aarjaneiro/planetmath huggingface dataset:
proof of criterion for conformal mapping of Riemannian spaces

proof of criterion for conformal mapping of Riemannian spaces

In this attachment, we prove that the a

mapping

f

of Riemannian (or
pseudo-Riemannian) spaces

(

M

,

g

)

and

(

N

,

h

)

is

conformal

if and only if

f

*

⁢

h

=

s

⁢

g

for some scalar field

s

(on

M

).

The key observation is that the angle

A

between curves

S

and

T

which

intersect

at a point

P

is determined by the

tangent vectors

to these two curves
(which we shall term

s

and

t

) and the metric at that point, like so:

cos

⁡

A

=

g

⁢

(

s

,

t

)

g

⁢

(

s

,

s

)

⁢

g

⁢

(

t

,

t

)

Moreover, given any tangent vector at a point, there will exist at least one
curve to which it is the

tangent

. Also, the tangent vector to the image of
a curve under a map is the pushforward of the tangent to the original curve
under the map; for

instance

, the tangent to

f

⁢

(

S

)

at

f

⁢

(

P

)

is

f

*

⁢

s

. Hence,
the mapping

f

is conformal if and only if

g

⁢

(

u

,

v

)

g

⁢

(

u

,

u

)

⁢

g

⁢

(

v

,

v

)

=

h

⁢

(

f

*

⁢

u

,

f

*

⁢

v

)

h

⁢

(

f

*

⁢

u

,

f

*

⁢

u

)

⁢

h

⁢

(

f

*

⁢

v

,

f

*

⁢

v

)

for all tangent vectors

u

and

v

to the manifold

M

. By the way pushforwards
and

pullbacks

work, this is

equivalent

to the condition that

g

⁢

(

u

,

v

)

g

⁢

(

u

,

u

)

⁢

g

⁢

(

v

,

v

)

=

(

f

*

⁢

h

)

⁢

(

u

,

v

)

(

f

*

⁢

h

)

⁢

(

u

,

u

)

⁢

(

f

*

⁢

h

)

⁢

(

v

,

v

)

for all tangent vectors

u

and

v

to the manifold

N

. Now, by elementary
algebra, the above equation is equivalent to the requirement that there
exist a scalar

s

such that, for all

u

and

v

, it is the case that

g

⁢

(

u

,

v

)

=

s

⁢

h

*

⁢

(

u

,

v

)

or, in other words,

f

*

⁢

h

=

s

⁢

g

for some scalar
field

s

.

Title

proof of criterion for conformal mapping of Riemannian spaces

Canonical name

ProofOfCriterionForConformalMappingOfRiemannianSpaces

Date of creation

2013-03-22 16:22:03

Last modified on

2013-03-22 16:22:03

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 30E20

Generated on Fri Feb 9 20:21:06 2018 by

LaTeXML
proof of Gauss’ mean value theorem

proof of Gauss’ mean value theorem

We can parameterize the circle by letting

z

=

z

0

+

r

⁢

e

i

⁢

ϕ

.
Then

d

⁢

z

=

i

⁢

r

⁢

e

i

⁢

ϕ

⁢

d

⁢

ϕ

. Using the

Cauchy integral formula

we can express

f

⁢

(

z

0

)

in the following way:

f

⁢

(

z

0

)

=

1

2

⁢

π

⁢

i

⁢

∮

C

f

⁢

(

z

)

z

-

z

0

⁢

𝑑

z

=

1

2

⁢

π

⁢

i

⁢

∫

0

2

⁢

π

f

⁢

(

z

0

+

r

⁢

e

i

⁢

ϕ

)

r

⁢

e

i

⁢

ϕ

⁢

i

⁢

r

⁢

e

i

⁢

ϕ

⁢

𝑑

ϕ

=

1

2

⁢

π

⁢

∫

0

2

⁢

π

f

⁢

(

z

0

+

r

⁢

e

i

⁢

ϕ

)

⁢

𝑑

ϕ

.

