Complex Derivative: A Basis Free Definition
PARTHA PRATIM DEY
Department of Computer Science
North South University
12 Kemal Ataturk Avenue, Banani, Dhaka
BANGLADESH
Abstract:- Complex variables as we know is very representation dependent, which many think it need not be.
Text books define complex numbers as x  iy using basis {1, i} which makes everything including complex
derivative, Cauchy-Riemann conditions etc. tied to that particular basis though {1, i} is not the only basis of C .
In this paper, we use Frechet derivative to construct a definition of complex derivative, which is basis free in the
sense that no basis, either explicit or implicit, has been used in the definition.
Key-Words:- Linear operator, matrix, basis, complex, derivative, Frechet derivative.
1 Introduction
Let R  ( x, y) | x, y  R} be a vector space
over R with usual addition and multiplication,
specifically
2
( x1 , y1 )  ( x2 , y2 )  ( x1  x2 , y1  y2 )
and
 ( x  y )  x  y
where   R . We define a multiplication between
the vectors ( x1 , y1 ), ( x2 , y 2 )  R 2 as follows:
( x1 , y1 )  ( x2 , y2 )  ( x1 x2  y1 y2 , y1 x2  x1 y2 )
The vetor space R 2 equipped with the multiplication
above is a field denoted by C and is called the field
of complex numbers. Hence a complex
number z  ( x, y ) is an ordered pair
where x and y are called real and imaginary parts of
z and denoted by Re z and Im z respectively.
Notice that we have not used any basis to define
our complex number.
Let now B be a basis for vector space R 2 and T be a
R -linear operator from R 2 to R 2 with
T (a1 )  1a1   2 a2
and
T (a2 )  1a1   2 a2 .
For any v  xa1  ya2 in R 2 , let [v] B denote
x 
 y ,
 
the transpose of the coordinate matrix v relative to
basis B and let [T ] B be the matrix such that the j th
column of [T ] B is [T ( a j )] B i.e.,
  
[T ] B   1 1  .
 2  2 
[T ] B is regarded as a representation of operator T
and is called the matrix of T with respect to
basis B . For such a matrix
[T (v)] B  [T ] B [v] B
for any v in R 2 , as
T (v )
 xT (a1 )  yT (a 2 )
 x( 1 a1   2 a 2 )  y ( 1 a1   2 a 2 )
 ( x 1  y1 )a1  ( x 2  y 2 )a 2
and
 x1  y1   1 1   x 
 x  y       y  .
2
 2
 2 2  
Next we quote a well-known theorem [ 1 ] that will
be used later to relate matrices of an operator T
with respect to various bases. The sketch of the
proof given below is ours.
Theorem (1.1) Let T be a R -linear operator
from R 2 to R 2 . If B '  {a, b} is a basis
of R 2 with a  (a1 , a2 ) and b  (b1 , b2 ) , then
[T ] B'  P 1[T ] B P
where
a b 
P 1 1
 a 2 b2 
and B is the standard basis comprising of
e1  (1,0), e2  (0,1) .
Proof. Notice that the equations below
a  a1e1  a2 e2
b  b1e1  b2 e2
is equivalent to the following matrix equation
1
a b1 
[e1 , e2 ]  [a, b] 1
 .
a2 b2 
Assume now that T (e1 )  1e1   2 e2 and
T (e2 )  1e1   2 e2 . Then
T (a )
 T (a1e1  a2 e2 )
 1 1  a1 
 e1 e2  
 
 2  2  a 2 
1
 1 1  a1 
   a 
 2 2  2 
a 
 [ a b P 1 [T ] B  1  .
a 2 
a b1 
 [ a b  1

a 2 b2 
Similarly one can show that
b 
T (b)  [a bP 1[T ] B  1  .
b2 
1
Hence [T ] B'  P [T ] B P .■
2 Matrix Form of a Multiplication
Transformation
Let M [w] denote the R - linear transformation
from C to C given by
M [ w]( z )  wz
where w, z  C and is called a multiplication
transformation. The collection M [w] | w  Cis a
subset of L (C , C ) and is denoted by C * . Since
M [ w](1,0)  (r , s)(1,0)  (r , s)  r (1,0)  s(0,1)
and
M [ w](0,1)  (r , s )(0,1)  ( s, r )   s (1,0)  r (0,1)
the matrix of M [w] with respect to the canonical
basis B  {(1,0), (0,1)} has the following form
r  s 
[ M ( w)] B  

s r 
whereas by Theorem (1.1) , the matrix of M [w]
with respect to an arbitrary
basis B '  {( a1 , a 2 ), (b1 , b2 )} resembles the
following
a b1 
[ M [ w]] B '   1

