ENGR 1405 Engineering Mathematics 1

Faculty of Engineering and Applied Science
2000 Fall

Problem Set 8
Questions

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Note:   if you see symbols like ³ or Þ in various places, then your browser is not reading the style sheet for this Web page properly (or the Symbol font is not installed on your computer).   The translation is:

-   =   - (minus)	Þ   =   Þ (implies)
Ö   =   Ö (square root)	p   =   p (pi)
Ð   =   Ð (angle)	q   =   q (theta)

1. Find the Maclaurin series expansion and discuss convergence of the series for the function

   f (x) = ln ( (1+x) / (1-x) )

2. Express each of the following as a single complex number in the Cartesian form   z = x + j y:

   1.   4 (5 - j3)   -   3 (2 - 7j)

   2.   (1 - 2j) (3 + 4j) / (4 + 3j)

   3.   (Ö2) e^jp/2 + 2 e^-jp/4

3. Express each of the following as a complex number in the Euler form   z = r e ^jq   or using the phasor notation   z = r Ðq   [which is an abbreviation for the polar form   z = r (cos q + j sin q) ]:

   1. ((Ö3) - j) (1 + jÖ3)) / (1 - j)

   2. Ö(12 - 9j)       (principal square root only)

4. Use deMoivre’s Theorem to express
   1. tan 4q   in the form of a ratio of functions which involve powers of   tan q;
   2. csc 4q   in the form of a ratio of functions which involve powers of   csc q.

5. By using deMoivre’s Theorem determine all distinct values of   z   in the Cartesian form   z = x + j y , where:

   1. z⁵ = -16(Ö3) + 16 j

   2. z^1/3 = 0.6 - 0.8 j

   3. z² + 12 z = -32 + 4 j

   4. z⁴ + (4 + 4j) z³ + (12 j) z² - (8 - 8j) z - 20 = 0

6. Express the complex number   z = u^w   in the form   z = x + jy   when

   u = 1 - j     and     w = (Ö3) - j

   [Hint:   Write the complex number   "u"   in the Euler form   u = r e ^{j(q + 2kp)} = e ^{(ln(r) + j(q + 2kp))},   then use the normal rules for multiplying exponents and then use de Moivre’s Theorem.]

7. Show that   j ^j = (Ö-1)^(Ö-1)   is real and find its principal value.

The solutions will appear in another part of this Web site.

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Created 2000 11 20 and modified 2000 11 20 by Dr. G.H. George

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