Title

proof of Gauss’ mean value theorem

Canonical name

ProofOfGaussMeanValueTheorem

Date of creation

2013-03-22 13:35:36

Last modified on

2013-03-22 13:35:36

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

18

Author

yark (2760)

Entry type

Proof

Classification

msc 30E20

Related topic

CauchyIntegralFormula

Generated on Fri Feb 9 20:21:11 2018 by

LaTeXML
Proof of Möbius transformation cross-ratio preservation theorem

Proof of Möbius transformation cross-ratio preservation theorem

From the definition of Möbius transform we get
that the image

w

k

of a point

z

k

is

w

k

=

a

⁢

z

k

+

b

c

⁢

z

k

+

d

From this we get

w

i

-

w

j

=

a

⁢

z

i

+

b

c

⁢

z

i

+

d

-

a

⁢

z

j

+

b

c

⁢

z

j

+

d

=

(

a

⁢

d

-

b

⁢

c

)

⁢

(

z

i

-

z

j

)

(

c

⁢

z

i

+

d

)

⁢

(

c

⁢

z

j

+

d

)

and by inserting this into the cross-ratios

(

w

1

-

w

2

)

⁢

(

w

3

-

w

4

)

(

w

1

-

w

4

)

⁢

(

w

3

-

w

2

)

=

(

a

⁢

d

-

b

⁢

c

)

⁢

(

z

1

-

z

2

)

(

c

⁢

z

1

+

d

)

⁢

(

c

⁢

z

2

+

d

)

⁢

(

a

⁢

d

-

b

⁢

c

)

⁢

(

z

3

-

z

4

)

(

c

⁢

z

3

+

d

)

⁢

(

c

⁢

z

4

+

d

)

(

a

⁢

d

-

b

⁢

c

)

⁢

(

z

1

-

z

4

)

(

c

⁢

z

1

+

d

)

⁢

(

c

⁢

z

4

+

d

)

⁢

(

a

⁢

d

-

b

⁢

c

)

⁢

(

z

3

-

z

2

)

(

c

⁢

z

3

+

d

)

⁢

(

c

⁢

z

2

+

d

)

=

(

z

1

-

z

2

)

⁢

(

z

3

-

z

4

)

(

z

1

-

z

4

)

⁢

(

z

3

-

z

2

)

Title

Proof of Möbius

transformation

cross-ratio

preservation

theorem

Canonical name

ProofOfMobiusTransformationCrossratioPreservationTheorem

Date of creation

2013-03-22 14:08:20

Last modified on

2013-03-22 14:08:20

Owner

Johan (1032)

Last modified by

Johan (1032)

Numerical id

5

Author

Johan (1032)

Entry type

Proof

Classification

msc 30E20

Generated on Fri Feb 9 20:21:19 2018 by

LaTeXML
proof of Rouché’s theorem

proof of Rouché’s theorem

Consider the

integral

N

⁢

(

λ

)

=

1

2

⁢

π

⁢

i

⁢

∮

C

f

′

⁢

(

z

)

+

λ

⁢

g

′

⁢

(

z

)

f

⁢

(

z

)

+

λ

⁢

g

⁢

(

z

)

⁢

𝑑

z

where

0

≤

λ

≤

1

. By the hypotheses, the

function

f

+

λ

⁢

g

is

non-singular

on

C

or on the

interior

of

C

and has no zeros on

C

. Hence, by the

argument principle

,

N

⁢

(

λ

)

equals the number of zeros (counted with

multiplicity

) of

f

+

λ

⁢

g

contained

inside

C

. Note that this means that

N

⁢

(

λ

)

must be an integer.

