aarjaneiro/planetmath huggingface dataset:
quadratic equation in ℂ

quadratic equation in

ℂ

The

quadratic formula

x

=

-

b

±

b

2

-

4

⁢

a

⁢

c

2

⁢

a

for solving the

quadratic equation

a

⁢

x

2

+

b

⁢

x

+

c

=

0

(1)

with real

coefficients

a

,

b

,

c

is valid as well for all

complex

values of these coefficients (

a

≠

0

), when the square root is determined as is presented in the parent entry (

http://planetmath.org/TakingSquareRootAlgebraically

).

Proof.

Multiplying (1) by

4

⁢

a

and adding

b

2

to both sides gives an

equivalent

(

http://planetmath.org/Equivalent3

)

equation

4

⁢

a

2

⁢

x

2

+

4

⁢

a

⁢

b

⁢

x

+

4

⁢

a

⁢

c

+

b

2

=

b

2

or

(

2

⁢

a

⁢

x

)

2

+

2

⋅

2

⁢

a

⁢

x

⋅

b

+

b

2

=

b

2

-

4

⁢

a

⁢

c

or furthermore

(

2

⁢

a

⁢

x

+

b

)

2

=

b

2

-

4

⁢

a

⁢

c

.

Taking square root algebraically yields

2

⁢

a

⁢

x

+

b

=

±

b

2

-

4

⁢

a

⁢

c

,

which implies the quadratic formula.

Note.

A quadratic formula is meaningful besides

ℂ

also in other fields with

characteristic

≠

2

if one can find the needed “square root” (this may require a

field extension

).

Title

quadratic equation in

ℂ

Canonical name

QuadraticEquationInmathbbC

Date of creation

2013-03-22 17:36:36

Last modified on

2013-03-22 17:36:36

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Theorem

Classification

msc 30-00

Classification

msc 12D99

Synonym

quadratic equation

Related topic

QuadraticFormula

Related topic

DerivationOfQuadraticFormula

Related topic

CardanosDerivationOfTheCubicFormula

Generated on Fri Feb 9 20:12:39 2018 by

LaTeXML
radical

radical

The

is called a

radical

. The of the radical symbol is the first letter “

r

” of the Latin

radix

, which “root (

http://planetmath.org/NthRoot

)”.

There is other terminology that is related to radicals: In the

expression

x

3

,

x

is the

radicand

and

3

is the

index

.

The index of a radical is usually an integer at least as large as two; however, if the index is two, it is usually not written. Moreover, if a radical does not have a written index, it is understood to be two, in which case the radical denotes a square root.

A

radical expression

is an expression that involves a radical.

One can use

fractional powers

instead of the radical symbol via

x

n

=

x

1

n

. This practice of avoiding the radical symbol may be due to typesetting issues.

Title

radical

Canonical

name

Radical12

Date of creation

2013-03-22 17:06:00

Last modified on

2013-03-22 17:06:00

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

22

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 30-00

Classification

msc 12D99

Classification

msc 00A05

Synonym

radical symbol

Related topic

Radical5

Related topic

EvenEvenOddRule

Related topic

NthRoot

Defines

radicand

Defines

index

Defines

radical expression

Generated on Fri Feb 9 20:12:42 2018 by

LaTeXML
region

region

A

region

is a nonempty

open subset

of

ℂ

. Note that this definition is a of that of domain (

http://planetmath.org/Domain2

) (as defined in

complex analysis

) to the

complex plane

. Some people prefer to use “region” instead of “domain” to avoid confusion with other mathematical definitions of domain. (The set theoretic definition of domain (

http://planetmath.org/Domain

) is also used in complex analysis.)

Regions play a major role in complex analysis since every nonempty open subset of

ℂ

is the union of countably many

connected components

, each of which is a region.

Title

region

Canonical name

Region

Date of creation

2013-03-22 11:56:19

Last modified on

2013-03-22 11:56:19

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

11

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 30-00

Related topic

Complex

Related topic

Domain2

Generated on Fri Feb 9 20:12:44 2018 by

LaTeXML
simple boundary point

simple boundary point

Definition

.

