CH-AP. 9 Vector Differential Calculus. Grad, Div, Curl
Flow of a Compressible Fluid. Physical Meaning of the Divergence
We consider the motion of a fluid in a region R having no sources or sinks in R, that is, no points at which
ﬂuid is prod
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):
— (m - l)! g
= — — [(z - Zo)mf(Z)]- l
(m — 1)! dz
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”; =
where all the coefﬁcients are now given by a single integral formula, namely,
, — dz*
2m c (z* - 2:0)”+1
(2’) (n = 0, :1, :2, - - -).
We prove L‘aurent’s theorem. (a) The nonnegative powers are those of a Tay
SEC. 15.5 Uniform Convergence.
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
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
(7*) < 1 (Fig. 364).
Fig. 364. Cauchy formula (6)
To (7) we now apply the sum formula for a ﬁnite geometric sum
n+1 1 n+1