Handbook of Conformal Mappings and Applications

This article was downloaded by: 10.3.98.104On: 08 May 2022Access details: subscription numberPublisher: CRC PressInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: 5 Howick Place, London SW1P 1WG, UK

Handbook of Conformal Mappings and Applications

Prem K. Kythe

Linear and Bilinear Transformations

Publication detailshttps://www.routledgehandbooks.com/doi/10.1201/9781315180236-3

Prem K. KythePublished online on: 04 Mar 2019

How to cite :- Prem K. Kythe. 04 Mar 2019, Linear and Bilinear Transformations from: Handbook of Conformal Mappings and Applications CRC PressAccessed on: 08 May 2022https://www.routledgehandbooks.com/doi/10.1201/9781315180236-3

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3Linear and Bilinear Transformations

The central problem in the theory of conformal mapping is to determine a function f whichmaps a given region D ⊂ C conformally onto another region G ⊂ C. The function f doesnot always exist, and it is not always uniquely determined. The Riemann mapping theorem(§2.6, Theorem 2.15) guarantees the existence and uniqueness of a conformal map of Donto the unit disk U under certain specific conditions. First, we will introduce definitionsof certain curves, and some elementary mappings, before we will study linear and bilineartransformations.

3.1 Definition of Certain Curves

Some curves are defined as follows:

3.1.1 Line, or straight line, has the equation lx+my + p = 0, where l,m, p are real, andl2 +m2 > 0. Then ℜλz = p, or ℑµz = p, where λ = l+ im 6= 0, and µ = m+ i l = i λ.

The distance of a point z0 from the line lx+my + p = 0 is given by∣∣∣ℜλz0 − p

λ

∣∣∣.

The angle between the x-axis and the line lx+my+p = 0 is given by argλ+ π2 +nπ, n =

0,±1,±2, . . . .

3.1.2 Circle in the extended sense means a circle or a straight line. There is no distinctionbetween circles and lines in the theory of bilinear transformations. However, a circle in thez-plan is defined by |z − z0| = r with center at z0 and radius r; in the w-plane it is definedby |w − w0| = R with center at w0 and radius R.

3.1.3 Ellipse has the equation |z−z1|+|z−z2| = k, k > |z1−z2|; its foci are z1 and z2; major

axis is k; eccentricity is|z1 − z2|

k; exterior of the ellipse is defined as |z − z1|+ |z − z2| > k,

and interior as |z − z1|+ |z − z2| < k.

3.1.4 Hyperbola has the equation |z − z1| + |z − z2| = ±k, 0 < k < |z1 − z2|, where theplus or minus sign is used according to whether the branch is nearer to the focus z2 or it

is nearer to the focus z1. The real axis is k; eccentricity|z1 − z2|

k; equations of asymptotes

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50 3 LINEAR AND BILINEAR TRANSFORMATIONS

are (z1 − z2) e

±θ(z − z1 + z2

2

)= 0, cos θ =

k

|z1 − z2|;

the exterior, not containing foci is defined by∣∣|z − z1| − |z − z2|

∣∣ < k.

3.1.5 Rectangular Hyperbola is the hyperbola of §3.1.4 if k =|z1 − z2|√

2. Moreover, if

z2 = −z1 (i.e., the origin is the center of the curve), then the equation of the rectangularhyperbola is

ℜz2z21

=

1

2,

with foci ±z1. The exterior of the hyperbola is defined by

−∞ ≤ ℜz2z21

<

1

2.

Equations of asymptotes are (three forms):

ℜz2z21

= 0; ℜ

zz1

= ±ℑ

zz1

=

1

2; ℜz(1± i)

z1

= 0.

3.1.6 Parabola has the equation |z − z0| = |k − ℜλz||λ|−1, λ 6= 0, k ≷ ℜλz0. Ithas focus at z = z0; directrix ℜλz = k; latus rectum 2|k − ℜλz0||λ|−1; and vertex

z = z0 +k −ℜλz0

2λ| .

3.1.7 Cassini’s ovals or Cassinians, presented in Figure 3.1, are defined by |z−z1||z−z2| =α, α > 0; its foci are z1, z2. For α = 1

4 |z1 − z2|2 = 1, the curve is a lemniscate.

-1 1

-1

1

.5α =

α = 0

1α =

.9α =

.99α =

x

y

0

Figure 3.1 Cassini’s ovals and lemniscate.

Cassini’s ovals are plotted using the definition with rectangular coordinates:

F (x, y, α) =[(x+ α)2 + y2

] [(x− α)2 + y2

]= 1

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3.2 BILINEAR TRANSFORMATIONS 51

for values of α = 0, 0.5, 0.9, 0.99, and 1. For α = 0 the curve becomes the unit circle. Theregion does not remain simply connected for α = 1; it is a doubly connected region andits boundary curve is called the lemniscate.

3.1.8 Cardioid and Limacons. The cardioid, defined by r = 2a(1+cosθ), is the curve inFigure 3.2(a) that is described by a point P of a circle of radius a as it rolls on the outsideof a fixed circle of radius a. It is a special case of the limacon of Pascal, which has theequation r = b + a cos θ. As described in Figure 3.2(b), the line 0Q joining the origin toany point Q on a circle of diameter a passes through 0; then the limacon is the locus ofall points P such that PQ = b. Figure 3.2(b) represents the case when 2a > b, and Figure3.2(c) the case when 2a < b. If b ≥ 2a, the curve is convex. If b = a, the limacon becomesthe cardioid.

0x

y

a

P

Q

A

Ba

b

x x

y

P

P

a0 0

y

(a) (b) (c)

•

•

•

•

••

•

Figure 3.2 Cardioid and limacons.

3.2 Bilinear TransformationsThe bilinear (linear-fractional, or Mobius) transformation is of the form

w = f(z) =az + b

cz + d, (3.2.1)

where a, b, c, d are complex constants such that ad − bc 6= 0 (otherwise the function f(z)would be identically constant). If c = 0 and d = 1, or if a = 0, d = 0 and b = c, then

the function (3.2.1) reduces to a linear transformation w = az + b, or an inversion w =1

z,

respectively. The transformation (3.2.1) is also written as

w =a

c+

bc− ad

c(cz + d), (3.2.2)

that can be viewed as composed of the following three successive functions:

z1 = cz + d, z2 =1

z1, w =

a

c+bc− ad

cz2,

which shows that the mapping (3.2.1) is a linear transformation, followed by an inversionwhich is followed by another linear transformation. The bilinear transformation (3.2.1)maps the extended z-plane conformally onto the extended w-plane such that the pole at

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z = −d/c is mapped into the point w = ∞. The inverse transformation

z = f−1(w) =b− dw

−a+ cw(3.2.3)

is also bilinear defined on the extended w-plane, and maps it conformally onto the extendedz-plane such that the pole at w = a/c is mapped into the point z = ∞. Note that

f ′(z) =ad− bc

(cz + d)26= 0; also [f−1(w)]′ =

−ad+ bc

(cw − a)26= 0. A bilinear transformation carries

circles into circles (in the extended sense), as explained in §3.1.

Example 3.1. The bilinear transformation that maps the square of side 2, centered at

the origin of the z-plane onto the parallelogram shown in Figure 3.3 is given by w =az + b

cz + dsuch that the points 1 + i, 1 − i, −1 − i, and −1 + i are mapped onto the points 1 − 2i,

1 + 2i,1 + 2i

5, and −1 + 2i

5, respectively, as shown in Figure 3.3.

w - planez- plane

x

y

u

v

0 0

1+i

_1 i__1 i

_1+i

_21 i

21+ i

21+ i5

_ 21+ i5

••

•

•

•

•

•

•

Figure 3.3 Conformal and isogonal mappings.

Example 3.2. The transformation that maps the half-plane ℑz > ℜz onto theinterior of the circle |w − 1| = 3 is

w =(1 + 3i)z + 4i

z + i.

To prove it, let us regard the given half-plane as the ‘interior’ of the ‘circle’ through ∞defined by the line ℑz = ℜz. The three points in clockwise order are z1 = ∞, z2 = 0,and z3 = −1−i. The three points on the circle |w−1| = 3 in clockwise order are w1 = 1+3i,w2 = 4, and w3 = −2. The linear fractional transformation that maps three points in orderis defined by Eq (3.2.1), where from the images of ∞ and 0 we have a = 1+3i and b/c = 4,i.e., b = 4i and c = i. Substituting these into Eq (3.2.1), we get the desired mapping.

Since z0 = i is in the half-plane ℑz > ℜz, its image under the above map is w0 =

2.5+1.5 i; thus, as w0 − 1 = 1.5+1.5 i has a modulus 1.5×√2 ≈ 2.12132 < 3, we find that

w0 is in the interior of |w − 1| = 3 (see Figure 3.4).

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•

w - planez- plane

z0

•

•

•

•

i

∞1z

2z

z 3

w1

2w3w•

•

w0

••02_ 4

i1 3+

Figure 3.4 w =(1 + 3i)z + 4i

z + i.

3.2.1 Fixed Points. The fixed points of a mapping are those points at which w = z. Forexample, the mapping w = z3 has a fixed point z = w = 1. Let F1 and F2 be the fixedpoints of the bilinear transformation (3.2.1), ad− bc 6= 0. Then F1 and F2 are the roots ofthe equation

cF 2 + (d− a)F − b = 0.

We will ignore the identical transformation w = z, and consider the following cases:

(i) If F1 = F2 = ∞, we have w = z + b, b 6= 0, p, q real. Then (i) any line ℑbz = pis mapped onto itself, but onto no other ‘circle’ (in the extended sense); and (ii) the lineℜbz = q is mapped onto ℜbw = q − |b|2.(ii) If F1 = F2 6= ∞, we have

1

w − F=

1

z − F+ k,

where

c 6= 0, F =a− d

2c=

2b

d− a, k =

2c

a+ d.

The transformation is ‘parabolic’. The straight line ℑk(z − r) = 0 and every circletouching this straight line at F is mapped onto itself; no other circle has this property. Theset of ‘circles’ orthogonal to the above set is, as a whole, mapped onto itself.

(iii) If F1 6= F2 = ∞, we have w = F1 = α (z − F1), where α = a/d, α 6= 0, 1, c = 0, and

F1 =b

d− a. Let D denote the set of concentric circles centered at F1, and E denote the

set of straight lines through F1. Then the entire set D is mapped onto itself; so is E. Noneof the ‘circles’ other than those stated below is mapped onto itself:

(a) α real, but α 6= −1: then every line of E is mapped onto itself;

(b) α = −1: then every circle of D, and every line of E, is mapped onto itself;

(c) |α| = 1, but α 6= −1: then every circle of D is mapped onto itself;

(d) α not real, |α| 6= 1: then no circle or line is mapped onto itself (this is known as the

loxodromic transformation).

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(iv) If F1 6= F2, F1 6= ∞, F2 6= ∞, we have

w − F1

w − F2= α

z − F1

z − F2,

where

α =cF2 + d

cF1 + d=

(a+ d−

√(a− d)2 + 4bc

)2

4(ad− bc).

Let E denote a pencil of ‘circles’ through F1 and F2, and let D denote the set of ‘circles’orthogonal to the circles of E (co-axial system). Then the results are the same as in (iii)above, but replace circle or straight line there by ‘circle’.

