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Lecture 4 Complex Variables I (See Chapter 2 in Boas.)
Although it is not immediately obvious, an extremely important and useful extension
of our usual study of real functions of real variables - all “physical” quantities are real
after all – is to consider the corresponding complex functions of a complex variable.
So where do complex numbers come from? The underlying idea is that we want to
be able to make sense of fractional powers of negative numbers. Typically we first
see this issue raised in the context of quadratic equations (you should commit this
result to memory, if you have not already done so),
2
2
ax bx c
b b ac
x
a
Thus we want to understand what happens if the discriminate (the argument of the
square root) is less than zero,
2 b 4 ac 0. In particular, we need a definition of 1
(or by extension (^) 1
, 1 ). The first step is to give this quantity a symbol ( i.e ., a
label), i 1 (the symbol j is also sometimes used, typically in contexts where i is
the electric current). By definition we have the following properties
2 3 4
i 1 i 1, i i i , 1. ( 4. 2 )
Starting with two real numbers, x and y , we construct a complex number in the form
z x iy , where x is called the real part and y is called the imaginary part. Thus a
complex number is associated with two real numbers. The properties of complex
numbers are very similar to the (familiar) properties of two-dimensional vectors. The
algebra of complex numbers (addition,
subtraction and multiplication by real constants)
is identical to that of two-dimensional vectors,
but the multiplication of two complex numbers is
not identical to the multiplication of two vectors,
as we will see. As with the usual two-
dimensional vectors we can represent complex
numbers as points in a two-dimensional plane
were the real and imaginary parts play the roles
of components (as in the figure). This realization
of complex numbers is called the “rectangular
x
y
x,y
r
form”. The corresponding two-dimensional plane is called the complex plane. Such
plots of complex numbers are often called Argand diagrams.
As with usual two-dimensional vectors complex numbers can also be represented in
“cylindrical coordinates” or “polar form”. The length of the radius in this form is
called the modulus or absolute value of the complex number, r z mod z. The
corresponding polar angle is called the phase of the complex number,
2 2
1
cos
tan^ sin
cos sin.
r x y z
x r
y
y r
x
z r i
^
This structure is illustrated in the figure to the right.
Using the Euler formula from the last lecture we can
write the compact expression
Re cos
cos sin ,
Im sin
i
i
i
i
e
e i
e
z re
The choice of whether to use the rectangular form or the polar form depends on the
context, i.e ., we use the representation that simplifies the discussion (recall that we
are lazy and smart). One of the goals in this course is to learn to let the mathematics
do the work for us.
Note that in order for the above expressions to make sense we require that
the resulting complex number,
.
i cz cx icy cre
(^) ( 4. 6 )
- We can add or subtract complex numbers just like we add or subtract two-
dimensional vectors,
1 2 1 2 1 2
z z x x i y y. ( 4. 7 )
- We can multiply two complex numbers, needing only to be careful about the
factors of i. This process is typically simplest to consider using the polar form,
1 2 1 2
2
1 2 1 1 2 2 1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
1 2 1 2
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
cos ,
sin.
i i^ i
z z x iy x iy x x i x y y x i y y
x x y y i x y y x
r e r e r r e
x x y y r r
x y y x r r
^ ^
The real part of this expression is similar (except for the minus sign) to the usual
two-dimensional scalar product of 2 vectors, while the imaginary part is analogous
to the usual two-dimensional vector product of 2 vectors (again except for the minus
sign). For example, consider
4
1
i z i e
,
3
2
i z i e
. We have
the product
12
1 2
i
z z i e i
Note that multiplication of a complex number by it’s complex conjugate yields the
modulus squared,
2 2 2 2 zz x iy x iy x y r z , ( 4. 10 )
which really is like the usual scalar product. Likewise the product of one complex
number by the complex conjugate of another complex number is again like a scalar
product plus i times a vector product (but now with the usual signs),
1 2
2
1 2 1 1 2 2 1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
1 2
i
z z x iy x iy x x i x y y x i y y
x x y y i x y y x
r r e
- Finally we can also divide complex numbers, which again is typically simplest
using the polar form,
1 2
1 1 2
2 1 2 1 2 1 2 1 2
2 2 2 2
1
2 1 2^1 2 1 2^1
2 2
cos sin.
i
z z z
x x y y i y x x y
z z z r
r
e r r ir r
r r
Consider the specific example,
1
2 2
2
4 7 12
3
i i
i
i i i
z i
z i
i
e e
e
While the first version in rectangular form requires several steps, the polar version
essentially takes only a single step.