Since

C

is

compact

, both

|

f

|

and

|

g

|

attain minima and maxima on

C

. Hence there exist

positive

real

constants

a

and

b

such that

|

f

⁢

(

z

)

|

>

a

>

b

>

|

g

⁢

(

z

)

|

for all

z

on

C

. By the

triangle inequality

, this implies that

|

f

⁢

(

z

)

+

λ

⁢

g

⁢

(

z

)

|

>

a

-

b

on

C

. Hence

1

/

(

f

+

λ

⁢

g

)

is a

continuous function

of

λ

when

0

≤

λ

≤

1

and

z

∈

C

. Therefore, the

integrand

is a continuous function of

C

and

λ

. Since

C

is compact, it follows that

N

⁢

(

λ

)

is a continuous function of

λ

.

Now there is only one way for a continuous function of a real variable to assume only integer values – that function must be constant. In particular, this means that the number of zeros of

f

+

λ

⁢

g

inside

C

is the same for all

λ

. Taking the extreme cases

λ

=

0

and

λ

=

1

, this means that

f

and

f

+

g

have the same number of zeros inside

C

.

Title

proof of Rouché’s

theorem

Canonical

name

ProofOfRouchesTheorem

Date of creation

2013-03-22 14:34:26

Last modified on

2013-03-22 14:34:26

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 30E20

Generated on Fri Feb 9 20:21:21 2018 by

LaTeXML
proof of Simultaneous converging or diverging of product and sum theorem

proof of Simultaneous converging or diverging of product and sum theorem

From the fact that

1

+

x

≤

e

x

for

x

≥

0

we get

∑

n

=

1

m

a

n

≤

∏

n

=

1

m

(

1

+

a

n

)

≤

e

∑

n

=

1

m

a

n

Since

a

n

≥

0

both the

partial sums

and the partial products are

monotone increasing

with the number of terms. This concludes the proof.

Title

proof of Simultaneous converging or diverging of product and sum theorem

Canonical name

ProofOfSimultaneousConvergingOrDivergingOfProductAndSumTheorem

Date of creation

2013-03-22 13:35:57

Last modified on

2013-03-22 13:35:57

Owner

Johan (1032)

Last modified by

Johan (1032)

Numerical id

5

Author

Johan (1032)

Entry type

Proof

Classification

msc 30E20

Generated on Fri Feb 9 20:21:23 2018 by

LaTeXML
Riemannian manifolds category R_M

Riemannian manifolds category

R

M

Definition 0.1

.

A

category

ℛ

M

whose objects are all

Riemannian manifolds

R

and whose

morphisms

are mappings between Riemannian manifolds

m

R

is defined as the

category of Riemannian manifolds

.

0.1

Applications of Riemannian manifolds in mathematical physics

1.

The

conformal Riemannian subcategory

ℛ

C

of

ℛ

M

, whose objects are Riemannian manifolds

R

, and whose morphisms are

conformal mappings

of Riemannian manifolds

c

R

, is an important category for mathematical physics, in

conformal

theories.

2.

It can be shown that, if

(

R

1

,

g

)

and

(

R

2

,

h

)

are Riemannian manifolds, then
a map

f

:

R

1

→

R

2

is conformal (

http://planetmath.org/ConformalMapping

) iff

f

*

⁢

h

=

s

.

g

for some scalar field

s

(on

R

1

), where

f

*

is the

complex conjugate

of

f

.

0.1.1

Category of pseudo-Riemannian manifolds

The category of pseudo-Riemannian manifolds (

http://planetmath.org/PseudoRiemannianManifold

)

ℛ

P

that generalize

Minkowski spaces

M

k

is similarly defined by replacing the Riemanian

manifolds

R

in the above definition with

pseudo-Riemannian manifolds

R

P

. Pseudo-Riemannian manifolds

R

P

s were claimed to have applications in Einstein’s theory of general relativity (

G

⁢

R

), whereas the

subcategory

𝐌𝐢𝐧𝐤

of four-dimensional Minkowski spaces in

ℛ

P

plays the central role in special relativity (

S

⁢

R

) theories.

