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### Transcript of LESSON 79 – EXPONENTIAL FORMS OF COMPLEX NUMBERS HL2 Math - Santowski

• LESSON 79 EXPONENTIAL FORMS OF COMPLEX NUMBERSHL2 Math - Santowski

• Opening Exercise #1List the first 8 terms of the Taylor series for (i) ex, (ii) sin(x) and (iii) cos(x)

HENCE, write the first 8 terms for the Taylor series for eHENCE, write the first 8 terms for the Taylor series for ei. Simply your expression.

What do you notice?

• Opening Exercise #2If y() = cos() + i sin(), determine dy/d

Simplify your expression and what do you notice?

• EXPONENTIAL FORMorWhich function does this?

• EXPONENTIAL FORMSo (not proof but good enough!)If you find this incredible or bizarre, it means youare paying attention.Substituting gives Eulers jewel:which connects, simply, the 5 mostimportant numbers in mathematics.

• Forms of Complex Numbers

• Conversion ExamplesConvert the following complex numbers to all 3 forms:

• Example #1 - Solution

• Example #2 - Solution

• Further Practice

• Example 5

• Example 5 - Solutions

• Verifying Rules .. If

Use the exponential form of complex numbers to determine:

• Example 6

• Example 6 - Solutions

• And as for powers .. Use your knowledge of exponent laws to simplify:

• And as for powers .. Use your knowledge of exponent laws to simplify:

• So .. Thus ..

• And de Moivres Theorem .. Use the exponential form to:

• de Moivres Theorem

• Famous Equations ..

Show that

• Challenge Q

Show that

*For the 4th order example given, ask students to give values of the ak coefficientsThe graph is a good subject for discussion of the nature of higher order polynomialsDont forget to ask the students what a 1st order polynomial is.*For the 4th order example given, ask students to give values of the ak coefficientsThe graph is a good subject for discussion of the nature of higher order polynomialsDont forget to ask the students what a 1st order polynomial is.