james-ward-brown-complex-variables-and-applications-2009

Brown-chap12-v3 11/06/07 9:52am 439

sec. 125 Exercises 439

and lim H (R, ψ) = 1 � 2π F (φ) dφ.
(7)
R→∞ 2π 0
Y

r0 X

FIGURE 181

EXERCISES

1. Obtain the special case

1 �π
(a) H (R, ψ) = [P (r0, R, φ + ψ) − P (r0, R, φ − ψ)]F (φ) dφ ;
2π
0

1 �π
(b) H (R, ψ) = − [P (r0, R, φ + ψ) + P (r0, R, φ − ψ)]F (φ) dφ
2π
0

of formula (4), Sec. 125, for the harmonic function H (R, ψ) in the unbounded region

R > r0, 0 < ψ < π , shown in Fig. 182, if that function satisfies the boundary condi-
tion

lim H (R, ψ) = F (ψ) (0 < ψ < π )

R→r0
R>r0

on the semicircle and (a) it is zero on the rays BA and DE; (b) its normal derivative
is zero on the rays BA and DE.

Y

C

AB r0
D E X FIGURE 182

2. Give the details needed in establishing formula (1) in Sec. 125 as a solution of the
Dirichlet problem stated there for the region shown in Fig. 180.

3. Give the details needed in establishing formula (3) in Sec. 125 as a solution of the
boundary value problem stated there.

4. Obtain formula (4), Sec. 125, as a solution of the Dirichlet problem for the region
exterior to a circle (Fig. 181). To show that u(r, − θ) is harmonic when u(r, θ) is
harmonic, use the polar form

r2urr (r, θ ) + rur (r, θ ) + uθθ (r, θ ) = 0

of Laplace’s equation.

Brown-chap12-v3 11/06/07 9:52am 440

440 Integral Formulas of the Poisson Type chap. 12

5. State why equation (6), Sec. 125, is valid.
6. Establish limit (7), Sec. 125.

126. SCHWARZ INTEGRAL FORMULA

Let f be an analytic function of z throughout the half plane Im z ≥ 0 such that for
some positive constants a and M, the order property

(1) |zaf (z)| < M (Im z ≥ 0)

is satisfied. For a fixed point z above the real axis, let CR denote the upper half
of a positively oriented circle of radius R centered at the origin, where R > |z|
(Fig. 183). Then, according to the Cauchy integral formula (Sec. 50),

(2) f (z) = 1 � f (s) ds 1 � R f (t) dt
+.
2π i CR s − z 2π i −R t − z

y

z s x FIGURE 183
–R CR

tR

We find that the first of these integrals approaches 0 as R tends to ∞ since, in
view of condition (1),

�� f (s) ds � < M πR = πM
� � Ra(R − |z|) Ra(1 − |z|/R) .
�
� CR s−z �
�

Thus

(3) f (z) = 1 � ∞ f (t) dt (Im z > 0).

2π i −∞ t − z

Condition (1) also ensures that the improper integral here converges.∗ The number
to which it converges is the same as its Cauchy principal value (see Sec. 78), and
representation (3) is a Cauchy integral formula for the half plane Im z > 0.

∗See, for instance, A. E. Taylor and W. R. Mann, “Advanced Calculus,” 3d ed., Chap. 22, 1983.

Brown-chap12-v3 11/06/07 9:52am 441

sec. 127 Dirichlet Problem for a Half Plane 441

When the point z lies below the real axis, the right-hand side of equation (2)
is zero; hence integral (3) is zero for such a point. Thus, when z is above the real
axis, we have the following formula, where c is an arbitrary complex constant:

1 � ∞� 1 c�
(4) f (z) = + f (t) dt (Im z > 0).
2π i −∞ t − z t − z

In the two cases c = −1 and c = 1, this reduces, respectively, to

(5) 1 � ∞ yf (t) (y > 0)
f (z) = −∞ |t − z|2 dt
π

and

(6) 1 � ∞ (t − x)f (t) (y > 0).
f (z) = dt
π i −∞ |t − z|2

If f (z) = u(x, y) + iv(x, y), it follows from formulas (5) and (6) that the
harmonic functions u and v are represented in the half plane y > 0 in terms of the
boundary values of u by the formulas

(7) 1 � ∞ yu(t, 0) 1 � ∞ yu(t, 0) (y > 0)
u(x, y) = −∞ |t − z|2 dt = π −∞ (t − x)2 + y2 dt
π

and

(8) 1 � ∞ (x − t)u(t, 0) (y > 0).
v(x, y) = −∞ (t − x)2 + y2 dt
π

Formula (7) is known as the Schwarz integral formula, or the Poisson integral
formula for the half plane. In the next section, we shall relax the conditions for the
validity of formulas (7) and (8).

127. DIRICHLET PROBLEM FOR A HALF PLANE

Let F denote a real-valued function of x that is bounded for all x and continuous
except for at most a finite number of finite jumps. When y ≥ ε and |x| ≤ 1/ε, where
ε is any positive constant, the integral

� ∞ F (t) dt
I (x, y) = −∞ (t − x)2 + y2

converges uniformly with respect to x and y, as do the integrals of the partial
derivatives of the integrand with respect to x and y. Each of these integrals is
the sum of a finite number of improper or definite integrals over intervals where
F is continuous; hence the integrand of each component integral is a continuous
function of t, x, and y when y ≥ ε. Consequently, each partial derivative of I (x, y)

Brown-chap12-v3 11/06/07 9:52am 442

442 Integral Formulas of the Poisson Type chap. 12

is represented by the integral of the corresponding derivative of the integrand when-
ever y > 0.

If we write

y
U (x, y) = I (x, y),

π

then U is the Schwarz integral transform of F , suggested by expression (7),
Sec. 126:

(1) 1 � ∞ yF (t) (y > 0).
U (x, y) = −∞ (t − x)2 + y2 dt
π

Except for the factor 1/π, the kernel here is y/|t − z|2. It is the imaginary component
of the function 1/(t − z), which is analytic in z when y > 0. It follows that the kernel
is harmonic, and so it satisfies Laplace’s equation in x and y. Because the order
of differentiation and integration can be interchanged, the function (1) then satisfies
that equation. Consequently, U is harmonic when y > 0.

To prove that

(2) lim U (x, y) = F (x)
y→0
y>0

for each fixed x at which F is continuous, we substitute t = x + y tan τ in inte-
gral (1) and write

(3) U (x, y) = 1 � π/2 F (x + y tan τ ) dτ (y > 0).