The fixed points should, however, not be confused with the involutory transformation forwhich the mapping w = f(z) has the same form as the mapping z = f−1(w). An example

is the mapping w =1

z, or the mapping w =

z + 1

z − 1.

3.2.2 Linear Transformation. w = az + b, a 6= 0. The special cases are as follows:

(a) w = az: (i) if a > 0, then there is magnification by ratio a centered at 0; (ii) ifa = eiτ , τ real, then there is rotation by the angle τ about 0; and (iii) if a = meiτ , m > 0,and τ real, then there is rotation by the angle τ about 0, followed by a magnification ofratio m centered at 0.

(b) w = z + b: There is a translation by the vector 0b.

(c) w = az + b, a 6= 1: Then F0 =b

1− a, a = meiτ ,m > 0, τ real. There is rotation

about F0 by the angle τ , followed by magnification by ratio m centered at F0.

The correspondence of the above linear transformations between the z-plane and thew-plane is as follows:

(i) The line x = p is mapped onto the line ℜw − b

a

= p.

(ii) The line y = q is mapped onto the line ℑw − b

a

= q.

(iii) The line lx+my = p is mapped onto the line ℜ(l − im)

w − b

a

= p.

(iv) The line ℜaz + b = p is mapped onto the line u = P .

(v) The line ℑaz + b = q is mapped onto the line v = Q.

(vi) The line ℜ(l− im)(az + b) = p is mapped onto the line lu+mv = P .

(vii) The circle |z − z0| = r is mapped onto the circle |w − (aw0 + b) | = |a|R.(vii) The circle

∣∣∣z − w0 − b

a

∣∣∣ =r

|a| is mapped onto the circle |w − w0| = R.

3.2.3 Composition of Bilinear Transformations. We will consider the general bilineartransformation (3.2.1) with ad − bc 6= 0. Note that if ad − bc = 0, then we have w =const. The general bilinear transformation (3.2.1) consists of the following three successivetransformations:

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(i) translation of z: z1 = z + d/c;

(ii) inversion of z1: z2 = 1/z1; and

(iii) translation, rotation, and scale change of z2: w =a

c− ad− bc

c2z2.

Map 3.1. If a = −d, the transformation is involutory, i.e., w and z may be interchanged.

The magnification is∣∣∣dw

dz

∣∣∣ =∣∣∣ad− bc

(cz + d)2

∣∣∣, or

|f ′(z)|2 =(∂u∂x

)2+(∂v∂x

)2=(∂u∂y

)2+(∂v∂y

)2.

The scale or magnification factor m is then defined as m = |dw/dz| = |f ′(z)|, wherem = |m|, and M = 1/m. The critical points are z = −d/c and z = ∞; at these points thetransformation is not conformal in the usual sense. The details of the transformation areas follows:

(i) The points z = 0; −b/a; z∞ = −d/c are mapped onto the points w = b/d; 0; ∞.

(ii) The point at infinity z = ∞ is mapped onto the point w∞ = a/c.

Map 3.2. The bilinear transformation w = f(z) that transforms the three points zkinto the points wk, k = 1, 2, 3, can be expressed as (Saff and Snider [1976:329])

∣∣∣∣∣∣∣

1 z w zw1 z1 w1 z1w1

1 z2 w2 z2w2

1 z3 w3 z3w3

∣∣∣∣∣∣∣= 0.

Map 3.3. Let z be fixed and let ℜz ≥ 0. Define a sequence of bilinear transformations

T0(w) =a0

z + a0 + b1 + w, Tk(w) =

akz + bk+1 + w

, k = 1, . . . , n− 1,

such that each aj > 0 is real, and each bj is zero or purely imaginary for j = 0, 1, 2, . . . , n−1. We use induction to prove that the chain of transformations defined by ζ = S(w) =(T0 · · · Tn−2 Tn−1) (w) maps the half-plane ℜw > 0 onto a region contained in thedisk |ζ−1/2| < 1/2. Thus, using theWallis criterion, if P (z) = zn+c1z

n−1+c2zn−2+· · ·+cn

is a polynomial of degree n > 0 with complex coefficients ck = pk + i qk, k = 1, 2, . . . , n,if Q(z) = p1z

n−1 + i q2zn−2 + p3z

n−3 + i q4zn−4 + · · · , and if Q(z)/P (z) can be written as

a continued fraction, then we have Q(z)/P (z) = (T0 Tn−2 · · · Tn−1) (z) (Saff and Snider[1976:330]).

Theorem 3.1. (Schwarzian derivative) (i) The Schwarzian derivative (also called theSchwarz differential operator), defined by

w, z =

(w′′

w′

)′− 1

2

(w′′

w′

)2

=w′′′

w′ − 3

2

(w′′

w′

)2

,

is invariant under bilinear transformations. (Nehari [1952:199]; Wen [1992:76]).

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(ii) The Schwarzian derivative of the function w = w(z) that maps the upper half-planeℑz > 0 conformally onto a circular triangle with interior angles απ, βπ and γπ is givenby

w, z =1− α2

2z2+

1− β2

2(1− z)2+

1− α2 − β2 + γ2

2z(1− z).

(Nevanlinna and Paatero [1969:342]).

3.3 Cross-RatioA cross-ratio between four distinct finite points z1, z2, z3, z4 is defined by

(z1, z2, z3, z4) =z1 − z2z1 − z4

· z3 − z4z3 − z2

. (3.3.1)

If z2, z3, or z4 is a point at infinity, then (3.3.1) reduces to

z3 − z4z1 − z4

,z1 − z2z1 − z4

, orz1 − z2z3 − z2

,

respectively. The cross-ratio (z, z1, z2, z3) is invariant under bilinear transformations.

Theorem 3.2. A bilinear transformation is uniquely defined by a correspondence of thecross-ratios, i.e.,

(w,w1, w2, w3) = (z, z1, z2, z3) , (3.3.2)

which maps any three distinct points z1, z2, z3 in the extended z-plane into three prescribedpoints w1, w2, w3 in the extended w-plane. The cross-ratio (z, z1, z2, z3) is the image of zunder a bilinear transformation that maps three distinct points z1, z2, z3 into 0, 1,∞.

3.3.1 Symmetric Points. The points z and z∗ are said to be symmetric with respect toa circle C (in the extended sense) through three distinct points z1, z2, z3 iff

(z∗, z1, z2, z3) = (z, z1, z2, z3). (3.3.3)

The mapping that carries z into z∗ is called a reflection with respect to C. Two reflectionsobviously yield a bilinear transformation.

If C is a straight line, then we choose z3 = ∞, and the condition for symmetry (3.3.3)gives

z∗ − z1z1 − z2

=z − z1z1 − z2

. (3.3.4)

Let z2 be any finite point on the line C. Then, since |z∗ − z1| = |z − z1|, the points z

and z∗ are equidistant from the line C. Moreover, since ℑz∗ − z1z1 − z2

= −ℑ

z − z1z1 − z2

, the

line C is the perpendicular bisector of the line segment joining z and z∗. If C is the circle|z − a| = R, then

(z, z1, z2, z3) = (z − a, z1 − a, z2 − a, z3 − a)

=

(z − a,

R2

z1 − a,R2

z2 − a,R2

z3 − a

)

=

(R2

z − a, z1 − a, z2 − a, z3 − a

)

=

(R2

z − a+ a, z1, z2, z3

).

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3.3 CROSS-RATIO 57

Hence, in view of (3.3.3), we find that the points z and z∗ =R2

z − a+ a are symmetric, i.e.,

(z∗ − a)(z − a) = R2. (3.3.5)

Note that |z∗ − a||z − a| = R2; also, sincez∗ − a

z − a> 0, the points z and z∗ are on the same

ray from the point a (Figure 3.5). Also, the point symmetric to a is ∞.

z

z

a

*

.

R

•

•

Figure 3.5 Symmetry with respect to a circle.

A generalization of this result is as follows: If Γ denotes an analytic Jordan curvewith parametric equation z = γ(s), s1 < s < s2, then for any point z sufficiently close toΓ, the point

z∗ = γ (γ−1(z)) (3.3.6)

defines a symmetric point of z with respect to Γ (Sansone and Gerretsen, [1960: 103];Papamichael, Warby and Hough [1986]). Some examples are as follows:(i) If Γ is the circle x2 + y2 = a2/9, then z∗ = a2/9z.

(ii) If Γ is the ellipse(x+ a/2)2

a2+ y2 = 1, then

z∗ = −a2+

(a2 + 1)(z + a/2) + 2ia√a1 − 1− (z + a/2)

a2 − a.

(iii) If Γ1 is a cardioid defined by z = γ(s) =

(1

2+ cos

s

2

)eis, −π < s ≤ π, then from

(3.3.6) we cannot write an explicit expression for symmetric points with respect to Γ1.However, for any real t,

γ (± it) =

(1

2+ cosh

t

2

)e∓t

defines two real symmetric points with respect to Γ1, provided the parameter t satisfies the

equation γ(it) γ(−it) = a2, i.e.,1

2+ cosh

t

2= a which has the roots

t = 2 cosh−1(a− 1/2) = ±2 log ρ, (3.3.7)

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where ρ = a− 1

2+

√(a− 1

2

)2

− 1.

−1.5 −1 1 1.5

−1.5

−1

1

1.5

−1.5 −1 1 1.5

−1.5

−1

1

1.5

0 0

a = 1.8= 1.55a

x

y

x

y

Figure 3.6 Cardioid inside a circle.

(iv) If a doubly connected region Ω is bounded outside by the circle Γ2 = z : |z| = a, a >1.5 and inside by the cardioid Γ1 defined above in (iii) (see Figure 3.6), then it followsfrom (3.3.7) that there is one pair of real common symmetric points ζ1 ∈ Int (Γ1) andζ2 ∈ Ext (Γ2) such that ζ1 = a/ρ2 and ζ2 = aρ2, where ρ is defined in (3.3.7).

3.3.2 Symmetry Principle. The symmetry principle states that if a bilinear transfor-mation maps a circle C1 onto a circle C2, then it maps any pair of symmetric points withrespect to C1 into a pair of symmetric points with respect to C2. This means that bilineartransformations preserve symmetry.

A practical application of the symmetry principle is to find bilinear transformationswhich map a circle C1 onto a circle C2. We already know that the transformation (3.3.2)can always be determined by requiring that the three points z1, z2, z3 ∈ C1 map onto threepoints w1, w2, w3 ∈ C2. But a bilinear transformation is also determined if a point z1 ∈ C1

should map into a point w1 ∈ C2 and a point z2 6∈ C1 should map into a point w2 6∈ C2.Then, by the symmetry principle, the point z∗2 which is symmetric to z2 with respect to C1

is mapped into the point w∗2 which is symmetric to w2 with respect to C2, and then the

desired bilinear transformation is given by

(w,w1, w2, w∗2) = (z, z1, z2, z

∗2) . (3.3.8)

Let U+ denote the region |z| < 1, U− the region |z| > 1, and let C = |z| = 1 be theircommon boundary (Figure 3.7). Let f(z) be a function defined on U+. Then this functioncan be related to the function f∗(z) defined in U− in the same manner as in (2.6.3) forhalf-planes, except that now the conjugate complex points are replaced by points inversewith respect to the circle C according to the relation (3.3.5). Thus,

f∗(z) = f

(1

z

)= f

(1

z

). (3.3.9)

This relation is symmetrical, i.e.,

f(z) = f∗(1

z

), (f∗(z))∗ = f(z). (3.3.10)

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3.3 CROSS-RATIO 59

U

U

10

+

−

C

x

y

Figure 3.7 Unit disk.