Let

G

⊂

ℂ

be a region and

ω

∈

∂

⁡

G

(the

boundary

of

G

). Then
we call

ω

a

simple boundary point

if whenever

{

ω

n

}

⊂

G

is a sequence converging to

ω

there is a path

γ

:

[

0

,

1

]

→

ℂ

such that

γ

⁢

(

t

)

∈

G

for

0

≤

t

<

1

,

γ

⁢

(

1

)

=

ω

and there is a sequence

{

t

n

}

∈

[

0

,

1

)

such that

t

n

→

1

and

γ

⁢

(

t

n

)

=

ω

n

for all

n

.

For example if we let

G

be the open

unit disc

, then every boundary point is a simple boundary point. This

definition

is useful for studying boundary behaviour of Riemann maps (maps arising from the

Riemann mapping theorem

), and one can prove for example the following

theorem

.

Theorem

.

Suppose that

G

⊂

C

is a

bounded

simply connected region such
that every point in the boundary of

G

is a simple boundary point, then

∂

⁡

G

is a Jordan curve.

References

1

John B. Conway.
.
Springer-Verlag, New York, New York, 1995.

Title

simple boundary point

Canonical

name

SimpleBoundaryPoint

Date of creation

2013-03-22 14:23:23

Last modified on

2013-03-22 14:23:23

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

5

Author

jirka (4157)

Entry type

Definition

Classification

msc 30-00

Classification

msc 54-00

Generated on Fri Feb 9 20:12:50 2018 by

LaTeXML
square of sum

square of sum

The well-known for squaring a sum of two numbers or is

(

a

+

b

)

2

=

a

2

+

2

⁢

a

⁢

b

+

b

2

.

(1)

It may be derived by multiplying the

binomial

a

+

b

by itself.

Similarly one can get the squaring for a sum of three summands:

(

a

+

b

+

c

)

2

=

a

2

+

b

2

+

c

2

+

2

⁢

b

⁢

c

+

2

⁢

c

⁢

a

+

2

⁢

a

⁢

b

(2)

Its contents may be expressed as the

Rule.

The square of a sum is equal to the sum of the squares of all the summands plus the sum of all the double products of the summands in twos:

(

∑

i

a

i

)

2

=

∑

i

a

i

2

+

2

⁢

∑

i

<

j

a

i

⁢

a

j

.

This is true for any number of summands.  The rule may be formulated also as

(

a

+

b

+

c

+

…

)

2

=

(

a

)

⁢

a

+

(

2

⁢

a

+

b

)

⁢

b

+

(

2

⁢

a

+

2

⁢

b

+

c

)

⁢

c

+

…

(3)

which in the case of four summands is

(

a

+

b

+

c

+

d

)

2

=

(

a

)

⁢

a

+

(

2

⁢

a

+

b

)

⁢

b

+

(

2

⁢

a

+

2

⁢

b

+

c

)

⁢

c

+

(

2

⁢

a

+

2

⁢

b

+

2

⁢

c

+

d

)

⁢

d

.

(4)

One can use the idea of (3) to find the , when one tries to arrange the

polynomial

into the form of the right hand side (

http://planetmath.org/Equation

) of (3).

Title

square of sum

Canonical name

SquareOfSum

Date of creation

2013-03-22 15:32:03

Last modified on

2013-03-22 15:32:03

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Topic

Classification

msc 30-00

Classification

msc 26-00

Classification

msc 11-00

Related topic

SquareRootOfPolynomial

Related topic

DifferenceOfSquares

Related topic

HeronianMeanIsBetweenGeometricAndArithmeticMean

Related topic

ContraharmonicMeansAndPythagoreanHypotenuses

Related topic

CompletingTheSquare

Related topic

TriangleInequalityOfComplexNumbers

Generated on Fri Feb 9 20:12:52 2018 by

LaTeXML
taking square root algebraically

taking square root algebraically

For getting the

square root

of the

complex number

a

+

i

⁢

b

(

a

,

b

∈

ℝ

) purely algebraically, one should solve the

real part

x

and the

imaginary part

y

of

a

+

i

⁢

b

from the

binomial equation

(

x

+

i

⁢

y

)

2

=

a

+

i

⁢

b

.

(1)

This gives

a

+

i

⁢

b

=

x

2

+

2

⁢

i

⁢

x

⁢

y

-

y

2

=

(

x

2

-

y

2

)

+

i

⋅

2

⁢

x

⁢

y

.

