730
THEOREM 1
PROOF
CHAP.17 Conformal Mapping
Fig. 376. Images of x = const, y = const under w = z2
A mapping w = f(z) is called conformal if it preserves angles between oriented curves
in magnitude as well as in sense. Figure 377 shows what this means.

SEC. 16.3 Residue Integration Method 715
We see that b1 is now the coefﬁcient of the power (2: - 20W"1 of the power series .of
3(2) = (z - zo)mf(z). Hence Taylor’s theorem (Sec. 15.4) gives (5):
l
— (m - l)! g
l (I‘m—1
= — — [(z - Zo)mf(Z)]- l
(m — 1)! dz

664
Complex power series, in particular, Taylor series, are analogs of real power and Taylor
series in calculus. However, they are much more fundamental in complex analysis than
their real counterparts in calculus. The reason is that power series represen

SEC. 16.4 Residue Integration of Real Integrals 721
l l I 1
Res ,2 = = _ = _ [31-2/4 - _ em]!
"2‘ ﬂ ) l: (I + Z4), :|z=21 l: 433 ] z=zx 4 4
I l 1 1
Res . = = = _ —9ml4 = _ —m14
2:22 “2) l: (I + 14), ] Z=Zz [ 4'3 ] Z=22 4 e 4 8
(Here we used e”; =

SEC. 16.1
Laurent Series
PROOF
703
where all the coefﬁcients are now given by a single integral formula, namely,
1 f(Z*)
, — dz*
2m c (z* - 2:0)”+1
(2’) (n = 0, :1, :2, - - -).
an=
We prove L‘aurent’s theorem. (a) The nonnegative powers are those of a Tay

SEC. 15.5 Uniform Convergence.
Optional 69]
Uniform Convergence.
DEFINITION
We know that power series are absolutely convergent (Sec. 15.2, Theorem 1) and, as
another basic property, we now show that they are uniformly convergent. Since uniform
convergenc

SEC. 15.4 Taylor and Maclaurin Series 685
For later use we note that since z* is on C while 2 is inside C, we have
Z_Zo
(7*) < 1 (Fig. 364).
z*-zo
Fig. 364. Cauchy formula (6)
To (7) we now apply the sum formula for a ﬁnite geometric sum
n+1 1 n+1

670
THEOREM 8
PROOF
EXAMPLE 4
CHAP. 15 Power Series, Taylor Series
Ratio Test
[fa series zl + z2 + ' ' - with zn at 0 (n = l, 2, - - -) is such that lim
00
then: "H 2”
Zn+1
(a) [f L < l, the series converges absolutely.
(b) If L > I, the series diverges