Informal lecture notes for Nov. 27
I’ll assume you’re familiar with the review of complex numbers and their
algebra as contained in Appendix G of Stewart’s book, so we’ll pick up where
that leaves off.
1
Elementary complex functions
In onevariable real calculus, we have a collection of basic functions, like poly
nomials, rational functions, the exponential and log functions, and the trig
functions, which we understand well and which serve as the building blocks for
more general functions. The same is true in one complex variable; in fact, the
real functions we just listed can be extended to complex functions.
1.1
Polynomials and rational functions
We start with polynomials and rational functions. We know how to multiply
and add complex numbers, and thus we understand polynomial functions. To
be specific, a degree
n
polynomial, for some nonnegative integer
n
, is a function
of the form
f
(
z
) =
c
n
z
n
+
c
n

1
z
n

1
+
· · ·
+
c
1
z
+
c
0
,
where the
c
i
are complex numbers with
c
n
= 0. For example,
f
(
z
) = 2
z
3
+ (1

i
)
z
+2
i
is a degree three (complex) polynomial. Polynomials are clearly defined
on all of
C
.
A rational function is the quotient of two polynomials, and it is
defined everywhere where the denominator is nonzero.
Example:
The function
f
(
z
) =
z
2
+1
z
2

1
is a rational function. The denomina
tor will be zero precisely when
z
2
= 1. We know that every nonzero complex
number has
n
distinct
n
th roots, and thus there will be two points at which the
denominator is zero. It’s easy to see that those points are 1 and

1, and so
f
is
defined on
C
\ {
1
,

1
}
. If we want to compute
f
at some point, we just use our
rules for complex algebra. For instance, using that (1 +
i
)
2
= 1 + 2
i

1 = 2
i
,
we have
f
(1 +
i
) =
(1 +
i
)
2
+ 1
(1 +
i
)
2

1
=
1 + 2
i

1 + 2
i
=
1 + 2
i

1 + 2
i
·

1

2
i

1

2
i
=

1

2
i

2
i
+ 4
1 + 4
=
3
5

4
5
i.
1.2
The exponential function and logarithm
We’ve already seen the (complex) exponential function. Recall that we have the
useful formula
e
x
+
iy
=
e
x
(cos
y
+
i
sin
y
)
.
1
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Here we use our usual convention of writing a complex number as
z
=
x
+
iy
for
real
x
and
y
. One reason this formula is useful is that it allows us to actually
compute. We had originally defined
e
z
by a power series, and if we used that
definition directly, then evaluating something like
e
1+
πi
would require summing
an infinite series. Instead, using the above formula allows us to write
e
1+
πi
=
e
1
(cos
π
+
i
sin
π
) =

e.
Another reason this formula is useful is that it allows us to give a somewhat
intuitive interpretation of the exponential function. Notice that we have written
our “input” variable in Cartesian coordinates, but our “output” variable is in
polar coordinates. We see that
e
x
+
iy
is the complex number with modulus (or
r
, in polar coordinates)
e
x
and argument (or
θ
, in polar coordinates)
y
. If our
input is a real number, then
y
is zero and the output will be real and positive
(since the positive real axis has argument zero) and will have modulus
e
x
. Thus
we confirm that this agrees with the real exponential function for real input.
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 Spring '07
 NEEL
 Calculus, Algebra, Exponential Function, Complex Numbers, lim, Complex number

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