Comparing (see equality (

http://planetmath.org/EqualityOfComplexNumbers

)) the real parts and the imaginary parts yields the pair of real equations

x

2

-

y

2

=

a

,

2

⁢

x

⁢

y

=

b

,

which may be written

x

2

+

(

-

y

2

)

=

a

,

x

2

⋅

(

-

y

2

)

=

-

b

2

4

.

Note that the

x

and

y

must be chosen such that their

product

(

=

b

2

) has the same sign as

b

.  Using the

properties of quadratic equation

, one infers that

x

2

and

-

y

2

are the roots of the equation

t

2

-

a

⁢

t

-

b

2

4

=

0

.

The

quadratic formula

gives

t

=

a

±

a

2

+

b

2

2

,

and since

-

y

2

is the smaller root,

x

2

=

a

+

a

2

+

b

2

2

,

-

y

2

=

a

-

a

2

+

b

2

2

.  So we obtain the result

x

=

a

2

+

b

2

+

a

2

,

y

=

(

sign

⁢

b

)

⁢

a

2

+

b

2

-

a

2

(see the

signum function

).  Because both may have also the

a

+

i

⁢

b

=

±

(

a

2

+

b

2

+

a

2

+

(

sign

⁢

b

)

⁢

i

⁢

a

2

+

b

2

-

a

2

)

.

(2)

The result shows that the real and imaginary parts of the square root of any complex number

a

+

i

⁢

b

can be obtained from the real part

a

and imaginary part

b

of the number by using only algebraic operations, i.e. the

rational

operations

and the .  Apparently, the same is true for all roots of a complex number with index (

http://planetmath.org/NthRoot

) an

integer

power of 2.

In practise, when determining the square root of a non-real complex number, one need not to remember the (2), but it’s better to solve concretely the equation (1).

Exercise.

Compute

i

and check it!

Title

taking square root algebraically

Canonical name

TakingSquareRootAlgebraically

Date of creation

2015-06-14 16:31:35

Last modified on

2015-06-14 16:31:35

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

17

Author

pahio (2872)

Entry type

Derivation

Classification

msc 30-00

Classification

msc 12D99

Synonym

square root of complex number

Related topic

SquareRootOfSquareRootBinomial

Related topic

CasusIrreducibilis

Related topic

TopicEntryOnComplexAnalysis

Related topic

ValuesOfComplexCosine

Generated on Fri Feb 9 20:12:58 2018 by

LaTeXML
triangle inequality of complex numbers

triangle inequality of complex numbers

Theorem.

All

complex numbers

z

1

and

z

2

satisfy the

triangle inequality

|

z

1

+

z

z

|

≦

|

z

1

|

+

|

z

2

|

.

(1)

Proof.

|

z

1

+

z

2

|

2

=

(

z

1

+

z

2

)

⁢

(

z

1

+

z

2

)

¯

=

(

z

1

+

z

2

)

⁢

(

z

1

¯

+

z

2

¯

)

=

z

1

⁢

z

1

¯

+

z

2

⁢

z

2

¯

+

z

1

⁢

z

2

¯

+

z

1

¯

⁢

z

2

=

|

z

1

|

2

+

|

z

2

|

2

+

z

1

⁢

z

2

¯

+

z

1

⁢

z

2

¯

¯

=

|

z

1

|

2

+

|

z

2

|

2

+

2

⁢

Re

⁢

(

z

1

⁢

z

2

¯

)

≦

|

z

1

|

2

+

|

z

2

|

2

+

2

⁢

|

z

1

⁢

z

2

¯

|

=

|

z

1

|

2

+

|

z

2

|

2

+

2

⁢

|

z

1

|

⋅

|

z

2

¯

|

=

(

|

z

1

|

+

|

z

2

|

)

2

Taking then the nonnegative square root, one obtains the asserted

inequality

.

Remark.

Since the

real numbers

are complex numbers, the inequality (1) and its proof are valid also for all real numbers; however the inequality may be simplified to

|

x

+

y

|

2

≦

(

x

+

y

)

2

=

x

2

+

2

⁢

x

⁢

y

+

y

2

≦

x

2

+

2

⁢

|

x

|

⁢

|

y

|

+

y

2

=

|

x

|

2

+

2

⁢

|

x

|

⁢

|

y

|

+

|

y

|

2

=

(

|

x

|

+

|

y

|

)

2

.