Title

Riemannian manifolds category

R

M

Canonical name

RiemannianManifoldsCategoryRM

Date of creation

2013-03-22 18:25:13

Last modified on

2013-03-22 18:25:13

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

26

Author

bci1 (20947)

Entry type

Definition

Classification

msc 30E20

Classification

msc 18-00

Classification

msc 53B20

Classification

msc 53B21

Related topic

RiemannianMetric

Related topic

ConformalMapping

Related topic

ExampleOfConformalMapping

Related topic

PseudoRiemannianManifold

Related topic

IndexOfCategories

Related topic

EinsteinFieldEquations

Defines

category of pseudo-Riemannian manifolds

Defines

conformal Riemannian subcategory

Defines

conformal Riemannian manifold

Defines

conformal mapping

Defines

c

R

Generated on Fri Feb 9 20:21:25 2018 by

LaTeXML
Rouché’s theorem

Rouché’s theorem

Let

f

,

g

be analytic on and inside a

simple closed curve

C

. Suppose

|

f

⁢

(

z

)

|

>

|

g

⁢

(

z

)

|

on

C

. Then

f

and

f

+

g

have the same number of zeros inside

C

.

Title

Rouché’s theorem

Canonical name

RouchesTheorem

Date of creation

2013-03-22 13:35:39

Last modified on

2013-03-22 13:35:39

Owner

Johan (1032)

Last modified by

Johan (1032)

Numerical id

8

Author

Johan (1032)

Entry type

Theorem

Classification

msc 30E20

Generated on Fri Feb 9 20:21:26 2018 by

LaTeXML
second form of Cauchy integral theorem

second form of Cauchy integral theorem

Theorem

.

Let the

complex function

f

be

analytic

in a

simply connected

open domain

U

of the

complex plane

, and let

a

and

b

be any two points of

U

.  Then the

contour integral

∫

γ

f

⁢

(

z

)

⁢

𝑑

z

(1)

is

independent

on the path

γ

which in

U

goes from

a

to

b

.

Example.

Let’s consider the integral (1) of the

real part

function

defined by

f

⁢

(

z

)

:=

Re

⁢

(

z

)

with the path

γ

going from the point

O

=

(

0

,

0

)

to the point

Q

=

(

1

,

1

)

.  If

γ

is the

line segment

O

⁢

Q

, we may use the substitution

z

:=

(

1

+

i

)

⁢

t

,

d

⁢

z

=

(

1

+

i

)

⁢

d

⁢

t

,

0

≦

t

≦

1

,

and (1) equals

∫

0

1

t

⋅

(

1

+

i

)

⁢

𝑑

t

=

1

2

+

1

2

⁢

i

.

Secondly, we choose for

γ

the

broken line

O

⁢

P

⁢

Q

where

P

=

(

1

,

0

)

.  Now (1) is the sum

∫

O

⁢

P

Re

⁢

(

z

)

⁢

𝑑

z

+

∫

P

⁢

Q

Re

⁢

(

z

)

⁢

𝑑

z

=

∫

0

1

x

⁢

𝑑

x

+

∫

0

1

i

⁢

𝑑

y

=

1

2

+

i

.

Thus, the integral (1) of the function depends on the path between the two points.  This is explained by the fact that the function

f

is not analytic — its real part

x

and

imaginary part

0 do not satisfy the

Cauchy-Riemann equations

.

Title

second form of Cauchy integral theorem

Canonical name

SecondFormOfCauchyIntegralTheorem

Date of creation

2013-03-22 15:19:39

Last modified on

2013-03-22 15:19:39

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Theorem

Classification

msc 30E20

Synonym

equivalent form of Cauchy integral theorem

Related topic

CauchyIntegralTheorem

Defines

example of non-analytic function

Generated on Fri Feb 9 20:21:28 2018 by

LaTeXML
simple example of composed conformal mapping

simple example of composed conformal mapping

Let’s consider the mapping

f

:

ℂ

→

ℂ

with

f

⁢

(

z

)

=

a

⁢

z

+

b

,

where

a

and

b

are

complex

and

a

≠

0

.