π −π/2

As a consequence, if

G(x, y, τ ) = F (x + y tan τ ) − F (x)

and α is some small positive constant,

� π/2

(4) π[U (x, y) − F (x)] = G(x, y, τ ) dτ = I1(y) + I2(y) + I3(y)

−π/2

where

� (−π/2)+α � (π/2)−α

I1(y) = G(x, y, τ ) dτ, I2(y) = G(x, y, τ ) dτ,

−π/2 (−π/2)+α

� π/2

I3(y) = G(x, y, τ ) dτ.

(π/2)−α

If M denotes an upper bound for |F (x)|, then |G(x, y, τ )| ≤ 2M. For a given
positive number ε, we select α so that 6Mα < ε; and this means that

Brown-chap12-v3 11/06/07 9:52am 443

sec. 127 Dirichlet Problem for a Half Plane 443

εε
|I1(y)| ≤ 2Mα < 3 and |I3(y)| ≤ 2M α < .
3

We next show that corresponding to ε, there is a positive number δ such that

ε whenever 0 < y < δ.
|I2(y)| < 3

To do this, we observe that since F is continuous at x, there is a positive number
γ such that

|G(x, y, τ )| < ε whenever 0 < y| tan τ | < γ .

3π

Now the maximum value of | tan τ | as τ ranges from

ππ
− + α to − α

22

is

�π �
tan − α = cot α.

2

Hence, if we write δ = γ tan α, it follows that 0 < y < δ.
εε

|I2(y)| < 3π (π − 2α) < 3 whenever
We have thus shown that

|I1(y)| + |I2(y)| + |I3(y)| < ε whenever 0 < y < δ.

Condition (2) now follows from this result and equation (4).
Formula (1) therefore solves the Dirichlet problem for the half plane y > 0,

with the boundary condition (2). It is evident from the form (3) of expression (1)
that |U (x, y)| ≤ M in the half plane, where M is an upper bound of |F (x)|; that is,
U is bounded. We note that U (x, y) = F0 when F (x) = F0 , where F0 is a constant.

According to formula (8) of Sec. 126, under certain conditions on F the func-
tion

(5) 1 � ∞ (x − t)F (t) (y > 0)
V (x, y) = −∞ (t − x)2 + y2 dt
π

is a harmonic conjugate of the function U given by formula (1). Actually, for-
mula (5) furnishes a harmonic conjugate of U if F is everywhere continuous, except
for at most a finite number of finite jumps, and if F satisfies an order property

|xaF (x)| < M (a > 0).

For, under those conditions, we find that U and V satisfy the Cauchy–Riemann
equations when y > 0.

Special cases of formula (1) when F is an odd or an even function are left to
the exercises.

Brown-chap12-v3 11/06/07 9:52am 444

444 Integral Formulas of the Poisson Type chap. 12

EXERCISES

1. Obtain as a special case of formula (1), Sec. 127, the expression

y � ∞� 1 1� (x > 0, y > 0)
U (x, y) = (t − x)2 + y2 − (t + x)2 + y2 F (t) dt
π0

for a bounded function U that is harmonic in the first quadrant and satisfies the
boundary conditions

U (0, y) = 0 (y > 0),

lim U (x, y) = F (x) (x > 0, x = xj ),

y→0
y>0

where F is bounded for all positive x and continuous except for at most a finite
number of finite jumps at the points xj (j = 1, 2, . . . , n).

2. Let T (x, y) denote the bounded steady temperatures in a plate x > 0, y > 0, with
insulated faces, when

lim T (x, y) = F1(x) (x > 0),

y→0
y>0

lim T (x, y) = F2(y) (y > 0)

x→0
x>0

(Fig. 184). Here F1 and F2 are bounded and continuous except for at most a finite
number of finite jumps. Write x + iy = z and show with the aid of the expression

obtained in Exercise 1 that

T (x, y) = T1(x, y) + T2(x, y) (x > 0, y > 0)

where y � ∞� 1 1�

T1(x, y) = π 0 |t − z|2 − |t + z|2 F1(t) dt,

y � ∞� 1 1�
T2(x, y) = π 0 |it − z|2 − |it + z|2 F2(t) dt.

y

T = F2( y)

T = F1(x) x FIGURE 184

3. Obtain as a special case of formula (1), Sec. 127, the expression

y � ∞� 1 1� (x > 0, y > 0)
U (x, y) = (t − x)2 + y2 + (t + x)2 + y2 F (t) dt
π0

for a bounded function U that is harmonic in the first quadrant and satisfies the
boundary conditions

Brown-chap12-v3 11/06/07 9:52am 445

sec. 128 Neumann Problems 445

Ux(0, y) = 0 (y > 0),

lim U (x, y) = F (x) (x > 0, x = xj ),

y→0
y>0

where F is bounded for all positive x and continuous except possibly for finite jumps
at a finite number of points x = xj (j = 1, 2, . . . , n).

4. Interchange the x and y axes in Sec. 127 to write the solution

1 � ∞ xF (t) (x > 0)
U (x, y) = −∞ (t − y)2 + x2 dt
π

of the Dirichlet problem for the half plane x > 0. Then write

F (y) = �1 when |y| < 1,
0 when |y| > 1,

and obtain these expressions for U and its harmonic conjugate −V :

1� y+1 y −1� 1 x2 + (y + 1)2
U (x, y) = arctan − arctan , V (x, y) = 2π ln x2 + (y − 1)2
πx x

where −π/2 ≤ arctan t ≤ π/2. Also, show that

1
V (x, y) + iU (x, y) = [Log(z + i) − Log(z − i)],

π
where z = x + iy.

128. NEUMANN PROBLEMS

As in Sec. 123 and Fig. 177, we write

s = r0 exp(iφ) and z = r exp(iθ ) (r < r0).

When s is fixed, the function
(1) Q(r0, r, φ − θ ) = −2r0 ln|s − z| = −r0 ln[r02 − 2r0r cos(φ − θ ) + r2]
is harmonic interior to the circle |z| = r0 because it is the real component of

−2 r0 log(z − s),

where the branch cut of log(z − s) is an outward ray from the point s. If, moreover,
r = 0,

− r0 � 2r2 − 2r0r cos(φ − θ ) �
r r02 − 2r0r cos(φ − θ ) + r2
(2) Qr (r0, r, φ − θ) =

= r0 [P (r0, r, φ − θ) − 1]
r

where P is the Poisson kernel (7) of Sec. 123.

