: Properties • Integration Example

POWER SERIES: Properties • Differentiation Example

POWER SERIES: Properties • Uniqueness

POWER SERIES: Properties • Uniqueness

POWER SERIES: Properties • Multiplication Taylor coefficients 前提