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Ch6.6 MappingCh6.7 Conformal Mapping
講者: 許永昌 老師
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ContentsConformal MappingMappings
TranslationRotationInversion
Branch Points and Multivalent Functions
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Conformal mapping ( 請預讀 P368~P370)
Mapping: zwThe mapping is conformal if angle and sense
of rotation are preserved by the mapping. If w=f(z) is analytic in a region R of the z-plane,
then the mapping of R onto its image in the w-plane is conformal, except at points where f ’(z)=0. Proof:
0analytic ' 0
arg arg arg arg ,
Therefore,
lim arg arg arg ' .
f i
f i f if i
zf z
f z f zww w z z
z z z
w z f z
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Conformal mapping (continue)Based on Cauchy-Riemann conditions, we get
2u=0=2v, u v=0.
They are orthogonal to each other The curves u=constant and v=constant are orthogonal to each other.
Example:w=z2=(x2-y2)+2ixyCode: z2_uv.m
y=sqrt(x2-u), y=v/(2x) Contour u= x2-y2 , v=2xy. -2 -1 0 1 2
-3
-2
-1
0
1
2
3
z-plane
-8 -6 -4 -2 0 2 4
-10
-8
-6
-4
-2
0
2
4
6
8
10
w-plane
Proper rotation
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Conformal Mapping (final)The mapping of w=z2.
From these figures, you will find that the contour lines of y=C and y=-C are the same in w-plane.
Reason: z=reiq and z’=rei q +ip . z2=r2ei2q=z’ 2.
Therefore, it has a two-to-one correspondence.
-2 -1 0 1 2
-3
-2
-1
0
1
2
3
z-plane
-4 -2 0 2 4-8
-6
-4
-2
0
2
4
6
8w-plane
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Mappings ( 請預讀 P360~P363)
Linear Transformation:Translation:
w=z+z0.
Rotation: w=cz=(rrc)ei(q+q
c).
Nonlinear Transformation:Inversion:
w=1/z=1/r e-iq.…
Code: mappings.m想像 w=z‘ 與 z 畫在同一個座標系
-4 -2 0 2 4 6
0
2
4
6
translation
-6 -4 -2 0 2 4
-2
0
2
4
rotation
-4 -2 0 2 4-3
-2
-1
0
1
2
inversion
-5 0 5 10 15 20
-10
-5
0
5
10
square
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ExerciseProve that w=1/z will map a straight line in z-
plane into a circle cross w=0.Try to add “z=z*(1+1i);x=real(z);y=imag(z);”
into the code mappings.m to see the result.
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Multivalent functions and Branch Points ( 請預讀 P363~P367)
Multivalent function:w=f(z), however, w is not unique for each z.
Example:w=sqrt(z)
If z=rei q ’=rei q +i2mp, w=?w=ln(z)
If z=rei q ’=reiq +i2mp, w=?
STOP TO THINK:Since
how can we say that f ’(z) does exist? analytic?
0
' lim and is a multivalent function,z
f z z f zf z f z
z
Hint: Restrict the allowed range of q’.
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Multivalent functions and Branch Points (continue)The cut line here joins the two branch point
singularities at 0 and .Dm: 2pm < q ’ < 2p (m+1 ).Sm : 2pm = q ’
Based on Morera’s theorem, we can get f(z) is analytic in zS0D0S1D1… SnDn.sqrt(z): n=1 and S0 = S2.ln(z): n=.We call this surface a Riemann surface.
* 莊 ( 土斤 ) 泰,張南岳,復變函數
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Multivalent functions and Branch Points (final)We can think that:
It is the basis for the entire calculus of residues.
Stop to think: Hint: Riemann surface.
200
00
1ln 2 .
i i
i
r e
r eCdz z iz
? Does it conflict with Morera's theorem?Czdz
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Homework6.6.26.6.36.6.56.6.66.6.76.7.16.7.4
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NounsConformal mapping:
P368Riemann surface: P366