An equation involving complex numbers, like a two-dimensional vector equation,
means that there are really 2 equations, one for the real part and one for the imaginary
part,
Re LHS Re RHS ,
Im LHS Im RHS.
1 0.5 0.5 1
x
1
0.
1
y
2 1 1 2
x
2
1
1
2
y
0 0
2 2
1 4 1 4
4
2 2
i i
i i
i i
i i
e e
e e i
z z
e e
e e i
This set exhausts the 4 distinct roots (note that
1 4 i 6 i 3 2 e e i
), although only 2 of the roots are real
numbers. The roots are always arrayed symmetrically
around a circle of radius r z
as in the figure.
A more interesting example is
1 3 3
1 3 3 3
1 3 3 5 3
i i
i i
i i i
e e
e e
e e e
These roots are illustrated in the figure at the right.
Note that in this case only one root is a real number.
In general for the quantity (^)
1 1
n n i z re
a) there are always n roots,
b) the roots lie evenly spaced on a circle of
radius
1 n
z.
c) the phases of the roots are given by
n , n 2 n , n 4 n , , n 2 (^) n (^1) n.
If our original function is just a polynomial in x , then the complex form is easy to
determine. For an initially real infinite power series (the bn^ are real) we can write
0 0 0
0
0
Re cos
Im sin
n n n in
n n n
n n n
n n n n n n
S x b x S z b z b r e
S z X z b r n
S z Y z b r n
For this expression to be useful both of the series must converge. Again we consider
first the question of absolute convergence, which treats both the real and imaginary
parts simultaneously,
0 0 0
n n n
n n n
n n n
S z b z b z b r
(^4.^20 )
If S z converges then S z , X z and Y z all converge absolutely. Note that
r cos n r and r sin n r.
When the complex series does not converge absolutely, it is still possible for X and/or
Y to converge conditionally due to the alternating signs. Consider the series
1
n
n (^) n
n
z^ z
S z z
n n
n
For
2 1
i z i e
, z 1 , the series S (^) i (^) diverges. On the other hand the real and
imaginary parts are
(^)
(^)
1 1
1 0
cos 1 2 Re ,
sin 1 2 Im ,
m
n m
m
n m
n
S i
n m
n
S i
n m
As a simple example consider the ratio sin z (^) 1 z . Both the numerator and
denominator converge for 1 2
z R R (the denominator is just a polynomial),
while the denominator vanishes at 2
z z 1. Hence the ratio converges, i.e ., the
ratio is well defined, for 3
z R 1.
Now we will remind ourselves of various useful complex functions/power series
expansions. Probably the most useful is the complex exponential, which is just like
the real case,
2 3
0
n
z
n
z z z
e z
n
As in the real case this converges everywhere, i.e ., the radius of convergence is
infinite, z R . Thus it follows from this convergent series representation that,
just as for the real case,
(^)
1 2
1 2
1
0 1 0
2 2
1 2
1 2
2 2
1 2 1 1 2 2
n n m
z z
n n m
z z
z z
d d z z z
e e
dz dz n n m
z z
e e z z
z z z z z z
e
ASIDE: This last result that the product of 2 exponentials is an exponential with
exponent equal to the sum of the two original exponents is true as long as the two
exponents commute, (^) 1 2 1 2 2 1
z , z z z z z 0 , i.e ., the exponents are “c” numbers
and not matrices.
As we noted in the previous chapter for an imaginary argument we have
^ ^ ^
cos sin
cos sin cos sin
cos , sin.
iy
iy iy
iy iy iy iy
e y i y
e e y i y y i y
e e e e
y y
i
These last exponential expressions for the sinusoidal functions are often useful. Note
that they lead directly to the usual double angle formulae
(^)
1 2 1 2
1 2 1 2 1 2 1 2 1 2 1 2
1 1 2 2 1 1 2 2
1 2
1 2 1 2
sin
sin cos cos sin ,
i i i i
i i i i i i i i i i i i
i i i i i i i i
e e
i
e e e e e e e e e e
i i i i
e e e e e e e e
i
(^)
1 2 1 2
1 2 1 2 1 2 1 2 1 2 1 2
1 1 2 2 1 1 2 2
1 2
2
1 2 1 2
cos
cos cos sin sin.
