Trasformation u =
The transformation is conformal for all z except z = 0, since
dw
dz
and is defined for all z except z = 0.
Chapter 3 Conformal Mapping and Transformations 63
1
w= u+2u =
Now, T+iy
2+y2
-y
i.e T?+y2 2+y2
u+2 2+y2
Also, (ar2 +y2)2
1
2+y
1
2+ 24?
-U
T 22 u2+v2
form
circles with centre at origin in the z-plane are of the
) The the
+y2= k. These circle transform in to u+ v= in
w-plane.
i.e.,Unit circles with the origin in the z-plane transforn
centre at
in to unit circle in the w-plane with centre at the origin.
Also, circles lying outside unit circle in the z-plane transformn
into circle with in the unit circle in the w-plane.
W-plane
Z plane
Fig. 3.6
64 Complex Analysis
+2/y+c = o
+y*+29T
form r
z-plane is ofthe
cirele in the
(i) Any in to
transforms
This +E) +"()+c=o
0
2fv+1
=
(u+v) +2gu
ie transtorm
in to circle in
in z-plane
then circle
Case (i) If c #0, through the
the u-plane. the z-plane passes
then circle
in +1=0
Ifc = 0, line 2gu -2fv
Case (ii) transforms in to a straight
the
origin get
in the w-plane. =0,
transtorms in
of the form
lz +my +n
in the z-plane
(i1) A line
to mu
U =0
u2+2
u?472 +
n(u+)+ lu
-
mv =0
i.e.
the origin.
i.e., a circle passing through
n(u2+v2)+lu-my 0
2gx+2fh-0
2gu-2fv+1=0
Fig. 3.7
Hence the lines in the z-plane transform in to circles passing
through the orgin in the w-plane. If n = 0, the straight line in
the z-plane passes through the origin, transform in to the
straig
line through the origin lu mu =0 in the
w-plane.
-
64 Complex Analysis
r + y + 2 9 z + 2 f y + c =
form
in the z-plane is of the
(i) Any circle
transforms in to
This
1 E)+2/(E)te-0
u+v)+2gu 2fv +1 =
0
i.e.,
transtorm in to circle in
then circle in z-plane
Case (i) Ifc # 0,
the w-plane.
then circle in the z-plane passes through thbe
Case Ifc
(ii) 0, =
line 2gu- 2fv +1 =0
transforms in to a straight
origin get the
in the w-plane.
0, transforms in
z-plane of the form lz +my+n
=
(ii) A line in the
to
mu
n = 0
u?+
n(u+)+lu- mv = 0
i.e.
ie., a circle passing through the origin.
P2gr-2 -Fu-=0
2g-210o
Fig. 3.7
Hence the lines in the
z-plane transform in to circles passins
through the orgin in the w-plane. If n 0, the straight line
=
the z-plane passes
through the origin. transform in to the straig
line through the origin lu
mv= 0 in the
-
w-plane.
Chapter 3 Conformal Mapping and Transformations 65
(iv) The line z= k transforms in to the circle
+- =0
The v-axis is a tangent to this circle at the origin.
The y-axis [i.e., r = 0] transforms in to the v-axis.
The line y = k1, transform in to the circle u2 + + 0
The u-axis touches this circle at the origin.
The T-axis, [i.e., y = 0 transforms in to the u-axis.
The half plane z > has its image in the region.
u2v2C (1)
i.e u2+<when c>0
2
i.e., u2- 4?+
C1 2c1 201
i.e., (2)
circle having it's center at(
Hence the point w is inside a
1
and radius
2c1
circle at the origin.
The v-axis is the tangent to this
W-plane
Z- plane
Fig. 3.8
66 Complex Analysis
Conversely, whenever u & v satisty the inequality & ci > 0, then
the inequality (1) follows and therefore a > c.
Consequently, every point nside the circle is the image of some
point in the half plane.