a 2 b2 
1
r
s

 s
r 
a1 b1 
a b  .
 2 2
Now we prove a theorem, which states given two
bases of C when the rings of matrices of
multiplication transformations are the same.
Theorem ( 2.1) Let [C * ] B and [C * ] B ' denote the
rings of matrices of multiplication transformations
with respect to bases B and B ' respectively,
where B is the canonical basis {(1,0), (0,1)} and B ' is
any other. Then [C * ] B = [C * ] B ' if and only if one of
the bases can be obtained from the other by
multiplication with a complex number.
Proof.Let B  {(1,0), (0,1)} and
B '  {( x1 , y1 ), ( y1 , x1 )} be two bases of C where
elements of B ' are obtained by multiplication of
each element of B by ( x1 , y1 ) i.e.,
( x1 , y1 )  ( x1 , y1 )  (1,0) and
( y1 , x1 )  ( x1 , y1 )  (0,1) .
Let M ( w)[ B ] and M ( w)[ B ' ] be representations
of w  (r , s ) with respect to
bases B and B ' respectively.
Then M ( w)[ B ' ]  P 1 M ( w)[ B ] P where
 x  y1 
P 1
.
 y1 x1 
Since
r  s
,
s r 
M (w)[ B ]  
we get
P 1 M ( w)[ B ] P
 1 0  
 0  1 
 P 1  r    P  P 1  s 
  P
0
0
1
1







y1  0  1
 x1
1 0
s
 r
 2


.
2 
0 1  x1  y1  y1 x1  1 0 
 x1  y1 

 y1 x1 
.
0
 ( x12  y12 )
1 0
s
 r   2

2  2
2
0 1  x1  y1  x1  y1 0

r  s 

.
s r 
Assume B ' to be the basis {( x1 , y1 ), ( x2 , y2 )} where
( x2 , y2 )  (1,0)  ( x2 , y2 ) , ( x1 , y1 )
 (0,1)  ( x2 , y2 )
Then x2  y1 and y 2   x1 . Thus B ' can be
rewritten as {( x1 , y1 ), ( y1 , x1 )} . Hence P becomes
 x1 y1 
y  x  .
1
 1
Then
r
[ M ( w)][ B ' ]  P 
s
1 0 0
= P 1 [ r 
 + s
0 1  1
1 0
0
 r
 sP 1 

0 1 
1
1
1 0
 r   s
0 1 
The equalities above are simplified to:
 x1 x2  y1 y 2  0
 2
2
2
2
 x1  y1  x2  y 2
Solving, we get
x2   y1 , x1  y2 or x2  y1 , x1   y2 , which
shows B ' is a multiplication of B by a complex
 s
P
r 
number. ■
 1
]P
0 
Then [C  ] B1  [C  ] B2 if and only if one of the bases
Theorem ( 2.2) Let B1 , B2 be two bases of C .
can be obtained from the other by multiplication
with a complex number.
Proof. Let B1  ( x1 , y1 ), ( x1' , y1' ) be a basis
 1
P
0 
0 1   r
1 0   s

 

s
.
r 
We now prove the converse.
Let B  {(1,0), (0,1)} and B '  {( x1 , y1 ), ( x2 , y 2 )} b
e two bases of C and [C * ] B  [C * ][ B ' ] . We want to
show that B  {( x1 , y1 ), ( x2 , y 2 )} can be obtained
from B  {(1,0), (0,1)} by multiplication of each of
its element with a complex number, which amounts
to one of the following:
a. (0,1)  ( x1 , y1 )  ( x2 , y2 ) i.e.,
'
( y1 , x1 )  ( x2 , y2 ) or
equivalently x2   y1 and y 2  x1
b. (0,1)  ( x2 , y2 )  ( x1 , y1 ) i.e.,
( y2 , x2 )  ( x1 , y1 ) or
equivalently x2  y1 and y 2   x1 .
Let w  (a, b)  C and [M (w)] B , [ M ( w)] B ' be the
matrix representations of w in
bases B and B ' respectively.
Then [ M ( w)] B '  [C * ] B ' and hence
[ M ( w)] B '  P 1[M ( w)] B P .
Since [C * ] B  [C * ] B' , we obtain P 1[ M ( w)] B P =
[ M ( w)] B '  [C] B , which implies the following.
a
b( x12  y12 )
b( x22  y 22 )
.

x1 y 2  y1 x 2 x1 y 2  y1 x2
b( y1 y 2  x1 x2 )
b( x1 x2  y1 y 2 )
,
a
x1 y 2  y1 x2
x1 y 2  y1 x2

of C and assume B2 be another where elements
of B2 are obtained by multiplication of each element
of B1 with an element w  (a, b) i.e.,
B2  {( ax1  by1 , bx1  ay1 ), (ax1'  by1' , bx1'  ay1' )}
Notice that
'
'
ax  by1 ax1  by1 
P2   1
'
'
bx1  ay1 bx1  ay1 
'
 a  b   x1 x1 
=
.