If f(z) is regular or meromorphic in U+, then f∗(z) is regular or meromorphic in U−.Also, if f(z) is a rational function defined by (2.6.4), then

f∗(z) =an z

−n + an−1 z−n+1 + · · ·+ a0

bm z−m + b−m+1 zm−1 + · · ·+ b0. (3.3.11)

Moreover, if f(z) has the power series expansion

f(z) =∞∑

k=−∞ak z

k z ∈ U+,

then

f∗(z) =∞∑

k=−∞ak z

−k, z ∈ U−. (3.3.12)

If f(z) has a zero (pole) of order k at z = ∞ (z = 0), so does f∗(z). Let us assume thatf(z) approaches a definite limit value f+(t) as z → t ∈ C from U+. Then f∗−(t) exists and

f∗−(t) = f−(1

t

)= f+(t), (3.3.13)

because 1/z → t from U+ as z → t from U−, and hence f∗(z) = f

(1

z

)= f

(1

z

)→ f+(t).

If f(z) is regular in U+ except possibly at infinity and continuous on C from the left, thenlet F (z) be sectionally regular and be defined by

F (z) =

f(z) for z ∈ U+,

f∗(z) for z ∈ U−.(3.3.14)

Then, F ∗(z) = F (z), and, as in (2.6.7),

F−(t) = F+(t), F+(t) = F−(t). (3.3.15)

Moreover, in view of the Schwarz reflection principle, if ℑf+(t) = 0 on some part of thecircle C, then f∗(z) is the analytic continuation of f(z) through this part of C.

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3.3.3 Special Cases. We will discuss the following special cases of the transformations(3.2.1) and (3.2.3).

Map 3.4. (Linear transformation w = az+ b, where a = a1+ ia2 and b = b1+ ib2. Thenw = u + iv = (α1 + ia2) (x + iy) + (b1 + b2) = (a1x− a2y + b1) + i (a1y + a2x+ b2). Themagnification factor and the clockwise rotation are

m =|dw||dx| = |a| =

√a21 + a22, (3.3.16)

α = argdwdz

= argm = arctan

(a2a1

). (3.3.17)

Let a = 1 + i and b = − 12 − i 32 . Then m =

√2 and α = π/4. The rectangular lines

u = ku and v = vk are represented in the z-plane by the rectangular lines y = x−(12 + ku

)

and y = −x+(32 + kv

), respectively. The transformation is presented in Figure 3.8.

w - planez- plane

x

y

u

v

1

23

−1

−3

−4−5

1

2−1−2−3

−4

10−3 −1−2

0

Figure 3.8 Linear transformation w = az + b.

In general, the mapping w = az + b translates by b, rotates by arga and magnifies (orcontracts) by |a|.

Example 3.3. Consider the transformation w = (1+i)z+(1+2i) =√2 eiπ/4z+1+2i in

the z-plane. It maps the rectangle bounded by the lines AB: x = 0; BC: y = 0; CD: x = 2;and DA: y = 1 in the z-plane onto the rectangle bounded by the lines A’B’: u + v = 3;B’C’: u− v = −1; C’D’: u+ v = 7; and D’A’: u− v = −3, respectively, such that the pointsz = (0, 1), (0, 0), (2, 0), (2, 2) are mapped onto the points w = (0, 3), (1, 2), (3, 4), (2, 5), re-spectively. The rectangle ABCD is translated by (1 + 2i), rotated by an angle π/4 in thecounterclockwise direction, and contracted by

√2.

Map 3.5. The transformation w = 2z − 2i maps the circle |z − i| = 1 onto the circle|w| = 2. The transformation that maps the disk |z − i| = 1 onto the exterior of the circle

|w| = 2 is w =2

z − i. In fact, let w = g(z) =

az + b

cz + d. Choose g(i) = ∞, so g(z) =

az + b

z − iwithout loss of generality. Take three points on the circle |z− i| = 1 as 0, 1+ i, and 2i; theng(0) = ib, g(1 + i) = a(1 + i), and g(2i) = 2a− ib. Since all these three points must lie on

|w| = 2, the simplest choice is a = 0 and b = 2. Then g(z) =2

z − i.

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3.3 CROSS-RATIO 61

Map 3.6. The inversion w =1

zis an involutory transformation with fixed points F1 = 1

and F2 = −1. The mapping from z-plane to w-plane, or inversely, is presented in Figure 3.9,where the radius 0A is 1; the angle ∠ z0z is bisected by the real axis; az : zB = Aw : Bw;p, q and r are real; and the points z and w are inverse with respect to the unit circle.

ℜ

w=

λ

0

ℜ

= 0_

λ z

A

Bz_

z

w

••

•

•

x =

0

y = 0

u =

0

v = 0

w - plane

z- plane

0

Figure 3.9 Transformation w =1

z.

There are five special cases under the inversion w =1

z. They are:

(a) The interior (exterior) of the circle of D in the z-plane is mapped onto the interior(exterior) of the same circle, with points z∞ = 0;∞; i;−i in the z-plane mapped into thepoints ∞;w∞ − 0;−i; i, respectively, in the w-plane.

(b) The line ℜλz = 0 in the z-plane is mapped onto the line ℜλw = 0 in the w-plane;and the circle |z| = r in the z-plane is mapped onto the circle |w| = 1/|z| in the w-plane.These two cases are geometrically obvious and no figures are needed for them.

The remaining three cases are presented in Figure 3.10.; details, with z = p, q, r, rp, rqreal, and w = P,Q,R,Rp, Rq real, are as follows:

(c) The line ℜλz = 0 is mapped onto the circle

∣∣∣w − λ

2p

∣∣∣ =∣∣∣λ

2p

∣∣∣.

See Figure 3.10(c).

(d) The circle |z − z0| = |z0| 6= 0 is mapped onto the line ℜz0 = 12 .

See Figure 3.10(d).

(e) The circle |z − z0| = r 6= |z0| is mapped onto the circle

∣∣∣w − z0|z0|2 − r2

∣∣∣ =r∣∣|z0|2 − r2

∣∣ .

See Figure 3.10(e).

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(c)

(d)

(e)

ℜ( ) = p

_λ z • _

λ 2 p/λ 2 p/R= | |

x

y

u

v

•x

y

u

vz0| |z _ z0= | |

z0

ℜw

=1/2z 0

•

x

y

z0

•

0w Rr

u

v

w - planez- plane


z.

Map 3.7. The transformation w =1

z, or z =

1

w, which maps the interior of a circle in

the z-plane onto the exterior of the circle in the w-plane and conversely, is shown in Figure3.11.

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3.3 CROSS-RATIO 63

w - planez - plane

x

y

u

v

0 01

_ i

•

1

•

A

A’

i

Figure 3.11 Circle in the z-plane onto the exterior of a circle.

Since

w = u+ iv =x

x2 + y2− y

x2 + y2i, z = x+ iy =

u

u2 + v2− v

u2 + v2i,

we get

u =x

x2 + y2, v = − y

x2 + y2.

A line through the origin in the z-plane is mapped onto a line through the origin in thew-plane. However, a line not passing through the origin in the z-plane is mapped onto acircle in the w-plane which passes through the origin, as the following example shows.

Example 3.4. The equation ax+ by+ c, c 6= 0 represents a straight line in the z-plane.Under the inversion map w = 1/z, this line is mapped onto a circle in the w-plane whichpasses through the origin. The details are as follows: The mapped curve is

au

u2 + v2− bu

u2 + v2+ c = 0.

Thus, au− bv + c(u2 + v2

)= 0. Dividing by c, we obtain

u2 + v2 +a

cu =

b

cv = 0,

which is the equation of the required circle.


z, which maps the interior of the circle |z − 1| = 1

in the z-plane onto the right plane u ≥ 12 , −∞ < v <∞ in the w-plane, is shown in Figure

3.12.

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w - planez- plane

x

y

u

v

0ABC

’A

B ʼ

ʼC

2• 1 2/ •

Figure 3.12 Circle in the z-plane onto a rectangle.

Map 3.9. The transformation that maps the exterior of the unit circle in the z-planeonto the interior of the unit circle in the w-plane is defined by

w = 1/z,

and is presented in Figure 3.13.

1

w - planez- plane

x

y

u

v

A

B

C

D

’A

B ʼ

ʼC

ʼD

1

Figure 3.13 Exterior of the unit circle onto the interior of the unit circle.

Map 3.10. The transformation w = k/z is known as the reciprocal or the inverse

transformation for real k. We have w = ReiΘ = k(r eiθ

)−1

=k

re−iθ, which gives R = k/r,

and Θ = −θ. The fixed points are: F1,2 = ±√k, and the critical points are z = 0,∞. It is

a useful transformation in electric transmission through a single wire, discussed in Example22.4 with k = 3600. Two particular cases for k = −α, α > 0, and k = 1 are given in Maps3.6, 3.7, 3.8, and 3.9, respectively.

Map 3.11. The transformation w = a/z maps the exterior of a circle of radius a:x2 + y2 ≥ a2, in the z-plane onto the unit disk |w| ≤ 1 in the w-plane.

Map 3.12. The transformation w = z +1

z, which maps the exterior of the upper

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3.3 CROSS-RATIO 65

semi-circle in the z-plane onto the upper-half of the w-plane, is presented in Figure 3.14.

1−

i

.

.

. ..A B

C

A’ B’ C’

2−21.

00

.w - planez - plane

x

y

u

v

Figure 3.14 Exterior of a circle in the z-plane onto the upper-half of the w-plane.

Map 3.13. The transformation w = z+1

z, or z =

w +√w2 − 1

2, which maps the interior

of the upper semi-circle in the z-plane onto the lower-half of the w-plane, is presented inFigure 3.15.

w - planez- plane

x

y

u

v

B

C

D

B ʼʼCʼD ’A•

A

•

_1

i

0 1• • •

• •0 2_2

Figure 3.15 Interior of the upper half circle onto the lower-half of the w-plane.

Map 3.14. The transformation w = z+1

z, which maps the annular region between two

upper semi-circles in the z-plane onto the upper-half of the ellipse in the w-plane, wherethe ellipse (B’C’D’) is defined by

( ku

k2 + 1

)2+( kv

k2 − 1

)2= 1,

is presented in Figure 3.16.

v

B ʼ

ʼC

ʼD ’AE ʼ ʼF •

z- plane

A B

C

D E

F

x

y

• 2_2 •_1 0 1• • •

•

•

i

i2

_2• • 20 5 /2•

i3 /2•

u

Figure 3.16 Annulus between two semi-circles onto the upper-half of the ellipse.

Map 3.15. If a = c = 1, d = −b = α, then w =z − α

z + α. For this bilinear transformation

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the above three successive cases are presented in Figure 3.17; these three cases may betreated as separate transformations (see cases 3.11 and 3.44 also).

w - plane

z- plane

u

v

0x

y

0

00

- planeζ

ξ - plane

Figure 3.17 w =z − α

z + α.