Also, the image of the half plane is the interior of the
Tegion. circular
The above transformation yields
the following result:
z-plane w-plane
) Circle not
through the origin ) Circle not through the origin
(ii) Circle through the origin (ii) Straight line not through the
origin
(i) Straight line not through (iii) Circle through the
the origin origin
(iv) Straight line through the (iv) Straight line through the
origin
origin
.Note
If 2 =
Te' & w =
Re'
Re' 1 e
1 i0
Tetd T
R= d= -0
i.e., Rr 1
Hence this transformation is
circle and reflection the real axis. equivalent to an inversion in unit
Transformation w = z2
This transformation w
z is =
mal everywhere except when analytic & so representation is
f(2) ==0 when z =0. contor
Now, u+iv =
W =
(z+ iy)
= + i y +i2xy
= ( - v ) + i(2zy)
u =
-y: v= 2æy
Chapter 3 Conformal Mapping and Transformations 67
Let a = k
(i)
u =k - y &v = 2ky
which is a
Eliminating y between u & v, we get v2 4k2(k2-u), =
k for different
parabola i.e., The system of the parallel lines c
=
the
values of k, transform in to a system of parabolas having
u-axis as the axis ot the parabola.
(ii) Let y =A
u =r - A*; v= 2Ar
Eliminating r between u & v, we have v 42(u + A*), which =
is a parabola.
Hence the system of lines y = A maps in to the family of parabolas
v2 4(u+ A*) having the u-axis as the axis of the parabolas.
yA2
X#k
yA
Fig. 3.9
above parabolas are orthogonal since
The two families of the
A are orthogonal of the z-plane
Tamilies of lines z =k and y
are confocal
and
two families of parabolas
We also note that the
the common focus is the origin.
0 is 0 (with u < 0)
The image of c
=
= v
i.e., The left half real axis. > 0). i.e., The right
half
0 is v =0 (with
u
The image of y =
real axis.
68 Complex Analysis
R and v =
A on the w-plane map in to
The straight lines
u
(iii) of rectanglar hyperbolas.
families
orthogonal
-y=k, 2xy =A
2
2xy, k
u=k u-k
V M
W-planee
Z plane
Fig. 3.10
These two families of rectangular hyperbolas have the same centre
(0,0) and the axes of the family of rectangular hyperbolas are the
asymptotes of the other family of rectangular hyperbolas,
(iv) Let z = re"
w = (re") = r*eai0 = Aeid
Then
= ro = 20
If 2 describes an are of a circle of radius r subtending an angle #
at (0,0), then w describes an arc or cirCie of radius r subtendin
an angle 20 at it's origin.
i.e., the angles at the origin are doubled under the man
ping.
. Upper half of the z-plane maps in to whole ot w-plane and frst
quadrant of z-plane maps into upper half of the w-plane
Evidently, +ve z-axis maps into the +ve u-axis
is and
and +ve
maps in to the -ve u-axis. y-axis
Conformal and Transformatious 69
Chapter 3 Mapping
v) Consicder
the circle with center at a and radius c, (a, c being renl)
in the z-plane,
.C. 2-ac> z-a = ce
(a t ce")?= a2 4 c2210 +2ace
2
-a + c4ce2 + 2ace" (Add c on both sides)
c(1+20) 4+2ace
=c(e"e") + 2ace"
= ce"(e" +e") + (2ac)e"
e"te)
=
2ce (a+ ccos0) (.2cos=
in the w-plane is w
=
c= Re'0] pole Then
Let w -
a + when A is chosen as
the
polar equation of the image
a - c, in the
pole & R, Q as polar co-ordinates.
Re= 2ce"|a + ccos 0
R 2cla + ccos0
limacon.
which is equation of
an
in particular, if a =C,
R = 2ala +acos6
= 2a(1+ cos )
of a cardioid.
which is an equation
R
a2-2
(a,0)
W-plane
Z-plane
Fig. 3.11