Because

f

′

⁢

(

z

)

≡

a

≠

0

,  the mapping is

conformal

in the whole

z

-plane.  Denote

a

:=

ϱ

⁢

e

i

⁢

α

(where

ϱ

,

α

∈

ℝ

) and

z

1

:=

ϱ

⁢

z

,

(1)

z

2

:=

e

i

⁢

α

⁢

z

1

,

(2)

w

:=

z

2

+

b

.

(3)

Then the mapping

z

↦

z

1

means a

dilation

in the complex plane, the mapping

z

1

↦

z

2

a

rotation

by the angle

α

and the mapping

z

2

↦

w

a

translation

determined by the vector from the origin to the point

b

.  Thus

f

is composed of these three consecutive mappings which all are conformal.

Figure 1:

The mapping from

z

to

z

1

Figure 2:

The mapping from

z

1

to

z

2

Figure 3:

The mapping from

z

2

to

w

Title

simple example of composed conformal mapping

Canonical name

SimpleExampleOfComposedConformalMapping

Date of creation

2013-03-22 16:47:25

Last modified on

2013-03-22 16:47:25

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Example

Classification

msc 30E20

Classification

msc 53A30

Generated on Fri Feb 9 20:21:30 2018 by

LaTeXML
simultaneous converging or diverging of product and sum theorem

simultaneous converging or diverging of product and sum theorem

Let

a

k

≥

0

. Then

∏

n

=

1

∞

(

1

+

a

n

)

⁢

and

⁢

∑

n

=

1

∞

a

n

converge or diverge simultaneously.

Title

simultaneous converging or diverging of product and sum theorem

Canonical name

SimultaneousConvergingOrDivergingOfProductAndSumTheorem

Date of creation

2013-03-22 13:35:55

Last modified on

2013-03-22 13:35:55

Owner

Johan (1032)

Last modified by

Johan (1032)

Numerical id

6

Author

Johan (1032)

Entry type

Theorem

Classification

msc 30E20

Generated on Fri Feb 9 20:21:32 2018 by

LaTeXML
sum of values of holomorphic function

sum of values of holomorphic function

Let

w

⁢

(

z

)

be a

holomorphic function

on a

simple closed curve

C

and inside it.  If

a

1

,

a

2

,

…

,

a

m

are inside

C

the simple zeros of a

function

f

⁢

(

z

)

holomorphic on

C

and inside, then

w

⁢

(

a

1

)

+

w

⁢

(

a

2

)

+

…

+

w

⁢

(

a

m

)

=

1

2

⁢

i

⁢

π

⁢

∮

C

w

⁢

(

z

)

⁢

f

′

⁢

(

z

)

f

⁢

(

z

)

⁢

𝑑

z

(1)

where the

contour integral

is taken anticlockwise.

The if some of the zeros are

multiple

and are counted with

multiplicities

(

http://planetmath.org/Pole

).

If the zeros

a

j

of

f

⁢

(

z

)

have the multiplicities

α

j

and the function has inside

C

also the poles

b

1

,

b

2

,

…

,

b

n

with the multiplicities

β

1

,

β

2

,

…

,

β

n

,  then (1) must be written

∑

j

α

j

⁢

w

⁢

(

a

j

)

-

∑

k

β

k

⁢

w

⁢

(

b

k

)

=

1

2

⁢

i

⁢

π

⁢

∮

C

w

⁢

(

z

)

⁢

f

′

⁢

(

z

)

f

⁢

(

z

)

⁢

𝑑

z

.

(2)

The special case

w

⁢

(

z

)

≡

1

gives from (2) the

argument principle

.

References

1

Ernst Lindelöf

:

Le calcul des résidus et ses applications à la théorie des fonctions

.  Gauthier-Villars, Paris (1905).