Brown-chap12-v3 11/06/07 9:52am 446

446 Integral Formulas of the Poisson Type chap. 12

These observations suggest that the function Q may be used to write an integral

representation for a harmonic function U whose normal derivative Ur on the circle
r = r0 assumes prescribed values G(θ ).

If G is piecewise continuous and U0 is an arbitrary constant, the function

(3) 1 � 2π Q(r0, r, φ − θ ) G(φ) dφ + U0 (r < r0)
U (r, θ ) =
2π 0

is harmonic because the integrand is a harmonic function of r and θ . If the mean
value of G over the circle |z| = r0 is zero, so that

� 2π
(4) G(φ) dφ = 0,

0

then, in view of equation (2),

Ur (r, θ ) = 1 � 2π r0 [P (r0 , r, φ − θ) − 1] G(φ) dφ
2π r
0

= r0 · 1 � 2π P (r0, r, φ − θ ) G(φ) dφ.
r 2π
0

Now, according to equations (1) and (2) in Sec. 124,

lim 1 � 2π P (r0, r, φ − θ ) G(φ) dφ = G(θ ).

r →r0 2π 0
r <r0

Hence

(5) lim Ur (r, θ) = G(θ )

r →r0
r <r0

for each value of θ at which G is continuous.
When G is piecewise continuous and satisfies condition (4), the formula

(6) U (r, θ ) = − r0 � 2π (r < r0),
2π ln[r02 − 2r0r cos(φ − θ ) + r2] G(φ) dφ + U0

0

therefore, solves the Neumann problem for the region interior to the circle r = r0 ,
where G(θ ) is the normal derivative of the harmonic function U (r, θ ) at the bound-

ary in the sense of condition (5). Note how it follows from equations (4) and (6) that
since ln r02 is constant, U0 is the value of U at the center r = 0 of the circle r = r0.

The values U (r, θ ) may represent steady temperatures in a disk r < r0 with
insulated faces. In that case, condition (5) states that the flux of heat into the

disk through its edge is proportional to G(θ ). Condition (4) is the natural phys-

ical requirement that the total rate of flow of heat into the disk be zero, since

temperatures do not vary with time.

A corresponding formula for a harmonic function H in the region exterior to

the circle r = r0 can be written in terms of Q as

Brown-chap12-v3 11/06/07 9:52am 447

sec. 128 Neumann Problems 447

(7) 1 � 2π
H (R, ψ) = − Q(r0, R, φ − ψ) G(φ) dφ + H0 (R > r0),
2π 0

where H0 is a constant. As before, we assume that G is piecewise continuous and
that condition (4) holds. Then

H0 = lim H (R, ψ)

R→∞

and

(8) lim HR(R, ψ) = G(ψ)
R→r0
R>r0

for each ψ at which G is continuous. Verification of formula (7), as well as special
cases of formula (3) that apply to semicircular regions, is left to the exercises.

Turning now to a half plane, we let G(x) be continuous for all real x, except
possibly for a finite number of finite jumps, and let it satisfy an order property

(9) |xaG(x)| < M (a > 1)

when −∞ < x < ∞. For each fixed real number t, the function Log|z − t| is har-
monic in the half plane Im z > 0. Consequently, the function

1� ∞
(10) U (x, y) = ln|z − t| G(t) dt + U0
π
−∞

= 1 �∞ ln[(t − x)2 + y2] G(t) dt + U0 (y > 0),
2π
−∞

where U0 is a real constant, is harmonic in that half plane.
Formula (10) was written with the Schwarz integral transform (1), Sec. 127, in

mind ; for it follows from formula (10) that

(11) 1 � ∞ y G(t) (y > 0).
Uy(x, y) = π −∞ (t − x)2 + y2 dt

In view of equations (1) and (2) in Sec. 127, then,

(12) lim Uy(x, y) = G(x)
y→0
y>0

at each point x where G is continuous.
Integral formula (10) evidently solves the Neumann problem for the half plane

y > 0, with boundary condition (12). But we have not presented conditions on
G which are sufficient to ensure that the harmonic function U is bounded as |z|
increases.

When G is an odd function, formula (10) can be written

(13) 1 � ∞ � (t − x)2 + y2 � (x > 0, y > 0).
U (x, y) = ln (t + x)2 + y2 G(t) dt
2π
0

Brown-chap12-v3 11/06/07 9:52am 448

448 Integral Formulas of the Poisson Type chap. 12

This represents a function that is harmonic in the first quadrant x > 0, y > 0 and
satisfies the boundary conditions

(14) U (0, y) = 0 (y > 0),

(15) lim Uy(x, y) = G(x) (x > 0).
y→0
y>0

EXERCISES

1. Establish formula (7), Sec. 128, as a solution of the Neumann problem for the region
exterior to a circle r = r0, using earlier results found in that section.

2. Obtain as a special case of formula (3), Sec. 128, the expression

1 �π
U (r, θ) = [Q(r0, r, φ − θ ) − Q(r0, r, φ + θ )] G(φ) dφ
2π
0

for a function U that is harmonic in the semicircular region r < r0, 0 < θ < π and
satisfies the boundary conditions

U (r, 0) = U (r, π ) = 0 (r < r0),

lim Ur (r, θ) = G(θ ) (0 < θ < π )

r →r0
r <r0

for each θ at which G is continuous.

3. Obtain as a special case of formula (3), Sec. 128, the expression

1 �π
U (r, θ) = [Q(r0, r, φ − θ ) + Q(r0, r, φ + θ )] G(φ) dφ + U0
2π
0

for a function U that is harmonic in the semicircular region r < r0, 0 < θ < π and
satisfies the boundary conditions

Uθ (r, 0) = Uθ (r, π ) = 0 (r < r0),

lim Ur (r, θ) = G(θ ) (0 < θ < π )

r →r0
r <r0

for each θ at which G is continuous, provided that

�π
G(φ) dφ = 0.