Title

triangle inequality of complex numbers

Canonical name

TriangleInequalityOfComplexNumbers

Date of creation

2013-03-22 18:51:47

Last modified on

2013-03-22 18:51:47

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Theorem

Classification

msc 30-00

Classification

msc 12D99

Synonym

triangle inequality

Related topic

Modulus

Related topic

ComplexConjugate

Related topic

SquareOfSum

Related topic

TriangleInequality

Generated on Fri Feb 9 20:13:01 2018 by

LaTeXML
Hadamard three-circle theorem

Hadamard three-circle theorem

Let

f

⁢

(

z

)

be a

complex analytic function

on the

annulus

r

1

≤

|

z

|

≤

r

3

. Let

M

⁢

(

r

)

be the

maximum

of

|

f

⁢

(

z

)

|

on the circle

|

z

|

=

r

. Then

log

⁡

M

⁢

(

r

)

is a

convex function

of

log

⁡

r

. Moreover, if

f

⁢

(

z

)

is not of the form

c

⁢

z

λ

for some

λ

, then

log

⁡

M

⁢

(

r

)

is a strictly convex (

http://planetmath.org/ConvexFunction

) as a

function

of

log

⁡

r

.

The

conclusion

of the

theorem

can be restated as

log

⁡

r

3

r

1

⁢

log

⁡

M

⁢

(

r

2

)

≤

log

⁡

r

3

r

2

⁢

log

⁡

M

⁢

(

r

1

)

+

log

⁡

r

2

r

1

⁢

log

⁡

M

⁢

(

r

3

)

for any three concentric circles of radii

r

1

<

r

2

<

r

3

.

Title

Hadamard three-circle theorem

Canonical name

HadamardThreecircleTheorem

Date of creation

2013-03-22 14:10:45

Last modified on

2013-03-22 14:10:45

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

7

Author

bbukh (348)

Entry type

Theorem

Classification

msc 30A10

Classification

msc 30C80

Related topic

MaximumPrinciple

Related topic

LogarithmicallyConvexFunction

Related topic

HardysTheorem

Generated on Fri Feb 9 20:13:03 2018 by

LaTeXML
analytic continuation by power series

analytic continuation by power series

Given a

holomorphic function

defined on some open set, one technique
for analytically continuing it to a larger set is by means of

power
series

. One picks a point of the region and constructs the

Taylor
series

of the

function

about that point. If it turns out that the

radius of convergence

of the Taylor series is large enough that it
contains points which are not in the original domain, one can
extend the function to a larger domain obtrained by adding these points.

Title

analytic continuation by power series

Canonical name

AnalyticContinuationByPowerSeries

Date of creation

2013-03-22 15:41:22

Last modified on

2013-03-22 15:41:22

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

4

Author

rspuzio (6075)

Entry type

Result

Classification

msc 30A99

Generated on Fri Feb 9 20:13:04 2018 by

LaTeXML
antiderivative of complex function

antiderivative of complex function

By the of a

complex function

f

in a domain

D

of

ℂ

, we every complex function

F

which in

D

satisfies

the condition

d

d

⁢

z

⁢

F

⁢

(

z

)

=

f

⁢

(

z

)

.

•

If

f

is a

continuous

complex function in a domain

D

and if the

integral

F

⁢

(

z

)

:=

∫

γ

z

f

⁢

(

t

)

⁢

𝑑

t

(1)

where the path

γ

z

begins at a

fixed point

z

0

of

D

and ends at the point

z

of

D

, is independent of the
path

γ

z

for each value of

z

, then (1) defines an

analytic function

F

with domain

D

.  This

function

is an

antiderivative

of

f

in

D

, i.e. (

http://planetmath.org/Ie

) at all
points of

D

, the condition

d

d

⁢

z

⁢

∫

γ

z

f

⁢

(

t

)

⁢

𝑑

t

=

f

⁢

(

z

)

is true.

