1. LESSON 79 – EXPONENTIAL FORMS OF COMPLEX NUMBERS HL2 Math - Santowski

2. Opening Exercise #1  List the first 8 terms of the Taylor series for (i) e x, (ii) sin(x) and (iii) cos(x)  HENCE, write the first 8 terms for the Taylor series for e θ  HENCE, write the first 8 terms for the Taylor series for e i θ. Simply your expression.  What do you notice?

3. Opening Exercise #2  If y( θ ) = cos( θ ) + i sin( θ ), determine dy/d θ  Simplify your expression and what do you notice?

4. EXPONENTIAL FORM or Which function does this?

5. EXPONENTIAL FORM So (not proof but good enough!) If you find this incredible or bizarre, it means you are paying attention. Substituting gives “Euler’s jewel”: which connects, simply, the 5 most important numbers in mathematics.

6. Forms of Complex Numbers

8. Conversion Examples  Convert the following complex numbers to all 3 forms:

9. Example #1 - Solution

10. Example #2 - Solution

11. Further Practice

12. Further Practice - Answers

13. Example 5

14. Example 5 - Solutions

15. Verifying Rules …..  If  Use the exponential form of complex numbers to determine:

16. Example 6

17. Example 6 - Solutions

18. And as for powers …..  Use your knowledge of exponent laws to simplify:

19. And as for powers …..  Use your knowledge of exponent laws to simplify:

20. So.. Thus …..

21. And de Moivres Theorem …..  Use the exponential form to:

22. de Moivres Theorem

23. Famous Equations …..  Show that

24. Challenge Q  Show that