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2.3 Cauchy's Theorem and Cauchy's Formula 117
In Exercises 9 to 12, evaluate the given integral using the technique of Example 10;
indicate which theorem or device you used to obtain your answer.
dz . . . . . . .
9. —2 , where y is any curve in Re 2 > 0 j
2.4 Consequences of Cauchy's Formula 123
Jf(z) dz = 0.
(b) There is an analytic function F in D with F’ = f throughout D. l
2.4 Consequences of Cauchy's Formula
Cauchy’s Formula has farreaching implications, allowing us to draw numerous
nonobvious conclu
2.3.1 The Cauchy—Goursat Theorem 121
1
(iii)’ diameter (1}) = 5 diameter (F)
(iv)’ 1 s 411,.
Since the diameters of the triangles {Aj} decrease to zero and since (i) holds,
there is a unique point 20 within all the Aj, and 20 lies in D. Thus, f is diﬂ‘ere
2.4 Consequences of Cauchy's Formula 125
To complete the proof of the theorem, we must show that its conclusion is
valid when we omit the assumption that f has a continuous derivative. However,
Theorem 2b of Section 2.3.1 implies that there is an analytic
2.4 Consequences of Cauchy's Formula 127
1 m1
z—3=1+;§(n +2)!(—1)"
(2 1)"
n!
co
=1+§ (—1)"(n + 2)(n +1)(z —1)".
=1
The series is valid for z — 1 < 1. 1:]
Theorem 1 has another signiﬁcant implication.
Suppose that f is analytic on a domain D and, furt
3.4 Conformal Mapping 209
Figure 3.11
See Figure 3.11. In particular, we obtain the two relations
IW'(to)l = f'(zo) Iz’(to) (1)
and
arg W'(to) = arg(f'(Zo) + arg(Z'(to) (2)
We make the natural assumption that f ’(zo) g6 0 and then note that the tran
10.
3.3 Linear Fractional Transformations 205
(a) T ﬁxes the points 0 and 1 and T(i) = 00.
(b) T maps the real axis onto itself and the imaginary axis onto the circle
rw—a=a
(c) T maps the real axis onto itself and the imaginary axis onto the circle
rw—a=
3.4 Conformal Mapping 215
46%
I k2 = 0.58
I. 1
71535?wa
“3.5%

— 1
Figure 3.16 The circles Arg = k1 and
z+1
Example 9 f (z) = log(z + r /z2 — R2), on the plane minus the segment [—R, R].
The function g(z) = z + t /z2 — R2 = w has
3.4 Conformal Mapping 213
y. = {2: u(z) = u(zoi} = {2: Re f(z) = u(zoh
= {g(w): Re w = u(zo), w 6 0},
so yl is precisely the range of the function g(w) on the set {w e (2: Re w = u(zo)} and
hence is a smooth arc. Likewise, the set yz, consisting of those
3.4 Conformal Mapping 211
Example 2 The function h(z) = sin 2 is conformal at all points except 2 = 12/2 + rm,
n = 0, i 1, ., which are the zeros of h’(z) = cos 2. E1
Example 3 The function 9(2) = 1/2 is conformal at all points 2 except 2 = 0, where
it is
3.3 Linear FractionalTransformations 207
19. (a) If 2 is a point in L or C, show that z“ = z.
(b) If L is the real axis, show that 2* = E, the complex conjugate of z.
(c) Show that (z*)* = z for any L, C, and z.
20. If C is the circle IC — {0 = r, show t
3.4 Conformal Mapping 217
r(1 + cos 0) = constant
r(1 — cos 0) = constant,
respectively. A few of these curves are sketched in Figure 3.19. E]
Figure 3.19 The curves (1 + cos 0) = c, and (1 — cos 0) = c2. 218
Chapter 3 Analytic Functions as Mappings
You
3.4.1
3.4.1 Conformal Mapping and Flows 219
(d) Let F1 = f(y1) and F2 = f (yz). The angle from I] to F2 is
“In [arg(f(21) ‘ W0) _ arg(f(zz) ‘ Won = ma
Zx‘Zo
zz—vzo
12. Suppose that f is analytic on a convex domain D and that Re( f ’(z) > 0 for all
z e D.
3.4.1 Conformal Mapping and Flows 223
2. Show that the function H (z) = z“, 0 < a < 2, maps the upper halfplane U onto
the plane minus a “wedge” of angle 7r(2 — a) (Fig. 3.23). Use H to streamline this
.(2  a)
3. Show that the function K (2) = z/(l — z)
3.4.1 Conformal Mapping and Flows 221
L W$
(b)
Figure 3.21
ﬂow problem can be handled by consideration of appropriate conformal mappings,
as we shall now show.
Our objective is to ﬁnd the path described by a particle as it moves through
the domain D when
2.3.1 The Cauchy—Goursat Theorem 119
e. z 1 — e. ’z' e — z
1 21: . eh; I . .
(c) — (e") t dt = 0,1ff(w)is analytic for w < 1 + a
21: o 1— e‘
1 21: . eir
_ n _ d =
(d) 2” f(e )e. _ Z r f(z)
(e) Add (c) and (d); then use (a) and (b) to conclude
2.3 Cauchy's Theorem and Cauchy's Formula 115
as R + 00. Setting the real and imaginary parts of the resulting expression separately
equal to zero, we ﬁnd that
e”2 I e"‘2 cos(2[3x) dx = 4 (2)
0
and
co 2 ﬂ
eBZJ‘ e“ sin(2f3x) dx =J e'2 dt. El (3)
0 0
The n
2.3 Cauchy's Theorem and Cauchy's Formula 109
closed curves to explore their other possible conﬁgurations. The crux of the matter
is that any closed loop in F surrounds only points of D, since D is simplyconnected.
Given that this is the case, repeated a
2.3 Cauchy's Theorem and Cauchy's Formula 113
Solution Again we use the substitution 2 = e”. This yields
1
1—2acos6+a2=1+a2—a<z+;),
SO
d0 dz dz
1—2acos0+a2 _iz(1+az_a<z+1)_i(—azz+(1+a2)z—a)'
Z
Now
—a22 + (1 + a2)z —a= —a(z —$>(z — a).
The point 1/12 is ou
2.3 Cauchy's Theorem and Cauchy's Formula 111
Cauchy's Formula
THEOREM 4 Cauchy's Formula Suppose that f is analytic on a domain D and that y is a
piecewise smooth, positively oriented simple closed curve in D whose inside Q also
lies in D. Then
f(z) =