•

If

f

is an analytic function in a simply connected open domain

U

, then

f

has an antiderivative in

U

, e.g. (

http://planetmath.org/Eg

) the function

F

defined by (1) where the path

γ

z

is within

U

.  If

γ

lies within

U

and connects the points

z

0

and

z

1

, then

∫

γ

f

⁢

(

z

)

⁢

𝑑

z

=

F

⁢

(

z

1

)

-

F

⁢

(

z

0

)

,

where

F

is an arbitrary antiderivative of

f

in

U

.

Title

antiderivative of complex function

Canonical name

AntiderivativeOfComplexFunction

Date of creation

2014-02-23 15:09:20

Last modified on

2014-02-23 15:09:20

Owner

Wkbj79 (1863)

Last modified by

pahio (2872)

Numerical id

10

Author

Wkbj79 (2872)

Entry type

Definition

Classification

msc 30A99

Classification

msc 03E20

Synonym

complex

antiderivative

Related topic

Antiderivative

Related topic

CalculationOfContourIntegral

Generated on Fri Feb 9 20:13:06 2018 by

LaTeXML
antiholomorphic

antiholomorphic

A

complex function

f

:

D

→

ℂ

,  where

D

is a domain of the

complex plane

, having the

derivative

d

⁢

f

d

⁢

z

¯

in each point

z

of

D

, is said to be

antiholomorphic

in

D

.

The following conditions are

equivalent

(

http://

planetmath

.org/Equivalent3

):

•

f

⁢

(

z

)

is antiholomorphic in

D

.

•

f

⁢

(

z

)

¯

is

holomorphic

in

D

.

•

f

⁢

(

z

¯

)

is holomorphic in

D

¯

:=

{

z

¯

⋮

z

∈

D

}

.

•

f

⁢

(

z

)

may be to a

power series

∑

n

=

0

∞

a

n

⁢

(

z

¯

-

u

)

n

at each

u

∈

D

.

•

The

real part

u

⁢

(

x

,

y

)

and the

imaginary part

v

⁢

(

x

,

y

)

of the

function

f

satisfy

the equations

∂

⁡

u

∂

⁡

x

=

-

∂

⁡

v

∂

⁡

y

,

∂

⁡

u

∂

⁡

y

=

∂

⁡

v

∂

⁡

x

.

N.B. the of minus; cf. the Cauchy–

Riemann

equations (

http://planetmath.org/CauchyRiemannEquations

).

Example.

The function

z

↦

1

z

¯

is antiholomorphic in

ℂ

∖

{

0

}

.  One has

f

⁢

(

z

)

=

z

|

z

|

2

=

x

x

2

+

y

2

⏟

u

+

i

⁢

y

x

2

+

y

2

⏟

v

and thus

∂

⁡

u

∂

⁡

x

=

y

2

-

x

2

(

x

2

+

y

2

)

2

,

∂

⁡

v

∂

⁡

y

=

x

2

-

y

2

(

x

2

+

y

2

)

2

,

∂

⁡

u

∂

⁡

y

=

-

2

⁢

x

⁢

y

(

x

2

+

y

2

)

2

,

∂

⁡

v

∂

⁡

x

=

-

2

⁢

x

⁢

y

(

x

2

+

y

2

)

2

.

Title

antiholomorphic

Canonical

name

Antiholomorphic

Date of creation

2014-11-06 12:07:50

Last modified on

2014-11-06 12:07:50

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Definition

Classification

msc 30A99

Synonym

antiholomorphic function

Related topic

ComplexConjugate

Generated on Fri Feb 9 20:13:08 2018 by

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antiperiodic function

antiperiodic function

A special case of the

quasiperiodicity

(

http://planetmath.org/Period3

) of

functions

is the

antiperiodicity

.
An

antiperiodic

function

f

satisfies for a certain constant

p

the equation

f

⁢

(

z

+

p

)

=

-

f

⁢

(

z

)

for all values of the variable

z

.  The constant

p

is the

antiperiod

of

f

. Then,

f

has also other antiperiods, e.g.

-

p

, and generally

(

2

⁢

n

+

1

)

⁢

p

with any

n

∈

ℤ

.

The

antiperiodic function

f

is always as well periodic with period

2

⁢

p

, since

f

⁢

(

z

+

2

⁢

p

)

=

f

⁢

(

(

z

+

p

)

+

p

)

=

-

f

⁢

(

z

+

p

)

=

-

(

-

f

⁢

(

z

)

)

=

f

⁢

(

z

)

.

