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

Further Practice - Answers

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.