Study
guide for Complex Numbers
Why are we studying complex
numbers? First, they are a fun subject,
but really, we need them to solve linear differential equations in the
next chapter. In that chapter you, the student, should know how to
solve polynomial
equations like
(
Do what I did in lecture or complete the square first.)
To understand how to do that you must know what complex
multiplication does to absolute value and argument of a complex number.
(This is illustrated by de Moivre's formula)
Besides solving polynomial equations you will also have to know what
the exponential of
a complex number is, i.e. ez when z is complex.
And
you will have to know how to differentiate complex
valued
functions
such as t2e(2+3i)t with respect to a real
variable t.
Finally, some of the
applications are important, especially how to
use complex
numbers to solve trigonometric integrals.
Important problems to practice
From the notes:
* 73(3),74,76 — Make sure you know how to draw
complex numbers!
* 75 — practice de Moivre
* 82 —how do you differentiate?
* 83, 84, 85, 87 — practice using the formulas that
relate complex exponentials
and trig functions. There are a few problems in 84
((iv), (x) and (xii))
that can be done more easily using a substitution
(u=sin(x))