Title

sum of values of holomorphic function

Canonical name

SumOfValuesOfHolomorphicFunction

Date of creation

2013-03-22 19:15:30

Last modified on

2013-03-22 19:15:30

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Theorem

Classification

msc 30E20

Generated on Fri Feb 9 20:21:34 2018 by

LaTeXML
variant of Cauchy integral formula

variant of Cauchy integral formula

Theorem.

Let

f

⁢

(

z

)

be

holomorphic

in a domain

A

of

ℂ

.  If

C

is a closed

contour

not intersecting itself which with its domain is contained in

A

and if

z

is an arbitrary point inside

C

, then

f

⁢

(

z

)

=

1

2

⁢

i

⁢

π

⁢

∮

C

f

⁢

(

t

)

t

-

z

⁢

𝑑

t

.

(1)

Proof.

Let

ε

be any positive number.  Denote by

C

r

the circles with radius

r

and centered in

z

.  We have

∮

C

f

⁢

(

t

)

t

-

z

⁢

𝑑

t

=

∮

C

f

⁢

(

z

)

+

(

f

⁢

(

t

)

-

f

⁢

(

z

)

)

t

-

z

⁢

𝑑

t

=

∮

C

f

⁢

(

z

)

t

-

z

⁢

𝑑

t

⏟

I

+

∮

C

f

⁢

(

t

)

-

f

⁢

(

z

)

t

-

z

⁢

𝑑

t

⏟

J

.

According to the

corollary of Cauchy integral theorem

and its example, we may write

I

=

f

⁢

(

z

)

⁢

∮

C

d

⁢

t

t

-

z

=

2

⁢

i

⁢

π

⁢

f

⁢

(

z

)

.

If

0

<

r

<

some

⁢

r

0

,  we have

J

=

∮

C

r

f

⁢

(

t

)

-

f

⁢

(

z

)

t

-

z

⁢

𝑑

t

.

The continuity of

f

in the point

z

implies, that

|

f

⁢

(

t

)

-

f

⁢

(

z

)

|

<

ε

when

0

<

|

t

-

z

|

<

some

⁢

δ

ε

i.e. when

t

∈

C

r

⁢

and

⁢

0

<

r

<

some

⁢

r

1

.

(2)

If (2) is in , we obtain first

|

f

⁢

(

t

)

-

f

⁢

(

z

)

t

-

z

|

=

|

f

⁢

(

t

)

-

f

⁢

(

z

)

|

|

t

-

z

|

=

|

f

⁢

(

t

)

-

f

⁢

(

z

)

|

r

<

ε

r

,

whence, by the

estimation theorem of integral

,

|

J

|

≦

ε

r

⋅

2

⁢

π

⁢

r

=

2

⁢

π

⁢

ε

for

0

<

r

<

min

⁡

{

r

0

,

r

1

}

,

and lastly

|

1

2

⁢

i

⁢

π

⁢

∮

C

f

⁢

(

t

)

t

-

z

⁢

𝑑

t

-

f

⁢

(

z

)

|

=

|

1

2

⁢

i

⁢

π

⁢

J

|

≦

1

2

⁢

π

⋅

2

⁢

π

⁢

ε

=

ε

when

⁢

0

<

r

<

min

⁡

{

r

0

,

r

1

}

.

(3)

This result implies (1).

Title

variant of Cauchy integral formula

Canonical name

VariantOfCauchyIntegralFormula

Date of creation

2013-03-22 18:54:15

Last modified on

2013-03-22 18:54:15

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Theorem

Classification

msc 30E20

Synonym

Cauchy integral formula

Related topic

CauchyIntegralFormula

Related topic

CorollaryOfCauchyIntegralTheorem

Related topic

ExampleOfFindingTheGeneratingFunction

Related topic

GeneratingFunctionOfLaguerrePolynomials

Related topic

GeneratingFunctionOfHermitePolynomials

Generated on Fri Feb 9 20:21:45 2018 by

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zero as contour integral

zero as contour integral

Suppose that

f

is a

complex function

which is defined in some open
set

D

⊆

ℂ

which has a

simple zero

at some point

p

∈

D

. Then we have

p

=

1

2

⁢

π

⁢

i

⁢

∮

C

z

⁢

f

′

⁢

(

z

)

f

⁢

(

z

)

⁢

𝑑

z

where

C

is a

closed path

in

D

which encloses

p

but does not enclose or

pass through

any other zeros of

f

.

