holomorphic mapping of curve and tangent

holomorphic mapping of curve and tangent

Let

D

be a domain of the

complex plane

and the

function

f

:

D

→

ℂ

be

holomorphic

.  Then for each point

z

of

D

there is a corresponding point

w

=

f

⁢

(

z

)

∈

ℂ

;  we think that

z

and

w

both lie in their own complex planes,

z

-plane and

w

-plane.

Since

f

is

continuous

in

D

, if

z

draws a continuous curve

γ

in

D

then its image point

w

also draws a continuous curve

γ

w

.  Let

z

0

and

z

0

+

Δ

⁢

z

be two points on

γ

and

w

0

and

w

0

+

Δ

⁢

w

their image points on

γ

w

.

We suppose still that the curve

γ

has a

tangent line

at the point

z

0

and that the value of the

derivative

f

′

has in

z

0

a nonzero value

f

′

⁢

(

z

0

)

=

ϱ

⁢

e

i

⁢

ω

.

(1)

If the

slope angles

of the

secant lines

(

z

0

,

z

0

+

Δ

⁢

z

)

and

(

w

0

,

w

0

+

Δ

⁢

w

)

are

α

and

α

w

, then we have

Δ

⁢

z

=

k

⁢

e

i

⁢

α

,

Δ

⁢

w

=

k

w

⁢

e

i

⁢

α

w

,

and the difference quotient of

f

has the form

Δ

⁢

w

Δ

⁢

z

=

f

⁢

(

z

0

+

Δ

⁢

z

)

-

f

⁢

(

z

0

)

Δ

⁢

z

=

k

w

k

⁢

e

i

⁢

(

α

w

-

α

)

.

Let now

Δ

⁢

z

→

0

.  Then the point

z

0

+

Δ

⁢

z

tends on the curve

γ

to

z

0

and

lim

Δ

⁢

z

→

0

⁡

Δ

⁢

w

Δ

⁢

z

=

f

′

⁢

(

z

0

)

.

This implies, by (1), that

lim

Δ

⁢

z

→

0

⁡

k

w

k

=

ϱ

.

(2)

From this we infer, because

ϱ

≠

0

that, up to a multiple of

2

⁢

π

,

lim

Δ

⁢

z

→

0

⁡

(

α

w

-

α

)

=

ω

.

(3)

But the limit of

α

is the slope angle

φ

of the

tangent

of

γ

at

z

0

.  Hence (3) implies that

φ

w

=

lim

Δ

⁢

z

→

0

⁡

α

w

=

φ

+

ω

.

(4)

Accordingly, we have the

Theorem

1.

If a curve

γ

has a tangent line in a point

z

0

where the derivative

f

′

does not vanish, then the image curve

f

⁢

(

γ

)

also has in the corresponding point

w

0

a certain tangent line with a direction obtained by rotating the tangent of

γ

by the angle

ω

=

arg

⁡

f

′

⁢

(

z

0

)

.

If the curve

γ

is smooth, then also

γ

w

is smooth, and it follows easily from (2) the corresponding limit equation between the

arc lengths

:

lim

Δ

⁢

z

→

0

⁡

s

w

s

=

|

f

′

⁢

(

z

0

)

|

.

(5)

Conformality

If we have besides

γ

another curve

γ

′

emanating from

z

0

with its tangent, the

mapping

f

from

D

in

z

-plane to

w

-plane gives two curves and their tangents emanating from

w

0

.  Thus we have two equations (4):

φ

w

=

φ

+

ω

,

φ

w

′

=

φ

′

+

ω

By subtracting we obtain

φ

w

′

-

φ

w

=

φ

′

-

φ

,

(6)

whence we have the

Theorem 2.

The mapping created by the

holomorphic function

f

preserves the magnitude of the

angle between two curves

in any point

z

where

f

′

⁢

(

z

)

≠

0

.  The equation (6) tells also that the

orientation

of the angle is preserved.

The facts in Theorem 2 are expressed so that the mapping is

directly

conformal

.  If the orientation were reversed the mapping were called

inversely conformal

; in this case

f

were not holomorphic but

antiholomorphic

.

Title

holomorphic mapping of curve and tangent

Canonical name

HolomorphicMappingOfCurveAndTangent

Date of creation

2013-03-22 18:42:19

Last modified on

2013-03-22 18:42:19

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Topic

Classification

msc 53A30

Classification

msc 30E20

Defines

directly conformal

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