i i i i
i i i i i i i i i i i i
i i i i i i i i
e e
e e e e e e e e e e
e e e e e e e e
i
For a complex exponent a little arithmetic with the series yields
0!^0!^0!
cos sin.
n n m
z x iy
n n m
x iy x
x iy x iy
e e
n n m
e e e y i y
( 4. 27 )
arguments. What about sines and cosines of imaginary arguments, which are related
to real exponentials? We have the following definitions
2 4
3 5
2 2 2 2
cos 1
cosh ,
sin
sinh
cosh sinh ,
1 cosh sinh cosh 1 sinh ,
cosh sinh , sinh cosh.
i iy i iy (^) y y
i iy i iy y y
y
y y
e e e e y y
iy
y
e e e e y y
iy i i y
i
i y
e y y
e e y y y y
d d
y y y y
dy dy
(^)
These are the hyperbolic functions, which, like the sinusoidal functions, are defined
by power series that converge for all finite (complex) arguments. On the other hand
the behaviors of the hyperbolic functions themselves (with real arguments) are
different from the sinusoidal functions. Instead of exhibiting periodic, bounded
behavior, the hyperbolic functions are monotonic with ranges given by
1 cosh
cosh cosh ,
sinh
sinh sinh.
y
y y
y
y
y y
y
We also have the following relations
cos cos cos cos sin sin
cos cosh sin sinh ,
sin sin sin cos cos sin
sin cosh cos sinh ,
cos sin
cos cosh sinh sin sinh cosh
cos sin.
iz
y
z x iy x iy x iy
x y i x y
z x iy x iy x iy
x y i x y
e z i z
x y y i x y y
e x i x
We can now combine our new knowledge of complex exponentials and logarithms to
explore the behavior of complex powers and roots, being careful to include the issue
of multi-valued phases. Consider two complex numbers,
1 1 1 1 1
i z x iy r e
and
2 2 2 2 2
i z x iy r e
and evaluate
2 2 ln 1 3 3 3 1 3 3
3 2 1 2 1 2 1 1
3 2 1 2 1 2 1 1
cos sin ,
Re ln ln 2 ,
Im ln ln 2.
z z z x iy x
z e e e e y i y
x z z x r y n
y z z y r x n
Thus such an expression will in general be multi-valued. Let’s consider some
specific examples. First consider 1 1 2 2
x 1, y 1, x 1 4, y 0
(^) (^1 11) ln 2 2 3 ln 1 4 4 4 4
3
16
9 16 1 8
17 16
25 16
ln 2
i^ i^ in
i
i
i
i
x
i e e
n
y
e
e
e
e
(^)
^
^
2 3 1 1
2 1 1 1 3 1 1 2
ln RHS ln 2 ,
ln LHS ln 2 ln 2.
z z r i in
z r i in z r i in
Thus the left-hand-side has more values than the right-hand-side, i.e ., the cases when
1 2
n n n. The same warning applies to expressions like
3 3?^3 1 2 1 2
z z^ z z z z z and
^
(^3 2 ) 2? 1 1
z (^) z z z z z. As a specific example consider
^
^
1 1
(^1 2 ) 2
? 1 2 2
ln 2 2 2 2
(^2 2 ) ln 2 2 2
i
i i in i
i i in n i i i
i (^) i n in i in i i i n
i i e
i e e e
i e e e e
(^)
(^)
(^)
The first expression and the last expression agree only for 2
n 0. You must use care
for such expressions. Typically in physics applications the situation will help define
how to proceed, i.e ., which of the possible values contribute.
Actually this issue is not unfamiliar. Consider what happens when we invert an
exponential by taking a logarithm. In the complex variable world we must use
ln 2
z e z in
, which leads to the questions above, i.e ., new values appear that
were not present when we lived just on the real axis. Similarly for other inversions
we have
1 1 1
1 2 2
2
1 2 2
cos
ln 1 2 ,
iz iz
e e
z z z
z i z z n
and
1 1 1
1 2 2
2
1 2 2
sin
ln 1 2
iz iz
e e
z z z
i
z i iz z n
where the principal value is the logarithm alone (familiar from real analysis).