0

4. Let T (x, y) denote the steady temperatures in a plate x ≥ 0, y ≥ 0. The faces of the
plate are insulated, and T = 0 on the edge x = 0. The flux of heat (Sec. 107) into the
plate along the segment 0 < x < 1 of the edge y = 0 is a constant A, and the rest of
that edge is insulated. Use formula (13), Sec. 128, to show that the flux out of the
plate along the edge x = 0 is

A � 1�
ln 1 + y2 .
π

Brown-bapp01-v3 11/06/07 10:07am 449

APPENDIX

1

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The following list of supplementary books is far from exhaustive. Further references
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Conway, J. B.: “Functions of One Complex Variable,” 2d ed., 6th Printing, Springer-Verlag, New
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449

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1984.
Sokolnikoff, I. S.: “Mathematical Theory of Elasticity,” 2d ed., Krieger Publishing Company, Mel-

bourne, FL, 1983.
Springer, G.: “Introduction to Riemann Surfaces,” 2d ed., American Mathematical Society, Provi-

dence, RI, 1981.
Streeter, V. L., E. B. Wylie, and K. W. Bedford: “Fluid Mechanics,” 9th ed., McGraw-Hill Higher

Education, Burr Ridge, IL, 1997.
Taylor, A. E., and W. R. Mann: “Advanced Calculus,” 3d ed., John Wiley & Sons, Inc., New York,

1983.
Thron, W. J.: “Introduction to the Theory of Functions of a Complex Variable,” John Wiley & Sons,

Inc., New York, 1953.
Timoshenko, S. P., and J. N. Goodier: “Theory of Elasticity,” 3d ed., The McGraw-Hill Companies,

New York, 1970.
Titchmarsh, E. C.: “Theory of Functions,” 2d ed., Oxford University Press, Inc., New York, 1976.
Volkovyskii, L. I., G. L. Lunts, and I. G. Aramanovich: “A Collection of Problems on Complex

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Wen, G.-C.: “Conformal Mappings and Boundary Value Problems,” Translations of Mathematical

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Whittaker, E. T., and G. N. Watson: “A Course of Modern Analysis,” 4th ed., Cambridge University

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Wunsch, A. D.: “Complex Variables with Applications,” 3d ed., Pearson Education, Inc., Boston,

2005.

Brown-bapp02-v3 11/06/07 9:58am 452

APPENDIX

2

TABLE OF TRANSFORMATIONS
OF REGIONS
(See Chap. 8)

y v
B
A B′ A′
C x D′ u
D
C′
FIGURE 1
w = z2.

y v B′ A′
C′
B D′
CA u

D

x

FIGURE 2
w = z2.

452

Brown-bapp02-v3 11/06/07 9:58am 453

y Table of Transformations of Regions 453
DA
v
CB A′
cx
FIGURE 3

D′ C′ B′ w = z2;
u A B on parabola v2 = −4c2(u − c2).

y v
B
C A D′ u
x A′
D FIGURE 4
C′ w = 1/z.
B′

y v A′

C B′
A Bx u

FIGURE 5
C′ w = 1/z.

y v
DE
F

1

C B Ax F′ E′D′ C′B′ A′ u FIGURE 6
w = exp z.

Brown-bapp02-v3 11/06/07 9:58am 454

454 Table of Transformations of Regions app. 2

yv

ED C′

C

1 FIGURE 7
Bx D′ E′ A′ B′ u w = exp z.
A

y D v FIGURE 8
E C C′ w = exp z.
Bx F′
F
D′ E′ A′ B′ u
A

y v
E
A

DB 1
Cx
E′ D′ C′ B′ A′ u FIGURE 9
w = sin z.

y v
DA D′

CB 1

x C′ B′ A′ u FIGURE 10
w = sin z.

y v
D CB C′

E A 1
F x D′ E′ F′ A′ B′ u

FIGURE 11
w = sin z; BCD on line y = b (b > 0),

u2 v2
B C D on ellipse cosh2 b + sinh2 b = 1.

Brown-bapp02-v3 11/06/07 9:58am 455

Table of Transformations of Regions 455

y v
A B′

Bi C′ 1 A′
C x E′u

D D′ FIGURE 12
w = z−1.
E z+1

yv
D′

1 E′ 1
Ex A′ C′ u
A BC D
FIGURE 13

B′ w = i − z.
i +z

v
y F′

B

B′
F

C E GA G′ A′ C′ E′
1 R0 u
–1 x2 x1 1 x

D′
D

FIGURE 14

�

w z−a ; a 1+ x1x2 + (1 − x12)(1 − x22) ,
az − 1 x1 + x2
= =

�

R0 = 1 − x1x2 + (1 − x12)(1 − x22) (a > 1 and R0 > 1 when − 1 < x2 < x1 < 1).
x1 − x2

Brown-bapp02-v3 11/06/07 9:58am 456

456 Table of Transformations of Regions app. 2

y v
B′
B
E E′
D′ F′ A′
A CD F C′
R0 1 u
1 x2 x1 x

FIGURE 15

�
w = z − a ; a = 1 + x1x2 + (x12 − 1)(x22 − 1) ,
az − 1 x1 + x2

�
− 1 − (x12 − 1)(x22 − 1)
R0 x1x2 x1 − x2 (x2 <a < x1 and 0 < R0 < 1 when 1 < x2 < x1).

=

y v
C
E′ D′ C′ B′ A′ u
DB FIGURE 16
EA 1x 2

1
w=z+ .

z

yv

C 2
1
ED Ax E′ D′ C′ B′ A′ u
FIGURE 17 B

1
w=z+ .

z

yv
C C′

F D′ E′ 2
F′ A′ B′ u
D EA B
1 bx

FIGURE 18

1 u2 v2 = 1.
w = z + ; B C D on ellipse (b + 1/b)2 + (b − 1/b)2
z

Brown-bapp02-v3 11/06/07 9:58am 457

Table of Transformations of Regions 457

y v
D′ C′
B′

1

AB CD Ex D′ E′ A′ B′ u

FIGURE 19

w = Log z − 1; z = − coth w
z+1 .

2

y v D′
F′ E′
B A′ B′ C′
CA u

D E F1 x

FIGURE 20

w = Log z − 1;
z+1

ABC on circle x2 + (y + cot h)2 = csc2 h (0 < h < π ).

y v
v
F′
A′

CD E B′ E′

A F1 Bx c1 c2 u

C′ v D′

FIGURE 21

w = Log z + 1; centers of circles at z = coth cn, radii: csch cn (n = 1, 2).
z−1

Brown-bapp02-v3 11/06/07 9:58am 458

458 Table of Transformations of Regions app. 2

v

y D′ C′

A′ B′

–1 x1 1 G′

E F G A BC D x E′ F′ u

FIGURE 22
h

w = h ln 1 − h + ln 2(1 − h) + iπ − h Log(z + 1) − (1 − h) Log(z − 1); x1 = 2h − 1 .