In Figure 3.17, we start in the z-plane and translate the right half-plane by α in theξ-plane so that ξ = z + α; the right half-plane in the ξ-plane is mapped onto the circle inthe ζ-plane by ζ = 1/ξ = 1/(z + α); finally, the circle in the ζ-plane is mapped onto the

circle in the w-plane so that w = 1 + 2αζ = 1− 2α

z + α=z − α

zα.

We can also select suitable parameters for this transformation; for example, given twocircles, marked I with radius r′ and marked II with radius r′′, such that the distancebetween their centers is h, and the distance of their centers as c1 and c2 from the origin ofthe coordinates (see Figure 3.18), we have the case of translation along the real axis, since

w =z − α

z + α=x+ iy − α

x+ iy + α= ReiΘ, (3.3.18)

where

R =∣∣∣z − α

z + α

∣∣∣, R2 =(x − α)2 + y2

(x + α)2 + y2=x2 + y2 + α2 − 2αx

x2 + y2 + α2 + 2αx. (3.3.19)

The equation of circle I from Figure 3.18 is (x − c1)2 + (y2 = (r′)2, or x2 + y2 =

(r′)2 − c21 + 2xc1. Then the radius R′ of the map of circle I is

R′2 =r′2 − c21 + 2xc1 + α2 − 2αx

r′2 − c21 + 2xc1 + α2 + 2αx=r′2 − c21 + α2 + 2x(c1 − α)

r′2 − c21 + α2 + 2x(c1 + α). (3.3.20)

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3.3 CROSS-RATIO 67

hc1

c2

c1

c2

h

I

II

x

y

I

II

x

y

r"’r

’r

r"

I

II

R’

"R

w - plane

z- plane

u

v

Figure 3.18 w =w − α

w + α, Two circles.

If we select α and c1 such that the relation α2 = c21 − r′2 is satisfied, then R′ becomesindependent of x, i.e.,

R′ =(c1 − α

c1 + α

)1/2=

r′

c1 + α, r′ = (c1 − α)(c1 + α)1/2. (3.3.21)

The circle II can be treated similarly by selecting α22 = c22 − r′′2, thus finally giving

R′′ =r′′

c2 + α. (3.3.22)

If we substitute h = c2 − c1, then this transformation can be written as

w =z − α

z + αwhere α2 = c21 − r′

2, (3.3.23)

and

c1 =r′′2 − r′2 − h2

2h, c2 =

r′′2 − r′2 + h2

2h= c1 + h, R′ =

r′

c1 + α, R′′ =

r′′

c2 + α.

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A critical point for this transformation is z = −α, and the inverse of this transformation

is of the same type, i.e., z = α1 + w

1− w.

The general form of the bilinear transformation (3.2.1) includes as special cases all theintermediate transformations which are discussed below.

Map 3.16. The bilinear transformations w =z + 1

z − 1, z =

w + 1

w − 1, are involutory, with

p, q real, and the fixed-points F1,2 = 1±√2. The interior of the circle of D is mapped onto

the interior of the same circle; however, the exterior of the circle of E is mapped onto theinterior of the same circle (see Figure 3.19).

w - planez- plane

y = 0 v = 0

x =

0 1x

=

x =

p

u =

0 1u

=

0

y = q

0

Figure 3.19 Transformation w =z + 1

z − 1.

Other details for some particular cases, with z = p, q real and w = P,Q real, are asfollows:

(a) The points z = 0; z∞ = 1;∞ are mapped onto the points w = −1;∞;w∞ = 1.

(b) The half-plane x ≥ 0 is mapped onto the region |w| ≥ 1.

(c) The half-plane y ≥ 0 is mapped onto the half-plane v ≤ 0.

(d) The region |z| ≤ 1 is mapped onto the half-plane u ≤ 0.

(e) The line x = 1 in D is mapped onto the line u = 1.

(f) The line x = p, p 6= 1 is mapped onto the circle∣∣∣w − p

p− 1

∣∣∣ =1

|p− 1| .

(g) The line y = q, q 6= 0 is mapped onto the circle∣∣∣w − q − 1

q

∣∣∣ =1

|q| .

Map 3.17. The transformation w =1 + z

1− z= eiπ

z + 1

z − 1, is the inverse of the Map 3.16.

It maps the upper half-plane ℑz > 0 onto the unit disk B(0, 1).

Map 3.18. The transformation w =1 + z

1− z, or z = −1− w

1 + wmaps the unit upper semi-

circle in the z-plane onto the first quadrant of the w-plane, such that the points z = −1, 0, 1

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3.3 CROSS-RATIO 69

map into the points w = 0, 1,−1, respectively, as shown in Figure 3.20.

.. .

.

.A

A’B’

C

DD’.

••

• •

• •B •_1

i

0 1

i

0 1

w - planez- planeFigure 3.20 Upper semi-circle in the z-plane onto the first quadrant of the w-plane.

Map 3.19. The transformation w =z − 1

z + 1, or z =

1 + w

1− wmaps the right-half of the z-

plane onto the unit circle, such that the points z = 0, i,−imap into the points w = −1, i,−i,respectively. This map is shown in Figure 3.21(a).

B

A

A’

..

.

B’.

w - planez - plane

x

y

u

v

•

• • •1_1

i

_ i ..

.C’

C

i

_ i

B

A

A’

..

.

B’.

w - planez - plane

x

y

u

v

•

• • •1_1

i

_ i ..

.

C’

C

i

_ i

(a)

(b)

Figure 3.21 Right-half z-plane onto the unit circle.

Example 3.5. To find the linear fractional transformation that maps the interior of thecircle |z − i| = 2 onto the exterior of the circle |w − 1| = 3, we need only three points onthe first circle in clockwise order and three on the second circle in counterclockwise order.Let z1 = −i, z2 = −2 + i, and z3 = 3i and w1 = 4, w2 = 1+ 3i, and w3 = −2. Then we get

(w − 4)(3 + 3i)

(w + 2)(−3 + 3i)=

(z + i)(−2− 2i)

(z − 3i)(−2 + 2i),

or

w =z − 7i

z − i.

The center of the first circle is z = i, which, under this transformation, is mapped onto the

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point w = ∞, and thus it lies outside the second circle.

Map 3.20. The transformation w =1− z

1 + z= eiπ

z − 1

z + 1, is the inverse of the Map 3.18.

Thus, this transformation, or its inverse z =1− w

1 + w, also maps the right-half of the z-plane

onto the unit circle such that the points z = 0, i,−i map into the points w = 1,−i, i,respectively, as in Figure 3.21(b).

Map 3.21. The transformation w =1− z

1 + zmaps the exterior of the unit circleD : |z| > 1

onto the regionG : ℜw < 0, which is the left-half of the w-plane. To see this, first considerthe left of the regions D and G, and choose any three points on the unit circle that givenegative (clockwise) orientation, say, z1 = 1, z2 = −i, z3 = −1. Similarly the three pointsw1 = 0, w2 = i, w3 = ∞ in the left region of G. Then the bilinear transformation is givenby

w − 0

i− 0=

(z − 1)(−i+ 1)

(z + 1)(−i− 1),

which simplifies to the above transformation.

Map 3.22. The bilinear transformation w =z − i

z + i, z = i

1 + w

1− w, has the fixed-points

F1,2 = 12 (1−i)

(1±

√3), and p, q, P,Q real. Two particular cases (A) and (B) are presented

in Figure 3.22 and Figure 3.23, respectively.

I II III

VI

VII

VII

VIII

VIII

IV

V

w - planez- plane

y = 0 v = 0

x =

0

0

x =

1pp 2

x =

y = −1

y = 1q

2qy =

I

IIIII

IV

IV

V

VII

VIII

u =

0

u =

1

Figure 3.22 Transformation w =z − 1

z + 1(Case A).

The details of the case A, with z = p, q real and w = P,Q real, are as follows:

(a) The points z = 0, i,−i; z∞ = ∞ are mapped onto the points w = −1, 0,∞;w∞ = 1.

(b) The half-plane x > 0 is mapped onto the half-plane v < 0.

(c) The half-plane y > 0 is mapped onto the region |w| < 1.

(d) The region |z| < 1 is mapped onto the half-plane u < 0.

(e) The line x− y = 1 is mapped onto the line u− v = 1.

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3.3 CROSS-RATIO 71

(f) The line x = p, p 6= 0 in D is mapped onto the circle∣∣∣w −

(1− 1

p

)∣∣∣ =1

|p| .

(g) The line y = −1 is mapped onto the line u = 1.

(h) The line y = q, q 6= −1 is mapped onto the circle∣∣∣w − q

q + 1

∣∣∣ =1

|q + 1| .

III

III

IV

V

VI

VI

VII

VIII

IX

X

XIXII

VII

I

II

IIIIV

V

VIII

IX

X XI

XII

VII

y = 0

v = 0

x =

0

y = −1

u =

0

u =

1 pu

=

w - planez- plane

Figure 3.23 Transformation w =z − 1

z + 1(Case B).

The details of the case B, with z = p, q real and w = P,Q real, are as follows:

(a) The line y = −1 is mapped onto the line u = 1.

(b) The circle∣∣∣z − i p

1− p

∣∣ = 1

|1− p| is mapped onto the line u = P, P 6= 1.

(c) The circle∣∣∣z −

(− i− 1

q

)∣∣ = 1

|q| is mapped onto the line v = Q, Q 6= 0.

Map 3.23. The transformation w =z − i

z + i, or z = i

1 + w

1− w, which maps the upper-half

z-plane onto the circle, is presented in Figure 3.24.

A

A’

B

B’

C

C’w - planez - plane

x

y

u

v

•• • •

•

•

1_1 0

10

i

_ i

Figure 3.24 Upper-half z-plane onto a circle.

Map 3.24. The bilinear transformation w =i+ z

i− zis a composition of the following

three transformations: (i) w = −1

s(Map 3.6); (ii) s =

ζ − i

ζ + i(Map 3.23); and (iii) ζ = z

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(translation). It maps the upper half-plane ℑz >: 0 in the z-plane onto the unit disk|w < 1 in the w-plane (see Figure 3.25).

w - planez- plane

x

y

u

v

A B C D E’A

E ʼ

1B ʼ

ʼC

ʼD

•1

Figure 3.25 Upper half-plane onto the unit disk.

Map 3.25. The transformation

w(z) = i1− z

1 + z

maps the unit disk onto the upper half-plane ℑw > 0.

x

y

u

v

B ʼ

1_ 1

x

y

u

v

1_ 1

0

0

0

0

••

••w - planez- plane

(a)

(b)

Figure 3.26 w = i1− z

1 + z.

Since the lower boundary of the semi-disk, i.e., the interval [−1, 1], is perpendicular tothe upper semi-circle at the point 1, and since w is conformal at z = 1, the images of theinterval [−1, 1] and the semi-circle intersect at right angle. Since they both pass throughthe point −1, which is mapped onto ∞, we conclude that these images are perpendicularlines. Thus, using w(0) = i and w(i) = 1, the interval [−1, 1] in the z-plane is mappedonto the upper half of the imaginary v-axis, and the semi-circle is mapped on the righthalf of the real u-axis. Finally, testing the image of an interior point, say i/2, we find thatw(i/2) = 4/5 + 3i/5, which is a point in the first quadrant. Hence, the upper semi-diskin the z-plane is mapped onto the first quadrant u > 0, v > 0 of the w-plane, since the

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3.3 CROSS-RATIO 73

boundary is mapped onto boundary and interior points onto interior points (Figure 3.26).


w = −i z + i

z − i, or z = i

w − i

w + i,

maps the points z = i,−i, 0, respectively, onto the points w = ∞, 0, and a point with v > 0.Thus, this map guarantees that the circle in the z-plane transforms into a straight line,which passes through the origin w = 0 and the interior of the circle is mapped onto theupper half of the w-plane.

