closure properties of Cauchy-Riemann equations

closure properties of Cauchy-Riemann equations

The set of solutions of the

Cauchy-Riemann equations

is

closed under

a surprisingly large number of

operations

. For convenience, let
us introduce the notational conventions that

f

and

g

are

complex
functions

with

f

⁢

(

x

+

i

⁢

y

)

=

u

⁢

(

x

,

y

)

+

i

⁢

v

⁢

(

x

,

y

)

and

g

⁢

(

x

+

i

⁢

y

)

=

p

⁢

(

x

,

y

)

+

i

⁢

q

⁢

(

x

,

y

)

. Let

D

and

D

′

denote
open subsets of the

complex plane

.

Theorem 1

.

If

f

:

D

→

C

and

g

:

D

→

C

satisfy

the Cauchy-Riemann equations, so does

f

+

g

. Furthermore, if

z

∈

C

, then

z

⁢

f

satisfies the Cauchy-Riemann equations.

Proof.

This is an immediate

consequence

of the linearity of

derivatives

.
∎

Theorem 2

.

If

f

:

D

→

C

and

g

:

D

→

C

satisfy the
Cauchy-Riemann equations, so does

f

⋅

g

.

Proof.

Letting

h

and

k

denote the real and

imaginary parts

of

f

⋅

g

respectively, we have

∂

⁡

h

∂

⁡

x

-

∂

⁡

k

∂

⁡

y

=

∂

∂

⁡

x

⁢

(

u

⁢

p

-

v

⁢

q

)

-

∂

∂

⁡

y

⁢

(

u

⁢

q

+

v

⁢

p

)

=

u

⁢

∂

⁡

p

∂

⁡

x

+

p

⁢

∂

⁡

u

∂

⁡

x

-

v

⁢

∂

⁡

q

∂

⁡

x

-

q

⁢

∂

⁡

v

∂

⁡

x

-

u

⁢

∂

⁡

q

∂

⁡

y

-

q

⁢

∂

⁡

u

∂

⁡

y

-

v

⁢

∂

⁡

p

∂

⁡

y

-

p

⁢

∂

⁡

v

∂

⁡

y

=

u

⁢

(

∂

⁡

p

∂

⁡

x

-

∂

⁡

q

∂

⁡

y

)

-

v

⁢

(

∂

⁡

p

∂

⁡

y

+

∂

⁡

q

∂

⁡

x

)

+

p

⁢

(

∂

⁡

u

∂

⁡

x

-

∂

⁡

v

∂

⁡

y

)

-

q

⁢

(

∂

⁡

u

∂

⁡

y

+

∂

⁡

v

∂

⁡

x

)

=

0

and

∂

⁡

h

∂

⁡

y

+

∂

⁡

k

∂

⁡

x

=

∂

∂

⁡

y

⁢

(

u

⁢

p

-

v

⁢

q

)

+

∂

∂

⁡

x

⁢

(

u

⁢

q

+

v

⁢

p

)

=

u

⁢

∂

⁡

p

∂

⁡

y

+

p

⁢

∂

⁡

u

∂

⁡

y

-

v

⁢

∂

⁡

q

∂

⁡

y

-

q

⁢

∂

⁡

v

∂

⁡

y

+

u

⁢

∂

⁡

q

∂

⁡

x

+

q

⁢

∂

⁡

u

∂

⁡

x

+

v

⁢

∂

⁡

p

∂

⁡

x

+

p

⁢

∂

⁡

v

∂

⁡

x

=

u

⁢

(

∂

⁡

p

∂

⁡

y

+

∂

⁡

q

∂

⁡

x

)

+

v

⁢

(

∂

⁡

p

∂

⁡

x

-

∂

⁡

q

∂

⁡

y

)

+

p

⁢

(

∂

⁡

u

∂

⁡

y

+

∂

⁡

v

∂

⁡

x

)

+

q

⁢

(

∂

⁡

u

∂

⁡

x

-

∂

⁡

v

∂

⁡

y

)

=

0

.

∎

Theorem 3

.

If

f

:

D

→

D

′

and

g

:

D

′

→

C

satisfy the
Cauchy-Riemann equations, so does

f

∘

g

.

Proof.

Letting

h

and

k

denote the real and imaginary parts of

f

∘

g

respectively, we have

∂

⁡

h

∂

⁡

x

-

∂

⁡

k

∂

⁡

y

=

∂

∂

⁡

x

⁢

u

⁢

(

p

⁢

(

x

,

y

)

,

q

⁢

(

x

,

y

)

)

-

∂

∂

⁡

y

⁢

v

⁢

(

p

⁢

(

x

,

y

)

,

q

⁢

(

x

,

y

)

)

=

∂

⁡

u

∂

⁡

p

⁢

∂

⁡

p

∂

⁡

x

+

∂

⁡

u

∂

⁡

q

⁢

∂

⁡

q

∂

⁡

x

-

∂

⁡

v

∂

⁡

p

⁢

∂

⁡

p

∂

⁡

y

-

∂

⁡

v

∂

⁡

q

⁢

∂

⁡

q

∂

⁡

y

=

∂

⁡

u

∂

⁡

p

⁢

(

∂

⁡

p

∂

⁡

x

-

∂

⁡

q

∂

⁡

y

)

+

∂

⁡

q

∂

⁡

y

⁢

(

∂

⁡

u

∂

⁡

p

-

∂

⁡

v

∂

⁡

q

)

+

∂

⁡

u

∂

⁡

q

⁢

(

∂

⁡

p

∂

⁡

y

+

∂

⁡

q

∂

⁡

x

)

-

∂

⁡

p

∂

⁡

y

⁢

(

∂

⁡

u

∂

⁡

q

+

∂

⁡

v

∂

⁡

p

)