y v
AE
D′ FIGURE 23
D E′ C′
� z �2 1 − cos z
BC A′ B′ 1 u tan .
x

w = =
2 1 + cos z

y E v u
GF x A′
H FIGURE 24
A D B′ C′ D′
B G′ F′ E′ 1 coth z ez + 1
w = = .
C H′ 2 ez −1

y E B′ v A′
GF x
H C′ D′ u
A D F′ E′ H′
B G′
FIGURE 25
C
� z� .
w = Log coth
2

Brown-bapp02-v3 11/06/07 9:58am 459

Table of Transformations of Regions 459

y v E′
D′ C′
1
A BC D E x A′ B′ u
FIGURE 26
w = π i + z − Log z.

y v
A′
C′
D′ B′

–1 Ex E′ u
A B CD

FIGURE 27 (z + 1)1/2
(z + 1)1/2
w = 2(z + 1)1/2 + Log − 1.
+1

y v
E′
– h2 1 Cx D′
DE FA B O B′ C′ u

F′ A′

FIGURE 28

i 1 + iht 1+t � z−1 �1/2
Log 1 − iht 1−t; z + h2 .
w = + Log t =
h

Brown-bapp02-v3 11/06/07 9:58am 460

460 Table of Transformations of Regions app. 2

y A′ v

–1 1 B′ hi D′ u
A B C Dx C′
FIGURE 29
w = h [(z2 − 1)1/2 + cosh−1 z].∗

π

v E′
y F′ D′ u

1h A′ B′

E FA B C Dx C′

FIGURE 30

� 2z − h − 1� 1 � (h + 1)z − 2h �
cosh−1 √ .
w = h−1 − h cosh−1 (h − 1)z

∗See Exercise 3, Sec. 122.

Brown-bindex-v2 11/27/07 4:50pm 461

INDEX

Absolute convergence, 186, Antiderivatives Bounded sets, 32
208–210 analytic functions and, 158 Branch cuts
explanation of, 142–149
Absolute value, 10 fundamental theorem of contour integrals and,
Accumulation points, 32–33 calculus and, 119 133–135
Additive identity, 4
Additive inverse, 4, 6 Arc explanation of, 96, 405
Aerodynamics, 391 differentiable, 124 integration along, 283–285
Algebraic properties, of complex explanation of, 122 Branches
simple, 122 of double-valued function,
numbers, 3–5 smooth, 125, 131, 146
Analytic continuation, 84–85, 338, 341–342, 346
Argument integrands and, 145, 146
87 principle value of, 16, 17, of logarithmic function,
Analytic functions 37
of products and quotients, 95–96, 144, 230, 328,
Cauchy–Goursat theorem 20–24 361
adopted to integrals of, of multiple-values function,
200 Argument principle, 291–294 246, 281, 284
Associative laws, 3 principal, 96, 102, 229–230
composition of, 74 of square root function,
derivatives of, 169–170 Bernoulli’s equation, 392 336–338
explanation of, 73–76, 229, Bessel functions, 207n Branch point
Beta function, 287 explanation of, 96, 350
231 Bierwirth, R. A., 269n indentation around, 280–283
isolated, 251 Bilinear transformation, at infinity, 352
properties of, 74–77 Bromwich integral, 299
real and imaginary 319–322 Brown, G. H., 269n
Binomial formula, 7, 8, 171 Brown, J. W., 79n, 207n, 270n,
components of, 366 Boas, R. P., Jr., 174n, 241n, 279n, 307n, 379n, 390n,
reflection principle and, 437n
322n Buck, J. R., 207n
85–87 Bolzano–Weierstrass theorem,
residue and, 238 Casorati–Weierstrass theorem,
simply connected domains 257 259
Boundary conditions, 367–370
and, 158 Boundary of S, 32 Cauchy, A. L., 65
uniquely determined, 83–85 Boundary points, 31–32, 329, Cauchy–Goursat theorem
zeros of, 249–252, 294
Analyticity, 73, 75–76, 215, 339 applied to integrals of
Boundary value problems, 365, analytic functions, 200
229, 231, 250, 405
Angle of inclination, 124, 356, 366, 376, 378, 379, 381, applied to multiply connected
437–439 domains, 158–160
358 Bounded functions, 173, 174
Angle of rotation, 356, 358–360
Angles, preservation of,

355–358

461

Brown-bindex-v2 11/27/07 4:50pm 462

462 Index

Cauchy–Goursat theorem Closed set, 32 flows around corner and
(continued) Closure, 32 around cylinder and,
Commutative laws, 3 395–397
applied to simply connected Complex conjugates, 13–14,
domains, 156–157 steady temperatures and,
421 373–375
explanation of, 151, 229, Complex exponents, 101–103
279, 430 Complex numbers steady temperatures in half
plane and, 375–377
proof of, 152–156 algebraic properties of, 3–5
residue and, 235–236 arguments of products and stream function and,
Cauchy integral formula 393–395
consequences of extension of, quotients of, 20–22
complex conjugates of, 13–14 temperatures in quadrant and,
168–170 convergence of series of, 379–381
explanation of, 164–165,
185–186 temperatures in thin plate
200, 429 explanation of, 1 and, 377–379
extension of, 165–168, 218, exponential form of, 16–18
imaginary part of, 1 two-dimensional fluid flow
248–249 polar form of, 16–17 and, 391–393
for half plane, 440 products and powers in
Cauchy principal value, 262, Conjugates
exponential form of, complex, 13–14, 421
270, 274 18–20 harmonic, 80–81, 363–366,
Cauchy product, 223 real part of, 1 443
Cauchy–Riemann equations roots of, 24–29
sums and products of, 1–7 Continuous functions
analyticity and, 75 vectors and moduli of, 9–12 derivative and, 59
in complex form, 73 Complex plane, 1 explanation of, 53–56, 406
explanation of, 65–66, 360, extended, 50
point at infinity and, 50–51 Contour integrals
371 Complex potential, 393, 394, branch cuts and, 133–135
harmonic conjugate and, 80, 426–427 evaluation of, 142
Complex variables examples of, 129–132
81, 364, 365, 443 functions of, 35–38 explanation of, 127–129
partial derivatives and, 66, integrals of complex-valued moduli of, 137–140
functions of, 122 upper bounds for moduli of,
67, 69, 70, 86, 365 Composition of functions, 74 137–140
in polar form, 70, 95 Conductivity, thermal, 373
sufficiency of, 66–68 Conformal mapping Contours
Cauchy’s inequality, 170, 172 explanation of, 357, 418 in Cauchy–Goursat theorem,
Cauchy’s residue theorem, harmonic conjugates and, 156–157
363–365 explanation of, 125
234–236, 238, 264, 281, local inverses and, 360–362 simple closed, 125, 150
283, 284, 294 preservation of angles and,
Chain rule, 61, 69, 74, 101, 355, 355–358 Convergence
363, 366–367, 426 scale factors and, 358–360 absolute, 186, 208–210
Chebyshev polynomials, 24n transformations of boundary circle of, 209, 211, 213–214,
Christoffel, E. B., 406 conditions and, 367–370 216
Churchill, R. V., 79n, 207n, transformations of harmonic of sequences, 181–184
270n, 279n, 299n, 301n, functions and, 365–367 of series, 184–189, 208–213,
307n, 379n, 390n, 437n Conformal mapping applications 250
Circle of convergence, 209, 211, cylindrical space potential uniform, 210–213
213–214, 216 and, 386–387
Circles electrostatic potential and, Conway, J. B., 322n
parametric representation of, 385–386 Cosines, 288–290
18 Critical point, of
transformations of, 314–317,
400 transformations, 357–358
Circulation of fluid, 391 Cross ratios, 322n
Closed contour, simple, 125, Curves
150, 235
Closed disk, 278 finding images for, 38–39
Closed polygons, 404 Jordan, 122, 123
level, 82
Cylindrical space, 386–387