'
b a   y1 y1 
Then
r
P 
s
1
2
'
 s
 x1 x1 
P2  

'
r 
 y1 y1 
 r  s  a
.

 s r  b
 b
a 
1
1
a
b

1
 b
.
a 
'
 x1 x1 

y
'
 1 y1 
1
'
'
 x x1   r  s   x1 x1 
 1

 

'
'
 y1 y1   s r   y1 y1 
r  s
 P11 
 P1 .
s r 
Hence [C  ] B1  [C  ] B2 .
We now prove the converse.
Suppose [C  ] B1  [C  ] B2 . Then for any r2 , s2  R ,
there exist r1 , s1  R such that
r
P 1
 s1
1
1
 s1 
r  s 2 
P1  P21  2

 P2
r1 
 s 2 r2 
or equivalently,
r  s1 
r2  s 2 
1
( P1 P21 ) 1  1
P
P

1
2

s r  .
 s1 r1 
 2 2 


This implies that [C ] B  [C ] B' where B is the
canonical basis and B ' comprises of the transposed
first and second columns of P1 P21 . Then by the
Theorem ( 2.2) , there is a complex
number (a, b) such that,
1
1 2
( first column of P P )
transposed
 (a, b)(1,0)
and
( second column of P1 P21 ) transposed  (a, b)(0,1) .
This gives
a  b 
P1 P21  

b a 
or
a  b
P1  
 P2 ,
b a 
which by the discussion in the very beginning of
the first part of this proof means the vectors of
basis B1 can be obtained from B2 by multiplication
of each of its elements with complex
number (a, b) .■
3 Matrix Form of Frechet Derivative
Next we discuss a derivative known as the Frechet
derivative. Since our definition of complex
derivative uses the notion of Frechet derivative, we
will say below a few words about Frechet
derivative.
Definition (3.1) . Let  be a norm and E be an
open set in R 2 with respect to this norm. Suppose
f maps E into R 2 , and z  E . If there exists a
linear transformation A of R 2 into R 2 such that
f ( z  h)  f ( z )  Ah  r (h)
where
lim
r ( h)
h
0
h0
then we say that f is differentiable (or Frechet
differentiable) at z , and we write
f  ( z)  A .
If f is differentiable at every z  E , we say that f
is differentiable in E .
We consider f that maps an open
set E  R 2 into R 2 and is differentiable at z  E .
Let B  {e1 , e2 } be the canonical basis of R 2 ,
where as usual e1  (1,0) and e2  (0,1) . The
components of f are real functions
u, v on E defined by
f ( z)  u( z)e1  v( z)e2 .
It is well-known [ 2 ] that
f * ( z )e1 
u
v
( z )e1  ( z )e2
x
x
and
u
v
( z )e1  ( z )e2 .
y
y
Thus by Theorem (1.1) , the Frechet derivative with
f * ( z )e 2 
respect to canonical basis assumes the following
form:
(1)
 u
 x ( z )
[ f * ( z )] B  
 v ( z )
 x
u 
( z)
y 
.
v

( z) 
y

We will use this important fact to prove our next
theorem.
Theorem (3.2) Let B '  {a, b} be a basis
for C over R and f ( z )  u ( x, y )a  v( x, y )b be a
function from an open set
U  C to C where z  xa  yb with
a  (a1 , a2 ), b  (b1 , b2 ) . If f has a Frechet
derivative f  (z ) at z  xa  yb , then
u x ( x, y) u y ( x, y ) 
[ f  ( z )] B'  

v x ( x, y) v y ( x, y ) 
where [ f  ( z)] B' is the matrix of Frechet
derivative f  (z ) with respect to basis B ' .
Proof. Set w1  xa1  yb1 , w2  xa2  yb2 . Then
w1b2  w2 b1
 w1a 2  w2 a1
and y 
.
a1b2  a 2 b1
a1b2  a 2 b1
Now f ( z )  u ( x, y )a  v( x, y )b can be rewritten
( 2) x 
as
f (w1 , w2 )
 (a1u( x, y), a2u( x, y))  (v( x, y)b1 , v( x, y)b2 )
 (a1u ( x, y )  b1v( x, y ))e1 
(a 2 u ( x, y )  b2 v( x, y ))
where e1  (1,0) and e2  (0,1) constitute the
canonical basis B of R 2 and x, y are as in
equation ( 2) above. The Frechet derivative of
f (w1 , w2 ) with respect to basis B denoted
by [ f  ( w1 , w2 )] B is then easily computed by (1) .
[ f  ( w1 , w2 )] B