Naturally, then there are all periods

2

⁢

n

⁢

p

with

n

∈

ℤ

.

Not all

periodic functions

are antiperiodic.

For example, the sine and

cosine functions

are antiperiodic with

p

=

π

, which is their absolutely least antiperiod:

sin

⁡

(

z

+

π

)

=

-

sin

⁡

z

,

cos

⁡

(

z

+

π

)

=

-

cos

⁡

z

The

tangent

(

http://planetmath.org/Trigonometry

) and

cotangent functions

are not antiperiodic although they are periodic (with the

prime period

π

; see

complex tangent and cotangent

).

The

exponential function

is antiperiodic with the antiperiod

i

⁢

π

(see

Euler relation

):

e

z

+

i

⁢

π

=

e

z

⁢

e

i

⁢

π

=

-

e

z

Title

antiperiodic function

Canonical name

AntiperiodicFunction

Date of creation

2015-12-16 15:19:14

Last modified on

2015-12-16 15:19:14

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Definition

Classification

msc 30A99

Related topic

PeriodicFunctions

Related topic

QuasiperiodicFunction

Defines

antiperiodicity

Defines

antiperiodic

Defines

antiperiod

Generated on Fri Feb 9 20:13:10 2018 by

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complex function

complex function

A

complex function

is a

function

f

from a subset

A

of

ℂ

to

ℂ

.

For every

z

=

x

+

i

y

∈

A

(

x

,

y

∈

ℝ

)

the

complex

value

f

⁢

(

z

)

can be split into its real and imaginary
parts

u

and

v

, respectively, which can be considered as
real functions of two real variables:

f

⁢

(

z

)

=

u

⁢

(

x

,

y

)

+

i

⁢

v

⁢

(

x

,

y

)

(1)

The functions

u

and

v

are called the

real part

and
the

imaginary part

of the complex function

f

,
respectively.  Conversely, any two functions

u

⁢

(

x

,

y

)

and

v

⁢

(

x

,

y

)

defined in some subset of

ℝ

2

determine via
(1) a complex function

f

.

If

f

⁢

(

z

)

especially is defined as a

polynomial

of

z

, then both

u

⁢

(

x

,

y

)

and

v

⁢

(

x

,

y

)

are polynomials of

x

and

y

with real coefficients.

Following are the notations for

u

and

v

that
are used most commonly (the parentheses around

f

⁢

(

z

)

may be
omitted):

u

⁢

(

x

,

y

)

=

Re

⁢

(

f

⁢

(

z

)

)

=

ℜ

⁡

(

f

⁢

(

z

)

)

v

⁢

(

x

,

y

)

=

Im

⁢

(

f

⁢

(

z

)

)

=

ℑ

⁡

(

f

⁢

(

z

)

)

The of mathematics concerning

differentiable

complex functions is called

function theory

or

complex analysis

.

Title

complex function

Canonical

name

ComplexFunction

Date of creation

2014-02-23 10:20:21

Last modified on

2014-02-23 10:20:21

Owner

Wkbj79 (1863)

Last modified by

pahio (2872)

Numerical id

12

Author

Wkbj79 (2872)

Entry type

Definition

Classification

msc 30A99

Classification

msc 03E20

Related topic

RealFunction

Related topic

Meromorphic

Related topic

Holomorphic

Related topic

Entire

Related topic

IndexOfSpecialFunctions

Related topic

ValuesOfComplexCosine

Defines

real part

Defines

imaginary part

Defines

function theory

Defines

complex analysis

Generated on Fri Feb 9 20:13:12 2018 by

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conformally equivalent

conformally equivalent

A region

G

is

conformally equivalent

to a set

S

if there is an

analytic

bijective function

G

to

S

.

Conformal equivalence is an

equivalence relation

.

Title

conformally equivalent

Canonical name

ConformallyEquivalent

Date of creation

2013-03-22 13:15:06

Last modified on

2013-03-22 13:15:06

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

6

Author

Koro (127)

Entry type

Definition

Classification

msc 30A99

Defines

conformal equivalence

Generated on Fri Feb 9 20:13:16 2018 by

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