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

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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

  • 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.