Phys374, Spring 2008, Prof. Ted Jacobson
Department of Physics, University of Maryland
Complex numbers—version 5/21/08
Here are brief notes about topics covered in class on complex numbers,
focusing on what is not covered in the textbook.
1
Complex algebra and geometry
1.
i
as a solution to
x
2
+ 1 = 0, that is,
i
=
√

1.
2. Complex numbers:
z
=
x
+
iy
, with
x
and
y
real numbers, called the
real and imaginary parts of
z
,
x
=
Re
(
z
) and
y
=
Im
(
z
).
3. Square roots of other negative numbers. E.g.
√

2 =
i
√
2.
4. Any quadratic equation
az
2
+
bz
+
c
= 0 has two roots, given by
z
= (

b
±
√
b
2

4
ac
)
/
2
a
. If
b
2

4
ac
is negative, the roots are complex.
5.
Fundamental theorem of algebra
: Every polynomial has at least one
root. This easily implies by induction that every
n
th order polynomial
can be factorized as
a
n
(
z

w
1
)(
z

w
2
)
···
(
z

w
n
). That is, it has
n
roots, some or all of which may coincide. For a proof of this see below.
6.
Complex conjugate
:
z
*
=
x

iy
.
7. Inverse of a complex number:
1
z
=
z
*
z
*
z
=
x

iy
x
2
+
y
2
.
(1)
8.
Modulus
:

z

=
√
z
*
z
=
p
x
2
+
y
2
.
9. Everything is nice: (
z
+
w
)
*
=
z
*
+
w
*
, (
zw
)
*
=
z
*
w
*
, (1
/z
)
*
= 1
/z
*
,

zw

=

z

w

,

z
*

=

z

.
10.
Exponential function
: The exponential function
f
(
x
) =
e
x
can be
deﬁned as the unique function equal to its own derivative and equal
to 1 when
x
= 0. To ﬁnd its Taylor series, write
f
(
x
) = 1 +
a
1
x
+
a
2
x
2
+
a
3
x
3
+
···
and impose equality of
df/dx
and
f
for all
x
. The
coeﬃcients of
x
n
must then be equal, which implies
a
n
= 1
/n
!. Thus
e
x
=
∑
∞
n
=0
x
n
/n
!. This deﬁnition can be generalized to any complex
number
z
, replacing
x
by
z
.
1
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View Full Document11. Key identity:
e
a
e
b
=
e
a
+
b
. One can prove this by writing out each
of the series expansions, multiplying term by term on the left hand
side, and expanding (
a
+
b
)
n
on the right hand side, then collecting
terms with equal powers of
a
and
b
on both sides. A much simpler
proof: deﬁne
f
(
a
) =
e
a
e
b
and
g
(
a
) =
e
a
+
b
, and note that
df/da
=
f
and
dg/da
=
g
. Also
f
(0) =
g
(0). So
f
and
g
are equal at one point
and satisfy the same ﬁrst order diﬀerential equation, so they are equal
everywhere!
12.
Complex plane
: Complex numbers can be viewed as points or vec
tors on a plane, with
Re
(
z
) on the horizontal axis and
Im
(
z
) on the
vertical axis. The distance from the origin to
z
is

z

.
13. Euler’s identity:
e
iθ
= cos
θ
+
i
sin
θ
. Two diﬀerent proofs: (a) Expand
both sides in a power series and show they are equal term by term,
or (b) note that both sides are equal to
i
times their own derivative
with respect to
θ
, so they satisfy the same ﬁrst order diﬀerential equa
tion. They are also obviously equal when
θ
= 0, so they are equal
everywhere.
14.
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 Fall '10
 Jacobson
 Physics, Complex Numbers, The Land, dz, Complex Plane

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