Brown-bindex-v2 11/27/07 4:50pm 463

Index 463

Definite integrals Double-valued functions two-dimensional, 391–393
of functions, 119–120 branches of, 338, 341–342, velocity of, 392–393
involving sines and cosines, 346 Flux, 373
288–290 Riemann surfaces for, Flux lines, 385
mean value theorem for, 436 351–353 Formulas
binomial, 7, 8, 171
Deformation of paths principle, Electrostatic potential Cauchy integral, 164–170,
159–160, 237–238 about edge of conducting
plate, 422–425 200, 218
Degenerate polygons, 413–414 explanation of, 385–386 de Moivre’s, 20
Deleted neighborhood, 31, 251, differentiation, 60–63, 74, 75,
Elements of function, 85
258, 259 Ellipse, 333 107, 111
de Moivre’s formula, 20, 24 Elliptic integral, 409, 411 Euler’s, 17, 28, 68, 104
Derivatives Entire functions, 73, 173 integration, 268–269, 279,
Equipotentials, 385, 393, 401
of branch of zc, 101–102 Essential singular points, 242 280–281, 283, 286, 289
directional, 74 Euler numbers, 227 Poisson integral, 429–431
first-order partial, 63–67, 69 Euler’s formula, 17, 28, 68, 104 quadratic, 289
of functions, 56–61 Even functions, 121 Schwarz integral, 440–441
of logarithms, 95–96 Expansion summation, 187, 194
of mapping function, 413, Fourier, Joseph, 373n
Fourier series, 208 Fourier integral, 270, 279n
416, 426 Maclaurin series, 192–195, Fourier series, 208
Differentiability, 66–68 Fourier series expansion, 208
Differentiable arc, 124 215, 233 Fourier’s law, 373
Differentiable functions, 56, 59 Exponential form, of complex Fractional transformations,
Differentiation formulas
numbers, 16–18 linear, 319–323,
explanation of, 60–63, 74, Exponential functions 325–327, 341, 416
75, 107 Fresnel integrals, 276
additive property of, 18–19 Functions. See also specific types
verification of, 111 with base c,103 of functions
Diffusion, 375 explanation of, 89–91 analytic, 73–77, 83–87, 158,
Directional derivative, 74 mappings by, 42–45 169–170, 200, 231, 238,
Dirichlet problem Extended complex plane, 50 249–252, 294
Exterior points, 31 antiderivative of, 158
for disk, 429, 432–435 Bessel, 207n
explanation of, 365, 366 Field intensity, 385 beta, 287
for half plane, 441–443 Finite unit impulse function, 436 bounded, 173, 174
for rectangle, 389–390 First-order partial derivatives branch of, 96, 229–230
for region exterior to circle, Cauchy–Riemann equations
Cauchy–Riemann equations and, 63–66
438–439 and, 66, 67, 69 of complex variables, 35–38
for region in half plane, 376 composition of, 74
for semicircular region, 438 explanation of, 63–65 conditions for differentiability
for semi-infinite strip, 383 Fixed point, of transformation, and, 66–68
Disk continuous, 53–56, 59, 406
closed, 278 324 definite integrals of, 119–120
Dirichlet problem for, 429, Fluid flow derivatives of, 56–61,
117–118
432–435 around corner, 395–396 differentiable, 56, 59
punctured, 31, 224, 240 around cylinder, 396–397 differentiation formulas and,
Distributive law, 3 in channel through slit, 60–63
Division, of power series, domain of definition of, 35,
417–419 84, 344, 345
222–225 in channel with offset,
Domains
420–422
of definition of function, 35, circulation of, 391
84, 344, 345 complex potential of, 393,

explanation of, 32 394
multiply connected, 158–160 incompressible, 392
simply connected, 156–158 irrotational, 392
union of, 85 in quadrant, 396