 

 w [a1u  b1v] w [a1u  b1v] 
1
2



 

 w [a 2 u  b2 v] w [a 2 u  b2 v]
2
 1

v
 u
a1 w  b1 w
1
1

v
 u
a 2 w  b2 w
1
1

a
 1
a 2
u
v 
 b1
w2
w2 

u
v 
a2
 b2
w2
w2 
a1
u 
w2 

v 
w2 
 u
b1   w1

b2   v
 w
 1
 u
 a b   x
 1 1 
 a 2 b2   v
 x
u 
1
y  a1 b1 

v  a 2 b2 
y 
using chain rule [2] in the last step.
Setting
a b 
P   1 1,
 a 2 b2 
Now we are ready for a definition of complex
derivative.
Definition ( 4.1) Let f be a map from an open
set E  C to C and f  be it's Frechet derivative
on E . We say that f has a complex derivative
f ' ( z ) (or f is C -differentiable) at an
element z in E if f * ( z )  C * i.e. f * ( z )  M ( w)
for some w C . We let w  M 1 ( f * ( z )) to be
f ' ( z) .
Recall that in the Introduction, we did not use any
basis when we defined the complex numbers. Now
notice that our definition of complex derivative
does not use a basis either. Nor do we use any
specific norm, as the definition of Frechet
derivative does not use any. Recall here that all
norms are equivalent in a finite dimensional vector
space and which is what makes the existence of
Frechet derivative independent of any norm.
The next theorem states the necessary and
sufficient conditions of complex differentiability of
a complex function f at a complex number z given
an arbitrary basis in C .
Theorem ( 4.2) Suppose B '  {a, b} is a basis of
C over R where a  (a1 , a2 ) and b  (b1 , b2 ) .
Let f be a function from an open set E  C to C
and is given by
f ( z )  u ( x, y )a  v( x, y )b
when z  xa  yb  E . Then f has a complex
derivative f ' ( z 0 ) at z 0  x0 a  y0 b if and only if
we thus obtain
 u
 x ( x, y )

[ f ( w1 , w2 )] B  P 
 v ( x, y )
 x
4 A New Definition of Complex
Derivative
u

( x, y ) 
y
 P 1 .
v

( x, y ) 
y

 u
 x ( x0 , y 0 )

 v ( x , y )
 x 0 0
u

( x0 , y 0 )
y
r  s
  P 1 
P
v
s r 


( x0 , y 0 ) 
y

Since by Theorem (1.1),
for some r , s  R . Moreover f ' ( z 0 )  (r , s ) .
[ f  (w1 , w2 )] B'  P 1[ f  (w1 , w2 )] B P , we obtain
Proof. Suppose f ' ( z 0 ) exists. Then
the desired equality. ■
f  ( z 0 )  M ( w) for some w  (r , s )  C and
f ' ( z 0 )  (r , s ) . Notice
that f  ( z 0 )  M ( w) implies
[ f  ( z )] B  [M (w)] B . By the last paragraph of
Theorem (3.2) , this amounts to:
u

 u
( x0 , y 0 )
(
x
,
y
)
0
0
 x
y
 P 1
P 
v

 v ( x , y )
( x0 , y 0 ) 
 x 0 0
y

r

s
 s
.
r 
To get the converse we just go backward. ■
Corollary ( 4.3) . ( Cauchy-Riemann Equations)
Let f ( z )  u ( x, y )1  v( x, y )i define a function
from C to C with the canonical
basis B  1, iwhere 1  (1,0) and i  (0,1) .
Then f has a complex derivative at z 0  x0 1 y 0 i
if and only if
u
v
( x0 , y 0 )  ( x0 , y 0 )
x
y
and
v
u
( x0 , y 0 )   ( x0 , y 0 ) .
x
y
Proof. By the theorem above
f ( z )  u ( x, y )  1  v( x, y )  i has a complex
derivative at z 0  x0 1 y 0 i if and only if
 u
 x ( x0 , y 0 )

 v ( x , y )
 x 0 0
u

( x0 , y 0 )
y

v

( x0 , y 0 ) 
y

r  s
 P 1 
P
s r 
for some r and s . Since in this
case P is 2  2 identity matrix, we obtain
 u
 x ( x0 , y 0 )

 v ( x , y )
 x 0 0
u

( x0 , y 0 )
y
r  s
 

v
 s r 
( x0 , y 0 ) 
y

which yields the desired equalities. ■
The corollary above is known in literature as
Cauchy-Riemann equations [ 3 ]. These equations
are used almost as a definition of C -
differentiability in textbooks. But notice that they
are not basis free, whereas our definition is.
References:
[ 1 ] Howard Anton and Chris Rorres, Elementary
Linear Algebra, John Wiley & Sons, Inc., 1994
[ 2 ] Walter Rudin, Principles of Mathematical
Analysis, McGraw-Hill International Editions, 1976
[ 3 ] Serge Lang, Complex Analysis, New York:
Springer, 1999