Map 3.27. The bilinear transformation w = f(z) =z

2z − 8maps the region Int (Γ),

where Γ = |z − 2| = 2, conformally onto the region ℜw < 0, such that the point z = 2goes into w = −1/2 (Saff and Snider [1976:317]).

Map 3.28. The bilinear transformations w =z

z − 1, z =

w

w − 1are involutory, with

z∞ = 1, w∞ = 1, the fixed-points F1 = 0, F2 = 2, and p, q real. The interior (exterior) ofcircles of D and E map onto exterior (interior) of circles. The details of different particularcases, with z = p, q real and w = P,Q real, are as follows:

(a) The half-plane y ≥ 0 is mapped onto the half-plane v ≤ 0.

(b) The half-plane x ≥ 0 is mapped onto the region∣∣w − 1

2

∣∣ ≥ 12 .

(c) The region |x| ≤ 1 is mapped onto the half-plane u ≤ 12 .

(d) The half-plane x ≤ 12 is mapped onto the region |w| ≤ 1.

(e) The line x = 1 is mapped onto the line u = 1, passing through w = 1.

(f) The line x = p, p 6= 1 is mapped onto the circle∣∣∣w − 2p− 1

2(p− 1)

∣∣∣ =1

|2p− 2| passing

through w = 1.

(g) The line y = q is mapped onto the circle∣∣w − (1 − 1

2q)∣∣ = 1

2|q| passing through w = 1.

Map 3.29. The function w = f(z) =z

z − (1 + i)maps the lens-shaped region bounded

by the circles |z−1| = 1 and |z− i| = 1 onto the region bounded by the rays argw = 3π/4and argw = 5π/4 such that f(0) = 0 and f(1 + i) = ∞.

Map 3.30. The transformation w =z

iz + 2maps the disk |z − i| = 1 onto the real line.

To see this, since the points z = 0, 1 + i, 2i lie on the given circle in the z-plane and sincethe real line in the w-plane passes through the points w = 0, 1,∞, so we choose h(z) such

that h(0) = 0, h(1 + i) = 1, and h(2i) = ∞. Then h(z) =z

iz + 2. Since h(i) = i, so h maps

the disk |z − i| < 1 onto the upper half-plane.

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Map 3.31. Circle onto circle. The transformation is

w − w0 = ℜeiτ

z − z0 − rα

αz − αz0 − r

, (3.3.24)

where τ is real, |α| 6= 1, and is presented in Figure 3.27.

w - planez- plane

•r

z0••

∞w R R

w0

• OR

| | < 1α| | > 1α

∞w•

w0

Figure 3.27 Circle onto circle.

Map 3.32. Mapping of the unit circle onto itself:

w = eiτz − α

αz − 1, τ real and |τ | 6= 1. (3.3.25)

Map 3.33. The transformation w = eiθz − z01− z0z

, which maps a circle onto the circle in

the w-plane such that the point z0 maps into the point w = 0, is shown in Figure 3.28.

0

zo

w - planez - plane

x

y

u

v

••

Figure 3.28 Circle onto a circle.

Map 3.34. The function f(z) = ρ eiαz − z1z − z2

maps the region enclosed by two arcs onto

the region enclosed by the sector shown in Figure 3.29, such that a fixed point ζ 6= z1,2 on

Γ1 goes into a point ω on the u-axis, where α = argζ − z1ζ − z2

and ρ = ω

∣∣∣ζ − z1ζ − z2

∣∣∣ (Pennisi[1963:321]).

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3.3 CROSS-RATIO 75

z1

L1

z2L2

α

α

1Γ

2Γ

ζ

ω

.

.0 0

−z enalpenalp −w

x u

vy

Figure 3.29 Crescent-shaped region.


1− z, z =

w − 1

w, with z∞ = 1;w∞ = 0; the fixed

points F1 = 12 + i

2

√3, and F2 =

1

F1, and p, q, P,Q are real. The interior of the circle in

the z-plane is mapped onto the interior of the circle in the w-plane. The details of thetransformation, with z = p, q real and w = P,Q real, are as follows:

(a) The half-plane y ≥ 0 is mapped onto the half-plane v ≥ 0.

(b) The half-plane x ≥ 0 is mapped onto the region∣∣w − 1

2

∣∣ ≥ 12 .

(c) The region |z| ≤ 1 is mapped onto the half-plane u ≥ 12 .

(d) The region |z − 1| ≤ 1 is mapped onto the region |w| ≥ 1.

II

I

III

I

II

III

II

w - planez- plane


1− z.

There are two special cases, one presented in Figure 3.30, and the other in Figure 3.31;their respective details, with z = p, q real and w = P,Q real, are as follows:

For Figure 3.30:

(a) The line x = 1 is mapped onto the line u = 0.

(b) The line x = p 6= 1 is mapped onto the circle∣∣∣w − 1

2(1− p)

∣∣∣ =1

2|1− p| .

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(c) The line y = q 6= 0 is mapped onto the circle∣∣∣w − 1

2q

∣∣∣ =1

2|q| .

u

v

x

y

1x

=

10

pu =

pu

=

= qv

= qv

w - planez- plane

w - planez- plane


1− z.

For Figure 3.31:

(a) The circle∣∣∣z −

(1− 1

2p

)∣∣∣ =1

2|p| is mapped onto the line u = P 6= 0.

(ib) The circle∣∣∣z −

(1 +

1

2q

)∣∣∣ =1

2|q| is mapped onto the line v = Q 6= 0.


z− a, a real, maps the line y = 0 and the circle

∣∣z + 12 i∣∣ = 1

2 onto the lines v = 0 and v = 1, respectively.

Map 3.37. The transformation w =2i r

z− a, a real, maps the line x = 0 and the circle

|z − r| = r > 0 onto the lines v = 1 and v = 1, respectively.

Map 3.38. The transformation is

w =r2

r2 − r1

(2r1z

+ i). (3.3.26)

It is presented in Figure 3.32.

The details of the transformation for this case with z = p, q real and w = P,Q real, areas follows:

(a) The points z = 0; ∞; 2ir1 are mapped onto the points w = ∞;ir2

r2 − r1; 0.

(b) The circle |z − ir1| = r1 is mapped onto the line v = 0.

(c) The circle |z − ir2| = r2 is mapped onto the line v = 1.

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3.3 CROSS-RATIO 77

(d) The line x = 0 is mapped onto the line u = 0.

(e) The line y = 0 is mapped onto the line v =r2

r2 − r1.

(f) The point z3 = ir1(1− eiθ

)is mapped onto the point w3 =

r2r2 − r1

cotθ

2.

(g) The small circle touching both circles and passing through z3 is mapped onto the circle∣∣w + w3 − 12 i∣∣ = 1

2 .

•

3w•2

w - planez- plane

2r2ri

y = 0

v = 0

u =

0x =

0

r1ir1

v = 1

θ

Figure 3.32 Two circles in outer contact.


w =i r2

r1 + r2

(1− 2r1

z

)(3.3.27)

is presented in Figure 3.33.

2

w - planez- plane

v = 0

u =

0

v = 1

| |z _ =i ρ ρ

y = 0

x =

0

••r12r−

2r

u =

τ

Figure 3.33 Two circles in outer contact.

The details of the transformation for this case with z = p, q real and w = P,Q real, areas follows:

(a) The points z = 0; ∞; 2r1 are mapped onto the points w = ∞;ir2

r1 + r2; 0.

(b) The circle |z − r1| = r1 is mapped onto the line v = 0.

(c) The circle |z + r2| = r2 is mapped onto the line v = 1.

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(d) The circle |z − iρ| = r2 is mapped onto the line u = τ , where τ = − r1r2ρ(r1 + r2)

.

(e) The lines x = 0; y = 0 are mapped onto the lines v =r2

r2 − r1; u = 0.

Map 3.40. The transformation that maps the upper-half of the unit circle in the z-planeonto the upper half-plane ℑw > 0 is defined by

w =(1 + z

1− z

)2,


0w - plane

z- planex

y

u

v

0 • ••••1

’A B ʼ ʼC ʼD• 1_C

•

•

1_ 1A

B

D ’A

Figure 3.34 Upper-half of the unit circle onto the upper half-plane.

Map 3.41. The transformation that maps the upper half plane ℑz > 0 onto around corner in the w-plane is w = (z + 1)α + (z − 1)α, 1 < α < 2; it is presented inFigure 3.35. The transformation maps the line segment y = 0,−1 ≤ x ≤ 1 in the z-planeonto the part −2α cos(α1)π ≤ u ≤ 2α, −2α sin(α − 1)π ≤ v ≤ 0, i.e., the part DCB, of

(v cos(απ) − u sin(απ))1/α

+(−v)1/α = 2 (− sin(απ))1/α

. Note that for α = 3π/2, the curveis the asteroid u2/3 + (−v)2/3 = 2.

III

w - planez- plane

AB

C

D

E

F

G

=v 0

_1z = 0z =ABCDE

F

G

••1z =•

•

x

y

2πα

iz = cot

•

•

•• 0w=

e iw 1= + α π2 αw =

2 αw = e iα π

arg 2παw =

arg παw =

I

II

w| | 2 α≥;

Figure 3.35 ℑz > 0 onto a round corner.

Map 3.42. The transformation that maps the sector of angle π/m of a circle in thez-plane onto the upper half-plane ℑw > 0 is defined by

w =(1 + zm

1− zm

)2,

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3.3 CROSS-RATIO 79


0

w - planez- plane

x

y

u

v

0 • ••••1

’A B ʼ ʼC ʼD• 1_

C

•

1A

B

D ’A

•

• /mπ

Figure 3.36 Sector of angle π/m of a circle onto the upper half-plane.


w =iz2 + 1

iz2 − 1, (3.3.28)

conformally maps the upper half-plane U = ℑz > 0 onto the unit disk C(0, 1). To

prove it, first note that the transformation z = g(w) =w − 1

w + 1(Map 3.17) maps the right

half-plane ℜw > 0 onto B(0, 1). Also, multiplication by i = eiπ/2, with z = h(w) = iwrotates the complex plane by π/2 and thus maps the right half-plane C(0, 12 ) onto the upper

half-plane U , while its inverse w = h−1(z) = −iz will map U to the right half-plane. Hence,to map the upper half-plane U onto the unit disk, the composition of these two mappingsgives the required map:

w = g h−1(z) =−iz − 1

−iz + 1=iz + 1

iz − 1. (3.3.29)

Similarly, we already know (see Maps 4.6) that the square map w = z2 maps the upperright quadrant 0 < argz < π/2 onto the upper half-plane U . Thus, composition of thismap with the previously constructed map, where we replace z by w in (3.3.29), we obtainthe final required mapping (3.3.28).

Map 3.44. The transformation that maps the lens-shaped region of angle π/m, wherethis region is bounded by two circular arcs, in the z-plane onto the upper half-plane ℑw >0 is defined by

w = e2mi arccot(p)(z + 1

z − 1

)m, (3.3.30)


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0

w - planez- plane

x

y

u

v

0 • ••••1

’A B ʼ ʼC ʼD

•

1_

C•

1A

B

’A1_

/mπp

D

•

•

Figure 3.37 Lens-shaped region of angle π/m onto the upper half-plane.