Brown-bindex-v2 11/27/07 4:50pm 464

464 Index

Functions. See also specific types Graphs, of functions, 38 Improper integrals
of functions (continued) Green’s theorem, 150–151 evaluation of, 262–264
explanation of, 261–262
double-valued, 338, 341–342, Half plane from Fourier analysis,
346, 351–353 Cuchy integral formula in, 269–272
440
elements of, 85 Dirichlet problem in, Impulse function, finite unit, 436
entire, 73, 173 441–443 Incompressible fluid, 392
even, 121 harmonic function in, 425 Indented paths, 277–280
exponential, 18–19, 42–45, mappings of upper, 325–329 Independence of path, 142, 147,
Poisson integral formula for,
103 441 394
finite unit impulse, 436 steady temperatures in, Inequality
gamma, 283 375–377
graphs of, 38 Cauchy’s, 170, 172
harmonic, 78–81, 365–367, Harmonic conjugates involving contour integrals,
explanation of, 80, 363–365
435, 437, 438, 442, 443 harmonic functions and, 137–138
holomorphic, 73n 80–81, 366 Jordan’s, 273, 274
hyperbolic, 106, 109–114 method to obtain, 81, 443 triangle, 11–12, 174
inverse, 112–114 Infinite sequences, 181
limits of, 45–52 Harmonic functions Infinite series
logarithmic, 93–96, 98–99, applications for, 78–80, 391, explanation of, 184
442, 443 of residues, 301, 307
144 bounded, 376 Infinite sets, 301
meromorphic, 291 explanation of, 78–79, 382, Infinity
multiple-valued, 37, 246, 281, 432 branch point at, 352
in half plane, 425 limits involving point at,
284 harmonic conjugate and,
near isolated singular points, 80–81, 366 50–52, 314
product of, 377 residue at, 237–239
257–260 in semicircular region, 437, Integral formulas
odd, 121 438 Cauchy, 164–168
piecewise continuous, 119, theories as source of, 79–80 Poisson, 429–431
transformations of, 365–367, Schwarz, 440–441
127, 137–138, 432–435 438 Integrals
polar coordinates and, values of, 395, 435 antiderivatives and, 142–149
Bromwich, 299
68–73 Heat conduction, 373. See also Cauchy–Goursat theorem
principal part of, 240 Steady temperatures
range of, 38 and, 150–156
rational, 37, 263 Hille, E., 125n Cauchy integral formula and,
real-valued, 36–38, 58–60, Holomorphic functions, 73n
Hoyler, C. N., 269n 164–170, 200, 218
125, 208 Hydrodynamics, 391 Cauchy principal value of,
regular, 73n Hyperbolas, 39–41, 331, 381
single-valued, 347–349, Hyperbolic functions 262, 270, 274
contour, 127–135, 137–140,
399 explanation of, 106, 109–111
square root, 349–350 identities involving, 110 142
stream, 393–395, 418–419 inverse of, 112–114 definite, 119–120, 288–290
trigonometric, 104–107, elliptic, 409, 411
Identities Fourier, 270
111–114 additive, 4 Fresnel, 276
zeros of, 106–107 involving logarithms, 98–99 improper, 261–264,
Fundamental theorem of algebra, Lagrange’s trigonometric, 23
multiplicative, 4 269–272
173, 174, 295 line, 127, 364
Fundamental theorem of Image of point, 38 Liouville’s theorem and
Imaginary axis, 1
calculus, 119, 142, 146 fundamental theorem of
algebra and, 173–174
Gamma function, 283 maximum modulus principle
Gauss’s mean value theorem, and, 176–178

175
Geometric series, 194
Goursat, E., 151

Brown-bindex-v2 11/27/07 4:50pm 465

Index 465

Integrals (continued) Laplace’s equation Maclaurin series
mean value theorem for, 120 harmonic conjugates and, examples illustrating, 227,
multiply connected domains 363, 364 233, 236, 247
and, 158–160 harmonic functions and, 78, explanation of, 190, 204
simply connected domains 377 Taylor’s theorem and,
and, 156–158 polar form of, 82, 439 192–195
theory of, 117
Laplace transforms Maclaurin series expansion,
Integral transformation, 432, applications of, 300–301 192–195, 215, 233
437, 442 explanation of, 299
inverse, 298–301, 309 Mann, W. R., 55n, 79n, 138n,
Integration, along branch cuts, 162n, 257n, 360n, 440n
283–285 Laurent series
coefficients in, 202–205, Mappings. See also
Integration formulas, 268–269, 207 Transformations
279, 280–281, 283, 286, examples illustrating, 224,
289 225, 231, 237, 245, 246, of circles, 400
278, 280 conformal, 355–370, 418
Interior points, 31 explanation of, 199, 201–202
Inverse residue and, 238, 240, 247 (See also Conformal
uniqueness of, 218–219 mapping)
of linear fractional derivative of, 413, 416, 426
transforms, 416 Laurent’s theorem explanation of, 38–42
explanation of, 197–198, 203 by exponential function,
local, 360–362 proof of, 199–202 42–45
Inverse functions, 112–114 isogonal, 357
Inverse hyperbolic functions, Lebedev, N. N., 141n, 283n linear fractional
Legendre polynomials, 141n, transformations as,
112–114 327–328
Inverse image, of point, 38 171n by logarithmic function, 328,
Inverse Laplace transforms, Leibniz’s rule, 222, 226 339
Level curves, 82 one to one, 39, 41–43, 320,
298–301, 309 l’Hospital’s rule, 283 327, 331, 332, 334, 338,
Inverse transform, 301n Limits 342, 345
Inverse transformation, 320, polar coordinates to analyze,
of function, 45–47 41–42
347, 354, 387, 401, 418 involving point at infinity, of real axis onto polygon,
Inverse trigonometric functions, 403–405
50–52, 314 on Riemann surfaces,
112–114 of sequence, 181, 184 347–350
Inverse z-transform, 207 theorems on, 48–50 of square roots of
Irrotational flow, 392 Linear combination, 77 polynomials, 341–346
Isogonal mapping, 357 Linear transformations by trigonometric functions,
Isolated analytic functions, 251 explanation of, 311–313 330–331
Isolated singular points fractional, 319–323, of upper half plane, 325–329
by 1/z,315–317
behavior of functions near, 325–327, 341, 416 by z2 and branches of z1/2,
257–260 Line integral, 127, 364 336–340
Lines of flow, 375 Markushevich, A. I., 157n,
explanation of, 229–231, Liouville’s theorem, 173–174, 167n, 242n
240–244, 247 Maximum and minimum values,
296 176–178
Isolated zeros, 251 Local inverses, 360–362 Maximum modulus principle,
Isotherms, 375, 377 Logarithmic functions 176–178
Mean value theorem, 120, 436
Jacobian, 360, 361 branches and derivatives of, Meromorphic functions, 291
Jordan, C., 122n 95–96, 144, 230, 361 Mo¨bius transformation,
Jordan curve, 122, 123 319–322
Jordan curve theorem, 125 explanation of, 93–95
Jordan’s inequality, 273, 274 identities involving, 98–99
Jordan’s lemma, 272–275 mapping by, 328, 339
Joukowski airfoil, 400 principal value of, 94
Riemann surface for,
Kaplan, W., 67n, 364n, 391n
348–349
Lagrange’s trigonometric
identity, 23