This follows from the

Cauchy residue theorem

. We have that the poles of

f

′

/

f

occur at the zeros of

f

and that the

residue

of a pole of

f

′

/

f

is

1

at a simple zero of

f

. Hence, the residue of

z

⁢

f

′

⁢

(

z

)

/

f

⁢

(

z

)

at

p

is

p

, so the above follows from the

residue theorem

.

Title

zero as contour integral

Canonical name

ZeroAsContourIntegral

Date of creation

2013-03-22 16:46:42

Last modified on

2013-03-22 16:46:42

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Corollary

Classification

msc 30E20

Generated on Fri Feb 9 20:21:47 2018 by

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Cauchy-Riemann equations

Cauchy-Riemann equations

The following system of

partial differential
equations

∂

⁡

u

∂

⁡

x

=

∂

⁡

v

∂

⁡

y

,

∂

⁡

u

∂

⁡

y

=

-

∂

⁡

v

∂

⁡

x

,

where

u

⁢

(

x

,

y

)

,

v

⁢

(

x

,

y

)

are real-valued functions defined on some
open subset of

ℝ

2

, was introduced by Riemann[1] as a
definition of a

holomorphic function

. Indeed, if

f

⁢

(

z

)

satisfies the
standard definition of a holomorphic function, i.e. if the

complex derivative

f

′

⁢

(

z

)

=

lim

ζ

→

0

⁡

f

⁢

(

z

+

ζ

)

-

f

⁢

(

z

)

ζ

exists in the domain of definition, then the real and

imaginary parts

of

f

⁢

(

z

)

satisfy the

Cauchy-Riemann equations

.
Conversely, if

u

and

v

satisfy the Cauchy-Riemann equations, and if their

partial derivatives

are

continuous

, then the complex valued

function

f

⁢

(

z

)

=

u

⁢

(

x

,

y

)

+

i

⁢

v

⁢

(

x

,

y

)

,

z

=

x

+

i

⁢

y

,

possesses a continuous complex derivative.

References

1.

D. Laugwitz,

Bernhard Riemann, 1826-1866:
Turning points in the Conception of
Mathematics

, translated by Abe Shenitzer. Birkhauser, 1999.

Title

Cauchy-Riemann equations

Canonical name

CauchyRiemannEquations

Date of creation

2013-03-22 12:55:36

Last modified on

2013-03-22 12:55:36

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

5

Author

rmilson (146)

Entry type

Definition

Classification

msc 30E99

Related topic

Holomorphic

Generated on Fri Feb 9 20:21:49 2018 by

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Cauchy-Riemann equations (complex coordinates)

Cauchy-Riemann equations (complex coordinates)

Let

f

:

G

⊂

ℂ

→

ℂ

be a

continuously differentiable function

in the real sense, using

ℝ

2

instead of

ℂ

, identifying

f

⁢

(

z

)

with

f

⁢

(

x

,

y

)

where

z

=

x

+

i

⁢

y

and we also write

z

¯

=

x

-

i

⁢

y

(the

complex conjugate

). Then we have the following

partial derivatives

:

∂

⁡

f

∂

⁡

z

:=

1

2

⁢

(

∂

⁡

f

∂

⁡

x

-

i

⁢

∂

⁡

f

∂

⁡

y

)

,

∂

⁡

f

∂

⁡

z

¯

:=

1

2

⁢

(

∂

⁡

f

∂

⁡

x

+

i

⁢

∂

⁡

f

∂

⁡

y

)

.