Map 3.45. The transformation w =z − a

az − 1, where a > 1, −1 < x2 < x1 < 1, R0 > 1,

and

a =1 + x1x2 +

√(1− x21) (1− x22)

x1 + x2,

R0 =1− x1x2 +

√(1− x21) (1− x22)

x1 + x2, (3.3.31)

maps the region between the two circles in the z-plane onto the annular region in thew-plane, as shown in Figure 3.38.

1

w - plane

z- plane

x

y

u

v

A

B

C

D

E

F

’A

B ʼ

ʼC

ʼD

ʼF

E ʼG•

R01• •

G ʼ1x•x2

Figure 3.38 Map 3.45.

Map 3.46. The transformation w =z − a

az − 1, where a and R0 are the same as in (3.3.31),

with x2 < a < x1, and 0 < R0 < 1 when 1 < x2 < x1, maps the region exterior to the twocircles in the z-plane onto the annular region in the w-plane, as shown in Figure 3.39.

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3.3 CROSS-RATIO 81

1

w - planez- plane

x

y

u

v

A

B

C DE

F ’A

B ʼ

ʼC ʼD ʼF

E ʼ

•R0 1• •

1x•2 •

Figure 3.39 Map 3.46.

Map 3.47. The transformation w = eiλz − z0z + z0

, where λ is a real number, maps the

upper half-plane −∞ < x <∞, 0 ≤ y <∞, in the z-plane onto the unit disk |w| ≤ 1 in thew-plane.

Map 3.48. The transformation w = eiλz − z01− z0z

, where λ is a real number, maps the

unit disk x2 + y2 ≤ 1, in the z-plane onto the unit disk |w| ≤ 1 in the w-plane.

Map 3.49. The transformation w = eiβz − α

αz − 1for some |α| < 1, −π < β < π, maps

the unit disk onto itself.

Map 3.50. Cassini’s ovals. There are four cases:

Map 3.50a. Either part of the interior of Cassini’s ovals |z − z1||z − z2| = C, with fociz1 = ±√

w1, is mapped onto the interior of the circle |w − w1| = C, C ≤ |w1|. This case ispresented in Figure 3.40, where w1is taken as a real and positive number.

x 0

w - planez- plane

u0

β

w 2= 4 z0

z z0=

2 β

Figure 3.40 Cassini’s ovals, Map 3.50a.

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Map 3.50b. Cassini’s ovals |z − z1||z − z2| = C, with foci z1 = ±√w1, is mapped onto

the circle |w − w1| = C, C > |w1|, counted twice.

Map 3.50c. The part x > 0 of the interior of Cassini’s ovals, and the part x < 0 of theinterior of Cassini’s ovals, both are mapped onto the interior of the circle cut from w = 0

to w′ = ℜw1 =(C2 −ℑw12

)1/2, i.e., point a. This case is presented in Figure 3.41.

y

w1•z1

w - planez- plane

0• •z1

a

b

c

abc

Figure 3.41 Cassini’s ovals, Map 3.50c.

Map 3.50d. (Generalized Cassini’s Ovals) The details of the transformation are asfollows:

(a) Each part of |z− z1||z − z2| · · · |z − zn| = C, with foci zj = e2iπj/n n√w0, j = 1, 2, . . . , n,

is mapped onto the circle |w − w0| = C, 0 < C ≤ |w0|.

(b) Closed curve |z − z1||z − z2| · · · |z − zn| = C (which is approximately circle for large C)is mapped onto the circle |w − w0| = C, counted n times, 0 < |w0| < C.

(c) Region bounded by this curve and the lines z = r eiθ, z = r ei(θ+2θ/n, θ fixed, is mappedonto the interior of |w − w0| = C, C > |w0|, cut from 0 to the point at which the linew = rn einθ meets the circle.

This transformation for n = 4 is presented in Figure 3.42.

•

••

•

Figure 3.42 Cassini’s ovals.

Map 3.51. Cardioid and Limacon. The mapping from z-plane onto the w-plane is

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3.3 CROSS-RATIO 83

presented in Figure 3.43.

x 0

w - planez- plane

u0

β

w 2= 4 z0

z z0=

2 β

Figure 3.43 Circle onto cardioid.

The details of this transformation are as follows:

(a) The circles

|z − z0| = |z0||z + z0| = |z0|

are mapped onto the cardioid w = 2z20(1 + cos θ) eiθ ,

−π ≤ θ < π.

(b) The circles

|z − z0| = c|z + z0| = c

, c > 0, z0 6= 0, are mapped onto the limacon w − w1 =

2c(z0 + c cos θ) eiθ , w1 = z20 − c2, −π ≤ θ < π.

Map 3.52. Cardioid and Generalized Cardioids (α > 1), and Lemniscates(0 < α < 1) (Figure 3.44).

••

x

y

•

0 = x

y = 01l

2l

wp wq

RqpR

)c/d -( = x

ℜ

)c/ d -( p = x

ℜ≠

y = (- d/c)ℑ

y = q (- d/c)ℑ≠

z- plane w - plane

• •

III

III III

III

0

∞z

Figure 3.44 Generalized cardioid.

The details of this transformation are as follows:

The line ℜe−iθ z = s, s > 0 is mapped onto each part of Rβ = s−1 cos(βθ + φ), |θ −α(2kπ − φ| ≤ απ/2, k = 0,±1,±2, . . . ; the curves are closed, with a cusp at 0, with

tangents w = Reiα(kπ+π2−φ), k = 0,±1,±2, . . . ; there are n parts if α−1 = n ∈ Z.

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3.4 Straight Lines and Circles

We will consider different cases of linear and bilinear transformations that map given ele-ments, like points, lines, circles, onto prescribed points, lines, and circles, using the cross-ratio. This classification of linear and bilinear transformations is based on Kober [1957:2-32].A consequence of the Riemann mapping theorem is that simply connected domains Dz andDw can always be mapped conformally onto each other. Three parameters may be chosenat will, based on the elements in w corresponding to similar elements in z, such as points,circles, rectangles, and so on. It is often convenient to start with a line, or a circle, or arectangle (or square) in the w plane and create an image in the z-plane by applying variousmapping functions z = g(w), where g(w) = f−1(w) are the inverse of the ‘forward’ mappingfunction w = f(z). Often a unit circle is used as an intermediate image in constructingsequential mappings. Some examples are as follows.

Map 3.53. Lines parallel to the axes. z∞ = −d/c, w∞ = a/c (see Figure 3.45).

••

x

y

•

0 = x

y = 01l

2l

wp wq

RqpR

)c/d -( = x

ℜ

) c/ d -( p = x

ℜ≠

y = (- d/c)ℑ

y = q (- d/c)ℑ≠

z- plane w - plane

• •

III

III III

III

0

∞z

Figure 3.45 Lines parallel to axes (direct map).

Different cases of this transformation, with z = p, q, r, rp, rq real and w = P,Q,R,Rp, Rqreal, are as follows:

(a) The line x = −ℜd/c is mapped onto the line l1 : ℜc(ad− bc

)(cw − a)

= 0.

(b) The line x = p, p 6= −ℜd/c is mapped onto the circle |w − wp| − Rp, where wp =acp+ 1

2

(ad+ bc

)

p|c|2 + ℜcd , and Rp =∣∣∣

ad− bc

2p|c|2 + 2ℜcd∣∣∣.

(c) The line y = −ℑd/c is mapped onto the line l2: ℑc(ad− bc

)(cw − a) = 0.

(d) The line y = q, q 6= −ℑd/c is mapped onto the circle |w − wq| = Rq, where wq =acp+ 1

2 i(ad− bc

)

q|c|2 + ℑcd , and Rq =∣∣∣

ad− bc

2q|c|2 + 2ℑcd∣∣∣.

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3.4 STRAIGHT LINES AND CIRCLES 85

The inverse mapping is presented in Figure 3.46.

p

•

•

w - plane

x•

u =

0

u = (a/c)ℜ

v =

(a

/c)

ℑ

z- plane

∞w

u

zq

qr zpu =

(a

/c)ℜ

v = (a/c)ℑ

rpu = p = (a/c)ℜ/•

•

•

v =

q

(

a/c)

ℑ= /

I

IIIII

I

II

III

∞z•

Figure 3.46 Lines parallel to axes (inverse map).

The details of different cases with z = p, q, r, rp, rq real and w = P,Q,R,Rp, Rq real, are asfollows:

(a) The line ℜc(ad− bc

)(cz + d) = 0 is mapped onto the line u = ℜa/c.

(b) The circle with center zp =12 (ad+ bc)− cdp

p|c|2 −ℜac and radius rp =∣∣∣

ad− bc

2P |c|2 − 2ℜac∣∣∣, is

mapped onto the line u = P, P 6= ℜa/c.(c) The line ℑc

(ad− bc

)(cz + d) = 0 is mapped onto the line v = ℑa/c.

(d) The circle with center zq =12 i (ad− bc)− cdq

q|c|2 − 2ℑac and radius rq =∣∣∣

ad− bc

2q|c|2 − 2ℑac∣∣∣, is

mapped onto the line v = Q, Q 6= ℑa/c.

Map 3.54. Other lines and circles. z∞ = −d/c, w∞ = a/c (see Figure 3.47.)

The details of the transformation with z = p, q, r, rp, rq real and w = P,Q,R,Rp, Rq real,are as follows:

(1a, b.) The line ℜλz = p passing through z∞ is mapped onto the line ℜΛw = Ppassing through w∞.

(2.) The line ℜλz = p not passing through z∞ is mapped onto the circle |w − w0| = Rpassing through w∞.

(3.) The circle |z − z0| = r passing through z∞ is mapped onto the line ℜΛw = P notpassing through w∞.

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(1.a)

(1.b)

(2)

(3)

ℜ ( ) = p_

λ z

ℜ ( ) = p_

λ z

ℜ ( ) = p_λ z

ℜ( ) = PΛ

_w

ℜ ( ) = P

Λ _w

ℜ ( ) = PΛ

_w

∞w

∞w

∞w

∞w

•

•

•

•

•

• •

•z _ z| | = r0

_| | =0w w R

w - planez- plane

∞z

∞z

∞z

∞z

Figure 3.47 Other lines and circles (direct map).

Map 3.55. The circle |z0 + d/c| = r not passing through z∞ is mapped onto the circle|w − w0| = R not passing through w∞ (see Figure 3.48).

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∞w•

_| | =0w w R

•w0R

’

’

•

z _ z| | = r0

•z0r

•

z _ z| | = r0

∞w•

_| | =0w w R

•w0R

’

’• rz0

z _ z| | = r0

•r •

_| | =0w w R

∞w

w - planez- plane

(4a)

(4b)

(4c)

∞z

∞z

∞z

Figure 3.48 Circles not passing through z∞.

The details of the transformation with z = p, q, r, rp, rq real and w = P,Q,R,Rp, Rq real,are as follows, where

w′0 =

(az0 + b)(az0 + d)− acr2

|cz0 + d|2 − |c|2r2 , R =r|ad− bd|

|cz0 + d|2 − |c|2r2 .

