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SEC.
95
is conformal at the point z = 1
+ i, where the half lines
y = x (x > 0) and x = 1 (x > 0)
intersect. We denote those half lines by C1 and C2, with positive sense upward, and
observe that the angle from C1 to C2 is n / 4 at their point of intersect
CHAPTER
CONFORMAL MAPPING
In this chapter, we introduce and develop the concept of a conformal mapping, with emphasis on connections between such mappings and harmonic functions. Applications
to physical problems will follow in the next chapter.
94. PRESE
Note that log z, defined on the sheet R1,represents the analytic continuation (Sec.
26) of the singlevalued analytic function
f (z) =In r
+ if3
(0 < 8 < 2n)
upward across the positive real axis. In this sense, log z is not only a singlevalued
function o
where
and where
= n.
Observe that G is analytic in the entire z plane except for the ray rl 3 0,
= 0; for then
Now F ( z ) = G ( z ) when the point z lies above or on the ray rl > 0,
ek = Ok(k= 1,2),When z lies below that ray, Ok = Ok 2n (k = 1 , 2 ) .Con
D'
D
C
B x
C'
A' X
B'
C"
FIGURE 119
w = Fo (sin 2 .
)
B"
of the logarithmic function is used, equation (5) yields the branch
of z 'I2, which corresponds to k = 1in equation (4). Since exp(ix) = 1, it follows that
Fl(z) Fo(z). values &Fo(z)thus represent
I
EXAMPLE 3. We need only recall the identity (Sec. 33)
I
to see that the transformation w = cos z can be written successively as
Hence the cosine transformation is the same as the sine transformation preceded by a
translation to the right through n / 2 u
EXERCISES
1. Find the linear fractional transformation that maps the points
g
onto the points wl = 1, w2 = i, w = 1.
Ans. w = (32 + 2i)/(iz
zl
= 2, z2 = i , z3 = 2
+ 6).
2. Find the linear fractional transformation that maps the points z l = i, z2 = 0,
and then complete the verification of the mapping illustrated in Fig. 13, Appendix 2, by
showing that segments of the x axis are mapped as indicated there.
2. Verify the mapping shown in Fig. 12, Appendix 2, where
Suggestion: Write the given transformatio
The first of these transformations is an inversion with respect to the unit circle
lz 1 = 1. That is, the image of a nonzero point z is the point Z with the properties
IZI=
1
IzI
and
argZ=argz.
Thus the points exterior to the circle lz I = 1are mapped on
86. LINEAR FRACTIONAL TRANSFORMATIONS
The transfomation
where a , b, c, and d are complex constants, is called a linearfractional transformation,
or Mobius transformation. Observe that equation (1) can be written in the form
and, conversely, any equation
CHAP. 7
EXAMPLE 1. Let us find the function f (t) that corresponds to
F(s) =
(5)
S
(s2
(a > 0).
+ a2)2
The singularities of F (s) are the conjugate points
sg=ai
and
so= ai.
Upon writing
F(s)=
# (s)
(s  ~
where @(s)=
i ) ~
(s
S
+ ai)2
I
we see that #(s)
282
APPLICATIONS
OF RESIDUES
CHAP. 7
The winding number can be determined from the number of zeros and poles of
f interior to C . The number of poles is necessarily finite, according to Exercise 1 1 ,
~ e c69. Likewise, with the understanding that f (z) i
00
2
4
cos2nt,
Ans. f (t) =  n ~ n=l n 2  1 .
C4
9. F (s) =
sinh(xs 'I2)
s2 sinh(s lI2)
(0 c x < 1).
x
00
1
2
Ans. f ( t ) = x(x2 1) + x t +
6
n3
.
(I)"+'
n3
en2n2r
sin nnx.
00
2
(lyfl
Ans. f (t) = sin nnt.
n n=l
n
11. F(s) =
sinh(xs)
s (s2 w2) co
10. Write f ( 2 ) = zn and g(z) = a.
prove that any polynomial
+ a l z + + anIznl and use Rouch6's theorem to
where n 2 1, has precisely n zeros, counting multiplicities. Thus give an alternative proof
of the fundamental theorem of algebra (Theorem 2, S
272
APPLICATIONS
OF RESIDUES
CHAP.
7
Also, since
f ( z ) = (Z (P (z)
 2i)2
where # ( z ) =
log z
(z
+ 2i)2'
the singularity z = 2i of f (z)is a pole of order 2, with residue
Equation (2) thus becomes

kp
f (z) dz 
lR
f
(z)
dz;
and, by equating the real
SEC.
EXERCISES 277
77
6. Show that
by integrating an appropriate branch of the multiplevalued function
over (a) the indented path in Fig. 97, Sec. 75; (b)the closed contour in Fig. 99, Sec. 77.
7. The betafunction is this function of two real variables:
352
CONFORMAL
MAPPING
CHAP.
9
is analytic in a domain D, then the realvalued functions u and v are harmonic in that
domain. That is, they have continuous partial derivatives of the first and second order
in D and satisfy Laplace's equation there:
We had