Sometimes these are written as

f

z

and

f

z

¯

respectively.

The classical

Cauchy-Riemann equations

are

equivalent

to

∂

⁡

f

∂

⁡

z

¯

=

0

.

This can be seen if we write

f

=

u

+

i

⁢

v

for real valued

u

and

v

and
then the

differentials

become

∂

⁡

f

∂

⁡

z

=

1

2

⁢

(

∂

⁡

u

∂

⁡

x

+

∂

⁡

v

∂

⁡

y

)

+

i

2

⁢

(

∂

⁡

v

∂

⁡

x

-

∂

⁡

u

∂

⁡

y

)

,

∂

⁡

f

∂

⁡

z

¯

=

1

2

⁢

(

∂

⁡

u

∂

⁡

x

-

∂

⁡

v

∂

⁡

y

)

+

i

2

⁢

(

∂

⁡

v

∂

⁡

x

+

∂

⁡

u

∂

⁡

y

)

.

In several complex dimensions, for a

function

f

:

G

⊂

ℂ

n

→

ℂ

which maps

(

z

1

,

…

,

z

n

)

↦

f

⁢

(

z

1

,

…

,

z

n

)

where

z

j

=

x

j

+

i

⁢

y

j

we generalize simply by

∂

⁡

f

∂

⁡

z

j

:=

1

2

⁢

(

∂

⁡

f

∂

⁡

x

j

-

i

⁢

∂

⁡

f

∂

⁡

y

j

)

,

∂

⁡

f

∂

⁡

z

¯

j

:=

1

2

⁢

(

∂

⁡

f

∂

⁡

x

j

+

i

⁢

∂

⁡

f

∂

⁡

y

j

)

.

Then the Cauchy-Riemann equations are given by

∂

⁡

f

∂

⁡

z

¯

j

=

0

for all

1

≤

j

≤

n

.

That is,

f

is

holomorphic

if and only if it

satisfies

the above equations.

References

1

Steven G. Krantz.
,
AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title

Cauchy-Riemann equations (

complex

coordinates)

Canonical name

CauchyRiemannEquationscomplexCoordinates

Date of creation

2013-03-22 14:24:28

Last modified on

2013-03-22 14:24:28

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Definition

Classification

msc 30E99

Related topic

CauchyRiemannEquations

Related topic

Holomorphic

Generated on Fri Feb 9 20:21:51 2018 by

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Cauchy-Riemann equations (polar coordinates)

Cauchy-Riemann equations (polar coordinates)

Suppose

A

is an open set in

ℂ

and

f

⁢

(

z

)

=

f

⁢

(

r

⁢

e

i

⁢

θ

)

=

u

⁢

(

r

,

θ

)

+

i

⁢

v

⁢

(

r

,

θ

)

:

A

⊂

ℂ

→

ℂ

is a

function

. If the derivative of

f

⁢

(

z

)

exists at

z

0

=

(

r

0

,

θ

0

)

. Then the functions

u

,

v

at

z

0

satisfy:

∂

⁡

u

∂

⁡

r

=

1

r

⁢

∂

⁡

v

∂

⁡

θ

∂

⁡

v

∂

⁡

r

=

-

1

r

⁢

∂

⁡

u

∂

⁡

θ

which are called

Cauchy-Riemann equations

in

polar form

.

Title

Cauchy-Riemann equations (

polar coordinates

)

Canonical name

CauchyRiemannEquationspolarCoordinates

Date of creation

2013-03-22 14:03:58

Last modified on

2013-03-22 14:03:58

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

8

Author

Daume (40)

Entry type

Definition

Classification

msc 30E99

Related topic

TangentialCauchyRiemannComplexOfCinftySmoothForms

Related topic

ACRcomplex

Generated on Fri Feb 9 20:21:53 2018 by

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