MAT 3730 Complex Variables

1. MAT 3730Complex Variables Section 1.4 The Complex Exponential http://myhome.spu.edu/lauw

2. Preview • Extension of the exponential function to the complex numbers • The Euler’s Formula • The De Moivre’s Formula (du mwA´vru )

3. The Complex Exponential

4. The Complex Exponential

5. Basic Property

6. Basic Property

7. Definition of eiy There are 2 ways to look at the definition of 1. Through the Maclaurin Series 2. Through the property

8. Definition of eiy There are 2 ways to look at the definition of 1. Through the Maclaurin Series 2. Through the property

9. Through the Maclaurin Series Suppose we want eiyto have the same Maclaurin series, then

10. Through the Maclaurin Series Suppose we want eiyto have the same Maclaurin series, then

11. Through the Maclaurin Series

12. The Euler’s Formula

13. Definition of Complex Exponential

14. Example 1

15. Example 2

16. Properties of Complex Exponential

17. Polar Form (Revisit)

18. Example 3

19. (du mwA´vru) Example 4 De Moivre’s Formula

20. Example 5

21. Next Class • Read Section 1.5 • We will look at how to find: • Powers zn • Roots z1/m