=

0

and

∂

⁡

h

∂

⁡

y

+

∂

⁡

k

∂

⁡

x

=

∂

∂

⁡

y

⁢

u

⁢

(

p

⁢

(

x

,

y

)

,

q

⁢

(

x

,

y

)

)

+

∂

∂

⁡

x

⁢

v

⁢

(

p

⁢

(

x

,

y

)

,

q

⁢

(

x

,

y

)

)

=

∂

⁡

u

∂

⁡

p

⁢

∂

⁡

p

∂

⁡

y

+

∂

⁡

u

∂

⁡

q

⁢

∂

⁡

q

∂

⁡

y

+

∂

⁡

v

∂

⁡

p

⁢

∂

⁡

p

∂

⁡

x

+

∂

⁡

v

∂

⁡

q

⁢

∂

⁡

q

∂

⁡

x

=

∂

⁡

u

∂

⁡

p

⁢

(

∂

⁡

p

∂

⁡

y

+

∂

⁡

q

∂

⁡

x

)

-

∂

⁡

q

∂

⁡

x

⁢

(

∂

⁡

u

∂

⁡

p

-

∂

⁡

v

∂

⁡

q

)

-

∂

⁡

u

∂

⁡

q

⁢

(

∂

⁡

p

∂

⁡

x

-

∂

⁡

q

∂

⁡

y

)

+

∂

⁡

p

∂

⁡

x

⁢

(

∂

⁡

u

∂

⁡

q

+

∂

⁡

v

∂

⁡

p

)

=

0

∎

Theorem 4

.

If

f

:

D

→

C

satisfies the Cauchy-Riemann equations,
and has non-vanishing Jacobian, then

f

-

1

also satisfies the Cauchy-Riemann equations.

Proof.

Let us denote the real and imaginary parts of

f

-

1

as

h

and

k

, respectively.
Then, by

definition

of

inverse function

, we have

u

⁢

(

h

⁢

(

x

,

y

)

,

k

⁢

(

x

,

y

)

)

=

x

v

⁢

(

h

⁢

(

x

,

y

)

,

k

⁢

(

x

,

y

)

)

=

y

.

Taking derivatives,

∂

⁡

u

∂

⁡

h

⁢

∂

⁡

h

∂

⁡

x

+

∂

⁡

u

∂

⁡

k

⁢

∂

⁡

k

∂

⁡

x

=

1

∂

⁡

u

∂

⁡

h

⁢

∂

⁡

h

∂

⁡

y

+

∂

⁡

u

∂

⁡

k

⁢

∂

⁡

k

∂

⁡

y

=

0

∂

⁡

v

∂

⁡

h

⁢

∂

⁡

h

∂

⁡

x

+

∂

⁡

v

∂

⁡

k

⁢

∂

⁡

k

∂

⁡

x

=

0

∂

⁡

v

∂

⁡

h

⁢

∂

⁡

h

∂

⁡

y

+

∂

⁡

v

∂

⁡

k

⁢

∂

⁡

k

∂

⁡

y

=

1

By the Cauchy-Riemann equations,

∂

⁡

u

/

∂

⁡

h

=

∂

⁡

v

/

∂

⁡

k

and

∂

⁡

u

/

∂

⁡

k

=

-

∂

⁡

v

/

∂

⁡

h

. Using these

relations

to re-express the
derivatives of

u

as derivatives of

v

, then subtracting the
fourth equation form the first equation and adding the second and
third equations, we obtain

∂

⁡

u

∂

⁡

h

⁢

(

∂

⁡

h

∂

⁡

x

-

∂

⁡

k

∂

⁡

y

)

+

∂

⁡

u

∂

⁡

k

⁢

(

∂

⁡

h

∂

⁡

y

+

∂

⁡

k

∂

⁡

x

)

=

0

∂

⁡

u

∂

⁡

h

⁢

(

∂

⁡

h

∂

⁡

y

+

∂

⁡

k

∂

⁡

x

)

-

∂

⁡

u

∂

⁡

k

⁢

(

∂

⁡

h

∂

⁡

x

-

∂

⁡

k

∂

⁡

y

)

=

0

.

With a little algebraic manipulation, we may conclude

(

(

∂

⁡

u

∂

⁡

h

)

2

+

(

∂

⁡

u

∂

⁡

k

)

2

)

⁢

(

∂

⁡

h

∂

⁡

y

+

∂

⁡

k

∂

⁡

x

)

=

0

(

(

∂

⁡

u

∂

⁡

h

)

2

+

(

∂

⁡

u

∂

⁡

k

)

2

)

⁢

(

∂

⁡

h

∂

⁡

x

-

∂

⁡

k

∂

⁡

y

)

=

0

.

Note that, by the Cauchy-Riemann equations, the Jacobian of

f

equals the common prefactor of these equations:

∂

⁡

(

u

,

v

)

∂

⁡

(

h

,

k

)

=

∂

⁡

u

∂

⁡

h

⁢

∂

⁡

v

∂

⁡

k

-

∂

⁡

u

∂

⁡

k

⁢

∂

⁡

v

∂

⁡

h

=

(

∂

⁡

u

∂

⁡

h

)

2

+

(

∂

⁡

u

∂

⁡

k

)

2

Hence, by

assumptions

, this quantity differs from zero and we may
cancel it to obtain the Cauchy-Riemann equations for

f

-

1

.
∎

Title

closure properties of Cauchy-Riemann equations

Canonical

name

ClosurePropertiesOfCauchyRiemannEquations

Date of creation

2013-03-22 17:44:20

Last modified on

2013-03-22 17:44:20

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

14

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 30E99

Related topic

TangentialCauchyRiemannComplexOfCinftySmoothForms

Related topic

ACRcomplex

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