(a) The circle |z − z0| = r not passing through z∞ is mapped onto the circle |w − w′0| = R

not passing through w∞.

(b) The circle |cz0 + d| > |c|r is mapped onto the circle |w − w0| = R, where w′0 =

az0 + b− cR2s

cz0 + d, R = r|s|, and s = ad− bc

|cz0 + d|2 − |cr|2 .

(c) The circle 0 < |cz0 + d| < |c|r is mapped onto the circle |w − w0| = R, where w′0, and

R, as in (4a).

(d) The point z0 − d/c = z∞ is mapped onto the point w0 = a/c = w∞; R =∣∣∣ad− bc

c2r

∣∣∣.

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Map 3.56. Three points onto three points. There are five cases (a)-(e) discussedbelow. Using the cross-ratio (3.3.1), the transformation is defined by (3.3.2). Then

Map 56a. This is defined by

w − w1

w − w2

w3 − w2

w3 − w1=z − z1z − z2

· z3 − z2z3 − z1

, (3.4.1)

where z1, z2, z3 are given and finite, and w1, w2, w3 are finite. This is presented in Figure3.49(a).

Map 3.56b. The transformation is of the form w,w1, w2, w3 = z, z1, z2,∞, i.e.,

w − w1

w − w2

w3 − w2

w3 − w1=z − z1z − z2

, (3.4.2)

where the three given points are z1, z2,∞ and w1, w2, w3. This case is presented in Figure3.49(b).

Map 3.56c. The transformation is of the form

w − w3

w3 − w1=z2 − z1z − z2

, (3.4.3)

where the three given points are z1, z2,∞ and w1,∞, w3. This case is presented in Figure3.49(c).

Map 3.56d. The transformation is of the form

w − w1

w1 − w2=

z − z1z1 − z2

, (3.4.4)

where the three given points are z1, z2,∞ and w1, w2,∞. This case is presented in Figure3.49(d).

Map 3.56e. The transformation is of the form

w − w1

w − w2=z − z1z − z2

z3 − z2z3 − z1

, (3.4.5)

where the three given points are z1, z2, z3 and w1, w2,∞. This case is presented in Figure3.49(e).

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•

w - planez- plane

•• •

•

•

•

••

•

• •

•

z1

z2

z3

w13w

2w

w1

3w

2w

OR

• •

•

• • •

•

•

w1

3w2w

w1

3w2w

OR

•

•

•

•

z1

z2

∞

•∞

•

•

•

z1

z2

∞

3w

w1

•

•

•∞

•

•

•

z1

z2

∞

w1

2w

• •

•

• • •

•

•

w1

3w2w

w1

3w2w

OR

•

•

•∞

w1

2w

Figure 3.49 Three points onto three points.

Map 3.57. Straight line onto straight line. The transformation is

λw = P +a(λz − p

)− i b

i c(λz − p

)+ d

, (3.4.6)

where λ, p,Λ and P are preassigned. This transformation is presented in Figure 3.50(a). Inthe case of the mapping of a line onto itself, the above holds except that λ = Λ and p = P ,i.e., ℜλz = p is the same as ℜΛw = P (Figure 50(b)).

The details for the case 12a, with z = p, q real and w = P,Q real, are as follows:

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If ad− bd > 0, then

(a) the half-plane I is mapped onto the half-plane I; and

(b) the half-plane II is mapped onto the region II.

If ad− bc < 0, then

(c) the half-plane I is mapped onto the half-plane II.

OR

ℜ

=_

λz

p

ℜΛ _

w=

P

ℜΛ

_w

= P

ℜ

=_

λ z

p

ℜΛ

_w

= P

ℜ

=_

λz

p(a)

(b)

w - planez- plane

w - planez- planeFigure 3.50 Straight line onto straight line.

Map 3.58. Angle onto itself, with arms interchanged. Given, z0, α 6= 0(−π < α < π), and τ , where α and tau are real, the case is presented in Figure 3.51. Forinvolutory transformation (w − w0)(z − z0) = b ei(α+2τ), b > 0, the regions I and II aremapped onto the regions I and II, respectively; but for b < 0, the region I is mapped ontothe region II. Other details for this case, with z = p, q real and w = P,Q real, are as follows:

The points z0; z1 = ∞; z2 = z0 + aeiτ , where a is real, a > 0 in Figure 3.51, are mapped

onto the points w0 = ∞;w1 = z0;w2 = z0 +b

aei(α+τ).

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z1

z2

w1

2w

III ℑ ( )z z

0_

= 0

e − iτ•

•

•

w - plane

z- plane

α

Figure 3.51 Angle onto itself.

Map 3.59. Straight line onto circle. The transformation is

w − w0 = ℜeiτ

λz − p+ β

λz − p− β

, τ real; ℜβ 6= 0, (3.4.7)

given in Figure 3.52.

•

w - planez- plane

w0

R•

ℜ

=_λ z

p∞z

Figure 3.52 Straight line onto circle.

Map 3.60. Circle onto straight line. The transformation is

Λw = P + i a+ bz − z0 + r eiτ

z − z0 − r eiτ, (3.4.8)

where a, b, τ are real, b 6= 0, given in Figure 3.53.

w - planez- plane

•

ℜΛ

_w

= P•

r

z0

∞w

Figure 3.53 Circle onto straight line.

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Map 3.61. Circle and line in-contact mapped onto two parallel lines.Given z0, z2,Λ, P1, P2, the transformation is

Λw = i a+ P1 +2 (z2 − z0) (P2 − P1)

z − z0, (3.4.9)

where a is real, and r = |z2 − z0|, λ = z2 − z0, and is presented in Figure 3.54.

I

I II

II

III

III

••

•

ℜ

_λ ( )

zz0

_

= 0

z’

z0

w - planez- plane

2z

ℜ

Λ_ w

=P’r ’

I

1

ℜ

Λ_ w

=P

2

ℜ

Λ_ w

=P

Figure 3.54 Circle onto circle.

The details of the transformation with z = p, q real and w = P,Q real, are as follows:

(a) The line ℜλz = ℜλz0 is mapped onto the line ℜΛw = P1.

(b) The circle |z − z2| = r is mapped onto the line ℜΛw = P2.

(c) The circle |z − z′| = r′ is mapped onto the line ℜΛw = P ′ = P1 +r

r′(p2 − p1).

Map 3.62. Two circles in-contact mapped onto two parallel lines. (InnerContact) Given r1 > 0, r2 > 0, z1, z2, and a real, the transformation is

Λw = P1 + i a+ γz + z0 − 2z1

z0 − z, (3.4.10)

where γ = r2P2 − P1

r1 − r2. The details are provided in Figure 3.55 and where z0 =

r1z2 − r2z1r1 − r2

.


(a) The circle |z − z1| = r1 is mapped onto the line ℜΛw = P1.

(b) The circle |z − z2| = r2 is mapped onto the line ℜΛw = P2.

(c) The circle |z − z′| = r′ touching at z0 is mapped onto the line ℜΛw = P ′, where

P ′ = P1 + (P2 − P1)r2r1

r′ − r1r2 − r1

.

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(d) The line ℜλ (z − z0) = 0 touching at z0, where λ = z1 − z2, is mapped onto the

line ℜΛw = P0, where P0 =P1r1 − P2r2r1 − r2

.

I

I

II

II

III

III

IV

IV

z _ z1| | = r1

2z _ z| | = r2

z _| | =z’ r’

•• •

•ℜ

_λ

(

)

zz 0_

= 0

z’z1

z2

p0

ℜ

Λ_w

=

p2

ℜ

Λ_w

=

p’

ℜ

Λ_w

=

p1

ℜ

Λ_w

=

IVIV

z0

w - planez- plane

Figure 3.55 Two circles in contact mapped onto two parallel lines (inner contact).

Map 3.63. Two circles in outer contact mapped onto parallel lines. (OuterContact) Given |z1 − z2| = r1 + r2, transformation is

Λw = P1 + i a+ γz + z0 − 2z1

z0 − z, (3.4.11)

where a is real, and γ = r2P1 − P2

r1 + r2. This case is presented in Figure 3.56, where z0 =

r1z2 + r2z1r1 + r2

.

I

I

II

III

IV

z _ z1| | = r1

z _| | =z’ r’

•

•

•z’ z1

z2p0

ℜ

Λ_w

=p’

ℜ

Λ_w

=

p1

ℜ

Λ_w

=

IV

z0

w - planez- plane

•

••

z _| | =z’’ ’’r

2z

z _| | = rz2 2

z’’

IIIII

V

VVI

VI

p2

ℜ

Λ_w

=

p

ℜ

Λ_w

=’’

Figure 3.56 Two circles in outer contact mapped onto two parallel lines (outer contact).


(a) The circle |z − z1| = r1 is mapped onto the line ℜΛw = P1.

(b) The circle |z − z2| = r2 is mapped onto the line ℜΛw = P2.

(c) circle |z − z′| = r′ touching at z0 is mapped onto the line ℜΛw = P ′, where P ′ =

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P2 + (P1 − P2)r1r′r′ + r2r1 + r2

.

(d) The circle |z − z′′| = r′′ touching at z0, where λ = z1 − z2, is mapped onto the

line ℜΛw + P ′′, where P ′′ = P1 + (P2 − p1)r2r′′r′′ + r1r1 + r2

.

(e) The line ℜλ(z − z0) = 0 touching at z0, where λ = z1 − z2, is mapped onto the

line ℜΛw = P0, where P′0 =

P1r1 + P2r2r1 + r2

.

Map 3.64. Two ‘circles’, intersecting at two points, mapped onto two

intersecting straight lines. Given z1; r1 > 0; z2; r2 > 0 and α = β, i.e.,Λ1

Λ2

z0 − z1z0 − z)2

is real, the transformation is presented in Figure 3.57, where |r1 − r2| < |z1 − z2| < r1 + r2,and z0, w0 are the points of intersection of the z- and w-plane, respectively.

w0∞w

I

II

III

IVr1 z

2

2r•

w - planez- plane

∞z

•

•

z1

z0

III

•• I

II

III

IV

1

ℜΛ

_ w= P1

p2

Λ

_w

=2

ℜβ

Figure 3.57 Two intersecting ‘circles’.

w0

I

II

III

r1•

w - planez- plane

∞z

•

•

z1

z0•

•

I

II

III

IV

α

•

I

II

III

IV

w0

w0

∞w

β

β

p

2 Λ _

w=

2

ℜ

1

ℜΛ

_ w=

P 1

p2

Λ_w

=2

ℜ

1

ℜΛ

_ w= P1

p 1

ℜ=

_ λz _ ℜ Λ

_z1 p| |< r1 |λ |

OR

ζ 0•

Figure 3.58 Two intersecting ‘circles’, alternative case.

An alternative situation is considered in Figure 3.58, where z0 and ζ0 are the points ofintersection of the two ‘circles’ in the z-plane; w0 is the point of intersection of ℜΛ1w = P1

and ℜΛ1w = P2. The required transformation is

w = w0 +KΛ1(z − z0)(z1 − ζ0)

(z − ζ0)(z0 − ζ0), K 6= 0 real. (3.4.12)

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The details of the alternative case with z = p, q real and w = P,Q real, are as follows:

(a) The points z = 0; ∞; 2r1 is mapped onto the points w = ∞;ir2

r1 + r2; 0.

(b) The circle |z − z1| = r1 is mapped onto the line ℜΛ1w = P1.

(c) The circle |z − z2| = r2 is mapped onto the line ℜΛ2w = P2.

(d) The circle |z + r2| = r2 is mapped onto the line ℜΛ2w = P2, or line ℜλz = p.

(e) The points z = z0; ζ0; z = ∞ are mapped onto the points w = w0; ∞; w∞ = w0 +

KΛ1z1 − ζ0z0 − ζ0

.

(f) The circle passing through z0, ζ0 is mapped onto the lines passing through w0.

(g) The circle orthogonal to |z − z0| = r1, and to |z − z2| = r2 or to ℜλz = p is mappedonto the circle with center w0.

Map 3.65. The transformation presented in Figure 3.59 is

w =K

σ

z(i σ − z1)

z − i σ, σ > 0,K > 0. (3.4.13)

∞w

z’

z0

z1

y = 0

v = 0

x =

0 p 2

ℜΛ

_ w=

•

•

ℑ w

=z1

0

z ʼʼz ʼʼʼ

z2

••

γδ

γ

w0 0=

w - planez- plane

σ

Figure 3.59 Two intersecting ‘circles’.

The details of the transformation, with z = p, q real and w = P,Q real, are as follows:

(a) The points ζ0 = iσ; z0 = 0; z′; z′′; z′′′ are mapped onto the points w = ∞;w0 =0;w′;w′′;w′′′.

(b) The point z = ∞ is mapped onto the w∞ =K

σ(i σ − z1) = −K

σz1.

(c) The circle |z − z1| = r1, ℑz1 = 12σ is mapped onto the line v = 0.

(d) The circle |z − z2| = r2, ℑz2 = 12σ is mapped onto the line ℜΛ2w = P2.

(e) The line x = 0 is mapped onto the line ℑwz1 = 0.

(f) The circle (z′, z′′, z′′′, ζ0) is mapped onto the line (w′, w′′, w′′′).

(g) The curvilinear triangle z0, z′, z′′ is mapped onto the right triangle w,w′, w′′.(h) The curvilinear triangle z0, z′′, z′′′, each with sum of angles, is mapped onto the right

triangle w0, w′′, w′′′.

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Map 3.66. circles and straight line, without common point, onto twoconcentric circles. The transformation, presented in Figure 3.60, is

w − w1 = R1 eiτ (z − z0)|λ| − λ(A+ σ)

(z − z0)|λ| − λ(A− σ), (3.4.14)

where τ is real, A =p−ℜλz0

|λ| , and σ =√A2 − ρ2 > 0 or σ = −

√A2 − ρ2.

ρz0

•

p

ℜ

=

_λ z

ℑ

_ λ(

)z

z 0

_

= 0

OR

• •

R

ww

ℑ

(

)0_

= 0

e−i τ

ww

ℑ

(

)0_

= 0

e−i τ

1R2R

1w1w

2w1w − || = 2R

w - planez- plane

| | A+ σ < ρρ>| | A+ σ

Figure 3.60 Circles and straight lines.


(a) The line ℜλz = p is mapped onto the circle |w − w0| = R1.

(b) The circle |z − z0| = ρ is mapped onto the circle |w − w1| = R2.

(c) The line ℑλ(z − z0) = 0 is mapped onto the line ℑe−iτ (w − w1) = 0.

Map 3.67. Two circles, without common point, onto two concentric cir-

cles. Given z1, z2, r1 > 0, r2 > 0, z1 6= z2, and w0, R1 > 0, R2 > 0, andR2

R1=r2r1

∣∣∣t

s− t

∣∣,where s, t are the real roots of the equations st = r21 , and (d−s)(d−t) = r22 , d = |z2−z1| > 0,the transformation is

w − w0 = tR1

r1eiθ

d(z − z1)− s(z2 − z1)

d(z − z1)− t(z2 − z1), θ real. (3.4.15)

This transformation is presented Figure 3.61.

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w0R2

•

• •

• •

•

•

z1

z1

z1

z2

z2

z2

1r

1r

1r

2r

2r

2rw - plane

z- plane

IIIIII

III III I

I

I II III

I

II

III

w w0_ = 1R| |

Figure 3.61 Transformation (3.2.35).

The details of this transformation, with z = p, q real and w = P,Q real, are as follows:

(a) The circle |z − z1| = r1 is mapped onto the circle |w − w0| = R1.

(b) The circle |z − z2| = r2 is mapped onto the circle |w − w0| = R2.

(c) The radical axis of these circles is mapped onto the circle |w − w0| = R1|t|/r1.(d) The point z∞ = z1 + (t/d)(z2 − z1) is mapped onto the point w = ∞.

(e) The point z0 = z1 + (s/d)(z2 − z1) is mapped onto the point w = w0.

(f) Any ‘circle’ passing through z∞ and z0 is mapped onto the line passing through w0.

Map 3.68. The transformation, presented in Figure 3.62, is

w = R1z − iσ

z + iσ, (3.4.16)

where σ =√k2 − ρ2, 0 < ρ < k.

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z0

w - planez- plane

y = 0

v = 0

u =

0

x =

0

ℜΛ

_ w= 0

•

•iσ

ρ

•

z z0_ =| | ’ ρ

1 2

w = R1

w = R2R2



(a) The points z = 0;∞; iσ;−iσ are mapped onto the points w = −R1;R1; 0;∞, respec-tively.

(b) The line x = 0 is mapped onto the line v = 0.

(c) The line y = 0 is mapped onto the circle |w| = R1.

(d) The circle |z−ik| = ρ′, where z′0 = iσR2

1 +R2

R21 −R2

, ρ′ =2σR1R

|R21 −R2| , and R1 6= R, is mapped

onto the circle |w| = R2 =R1ρ

|k + σ| .

(e) The circle |z| = σ is mapped onto the line u = 0.

(f) The circle∣∣z − σ

ℑΛℜΛ

∣∣ = σ∣∣ Λ

ℜλ∣∣, where ℜΛ 6= 0, intersecting x = 0 at ±iσ, is

mapped onto the line ℜΛw = 0.


w =tR1

r1

z − s

z − t, (3.4.17)

where s and t are roots of the equations st = r21 and (z2 − s)(z2 − t) = r22 , is presented inFigure 3.63.

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3.5 ELLIPSES AND HYPERBOLAS 99

•

w - planez- plane

•

0

ℜΛ

_ w=x

= 0

1r= 0z1

r1

=z r1| |

• •z2

0

ℜΛ w

=

z 5z =_| | ρ

4z _| |z = r

z 5z =_| | ρ

r1 w = 03w ∞w•

2R| | =w

R| | =w 1

s t• •=

t+ (

) /s

2x


The details of this transformation, with z = p, q real and w = P,Q real, are as follows:

(a) The circles |z − z1,2| = r1,2, z1 = 0, z2 > 0 are mapped onto the circles |w| = R1,2;R2

R1=r2r1

t

r2 − t.

(b) The points z0 = s; z∞ = t; z = ∞; and z3 =s+ t

2are mapped onto the points

w0 = 0,∞;w∞ =tR1

r1; and w3 = − tR1

r1.

(c) The circle |z − z0| = r, where z4 = r21R2

1 −R2

tR21 − sR2

1

and r =r1R1R|t− s||sR2

1 − tR21|

, is mapped onto

the circle |w| = R 6= ∞.

(d) The line x =s+ t

2is mapped onto the circle |w| = R1t

r1= w∞.

(e) The circle |z − z5| = ρ, where z5 =tΛ + sΛ

2ℜΛ and ρ =∣∣∣Λ(s− t)

2ℜΛ∣∣∣, ℜΛ 6= 0 is mapped

onto the line ℜΛw = 0.

(f) The circle |z − z5| = ρ is mapped onto the line ℜΛw = 0.

(g) The circle y = 0 is mapped onto the line v = 0.

(h) The curvilinear rectangular quadrilateral formed by |z| = r1, |z − z2| = r2 between|w| = R1, |w| = R and |z − z5| ≤ ρ, is mapped onto the half-ring left of ℜΛ = 0.

(i) The region |z − z2| ≤ r2 is mapped onto the region |w| ≥ R2.

3.5 Ellipses and Hyperbolas

For ellipses and hyperbolas, we will set t, t′, and τ fixed, 0 < t < t′ < 1, and 0 < τ < π/2;also l = t + 1/t, l′ = t′ + 1/t′, and σ = argk/α. The ellipse has been defined in §3.1.3,and the hyperbolas in §3.1.4.

Map 3.70. In the case of ellipses, we find that (i) the regions |z| < |k/α|t and |z| >|k/α|t−1 are both mapped onto the region |w + 2k|+ |w − 2k| > 2l|k| which is the exterior

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of the ellipses; (ii) the annular regions t|k/α| < |z| < |k/α| and t−1|k/α| > |z| > |k/α|are both mapped onto the region 2l|k| > |w + 2k| + |w − 2k| > 4|k| which is the exteriorof the same ellipses, excluding the segment joining the foci; and (iii) the annular regionst|k/α| < |z| < t′|k/α| and t−1|k/α| > |z| > (1/t′)|k/α| are both mapped onto the region2l|k| > |w + 2k| + |w − 2k| > 2l′|k| which is the annular region bounded by two confocalellipses.

Map 3.71. The transformation z = 12

[(a− b)w+

a+ b

w

]maps the exterior of an ellipse

with semi-axes a and b: (x/a)2 + (y/b)2 ≥ 1, in the z-plane onto the unit disk |w| ≤ 1 inthe w-plane.

Map 3.72. In the case of hyperbolas, we find that (i) the half-lines argz = σ+ τ andargz = σ − τ are both mapped onto the line |w+ 2k| − |w− 2k| = 4|k| cos τ which is onebranch of the hyperbola, with asymptotes argw = argk± τ; (ii) the half-lines argz =σ+τ+π and argz = σ−τ+π are both mapped onto the line |w+2k|−|w−2k| = −4|k| cos τwhich is the other branch of the hyperbola; (iii) the half-lines argz = σ + π/2 andargz = σ − π/2 are both mapped onto the line ℜkw = 0; (iv) the half-line argz = σis mapped onto the half-line |w| > 2|k|, argw/k = 0, counted twice; and (v) the half-lineargz = σ + π is mapped onto the half-line |w| > 2|k|, argw/k = π, counted twice.These properties are presented in Figure 3.64.

A

B

C

D

EF

G

J

K

L

M A

B

C

DE

F

G

J

K L

MP

R

S

PR

S

0

•

w - planez- plane

••

J

J

JC

CN

N

•

C

C

•

•

•J

J

J

C

•

••

•

•

••••

••

•••

•

arg z π=σ τ_ +

arg z =σ τ_

arg z =σ τ+

arg z =σ τ+ + π

2k

_2k

Figure 3.64 Mapping of different regions onto different regions of a hyperbola.

References used: Ahlfors [1966], Boas [1987], Carrier, Krook and Pearson [1966], Ivanov and

Trubetskov [1995], Kantorovich and Krylov [1958], Kober [1957], Kythe [1996], Nehari [1952], Nevanlinna

and Paatero [1969], Pennisi et al. [1963], Papamichael, Warby and Hough [1986], Saff and Snider [1976],

Schinzinger and Laura [2003], Silverman [1967], Wen [1992].

Documents

Handbook of Conformal Mappings and Applications