Brown-bindex-v2 11/27/07 4:50pm 466

466 Index

Moduli Point at infinity electrostatic, 385–386,
of contour integrals, 137–140 limits involving, 50–52, 314 422–425
explanation of, 10–12 neighborhood of, 51–52
unit, 325 residue at, 237–239 velocity, 392–393
Powers, of complex numbers, 19
Morera, E., 169 Poisson integral formula Power series
Morera’s theorem, 169, 215 for disk, 431
Multiple-valued functions, 37, explanation of, 429–431 absolute and uniform
for half plane, 441 convergence of, 208–211
246, 281, 284
Multiplication, of power series, Poisson integral transform, 432, continuity of sums of,
437 211–213
222–225
Multiplicative identity, 4 Poisson kernel, 431, 436 explanation of, 187
Multiplicative inverse, 4, 6, 19 Poisson’s equation, 371, 372 integration and differentiation
Multiply connected domain, Polar coordinates
of, 213–217
158–160 to analyze mappings, 41–42, multiplication and division of,
337
Negative powers, 195 222–225
Neighborhood convergence of sequences Principal branch
and, 183–184
deleted, 31, 251, 258, 259 of double-valued function,
explanation of, 31, 32 explanation of, 16 338
of point at infinity, 51–52 functions and, 36, 68–73
Nested intervals, 163 Laplace’s equation in, 433 of function, 96, 229–230
Nested squares, 163 Polar form of logarithmic function, 362
Neumann problems, 445–448 of Cauchy–Riemann of zc, 102
explanation of, 365 Principal part of function, 240
Newman, M.H.A., 125n equations, 70, 95 Principal root, 26
Nonempty open set, 32 of complex numbers, 16–17 Principal value
Numbers of Laplace’s equation, 82, of argument, 16, 17, 37
complex, 1–28 Cauchy, 262, 270, 274
pure imaginary, 1 439 of logarithm, 94
real, 101 Poles of powers, 102, 103
winding, 292 Punctured disk, 31, 224, 240
of functions, 249 Pure imaginary numbers, 1
Odd functions, 121 of order m,241 Pure imaginary zeros, 289
One to one mapping, 39, 41–43, residues at, 244–247, 253
simple, 241, 253, 302 Quadrant, temperatures in,
320, 327, 331, 332, 334, zeros and, 249, 252–255 379–381
338, 342, 345 Polygonal lines, 32
Open set Polygons Quadratic formula, 289
analytic in, 73 closed, 404
connected, 83 degenerate, 413–417 Radio-frequency heating, 269
explanation of, 32 mapping real axis onto, Range of function, 38
Oppenheim, A. V., 207n Rational functions
403–405
Parabolas, 337 Polynomials explanation of, 37
Partial derivatives improper integrals of, 263
Chebyshev, 24n Ratios, cross, 322n
Cauchy–Riemann equations of degree n, 36 Real axis, 1
and, 66, 67, 69, 70, 79, as entire function, 73 Real numbers, 101
86 fundamental theorem of Real-valued functions
derivative of, 60
first-order, 63–65 algebra and, 173, 174 example of, 58–59
second-order, 364 Legendre, 141n, 171n explanation of, 36–37
Partial sums, sequence of, 184 quotients of, 36–37 Fourier series expansion of,
Picard’s theorem, 242 square roots of, 341–346
Piecewise continuous functions, zeros of, 173, 174, 268, 208
identities and, 125
119, 127, 137–138, 296 properties of, 38
432–435 Positively oriented curve, 122 Rectangles
Potential Dirichlet problem for,

complex, 393, 394, 426–427 389–390
in cylindrical space, 386–387

Brown-bindex-v2 11/27/07 4:50pm 467

Index 467

Rectangles (continued) Riemann surfaces Taylor, 189–190, 192–195,
Schwarz–Christoffel for composite functions, 217–218, 224
transformation and, 351–353
412–413 for double-valued function, uniqueness of representations
351–353 of, 217–219
Rectangular form, powers of explanation of, 347–350
complex numbers in, Simple arc, 122
19–20 Roots Simple closed contour, 125, 150,
of complex numbers, 24–29
Reflection, 38 principal, 26 235
Reflection principle, 85–87 of unity, 28, 30 Simple poles
Regions
Rotation, 38 explanation of, 241, 253
in complex plane, 31–33 Rouche´’s theorem, 294–296 residue at, 302
explanation of, 32 Simply connected domains,
table of transformations of, Scale factors, 358–360
Schafer, R. W., 207n 156–158
452–460 Schwarz, H. A., 406 Single-valued functions,
Regular functions, 73n Schwarz–Christoffel
Removable singular point, 242, 347–349, 399
transformation Singular points
258 degenerate polygons and,
Residue applications essential, 242
413–417 isolated, 229–231, 240–244,
argument principle and, electrostatic potential about
291–294 247, 257–260
edge of conducting plate removable, 242, 258
convergent improper integral and, 422–425 Sink, 417, 418, 420
evaluation and, 269–272 explanation of, 405–407 Smooth arc, 125, 131, 146
fluid flow in channel through Sphere, Riemann, 51
definite integrals involving slit and, 417–419 Square root function, 349–350
sines and cosines and, fluid flow in channel with Square roots, of polynomials,
288–290 offset and, 420–422
triangles and rectangles and, 341–346
examples of, 301–306 408–413 Squares, 152
improper integral evaluation Schwarz integral formula, Stagnation point, 419
440–441 Steady temperatures
and, 261–267 Schwarz integral transform, 442
indentation around branch Second-order partial derivatives, conformal mapping and,
364 373–375
point and, 280–283 Separation of variables method,
indented paths and, 277–280 379 in half plane, 375–377
integration around branch cut Sequences Stereographic projection, 51
convergence of, 181–184 Stream function, 393–397,
and, 283–285 explanation of, 181
inverse Laplace transforms limit of, 181, 184 418–419
Series. See also specific type of Streamlines, 393–395, 397, 401,
and, 298–301, 309 series
Jordan’s lemma and, convergence of, 184–189, 419
208–213, 250 Summation formula, 187, 194
272–275 explanation of, 184 Sums
Rouche´’s theorem and, Laurent, 199, 201–205, 207,
218–219, 224, 225, 231, of power series, 211–213
294–296 237 of residues, 263
Residues Maclaurin, 190, 192–195,
204, 215, 233, 236 Taylor, A. E., 55n, 79n, 138n,
Cauchy’s theorem of, power, 187, 208–217, 162n, 257n, 360n, 440n
234–236, 238, 264, 281, 222–225
283, 284, 294 Taylor series
examples illustrating,
explanation of, 229, 231–234 192–195, 224, 250, 252,
infinite series of, 301, 307 405
at infinity, 237–239 explanation of, 189–190
at poles, 244–247 uniqueness of, 217–218
poles and, 253–255
sums of, 263 Taylor series expansion, 189,
Resonance, 309 192, 222
Riemann, G. F. B., 65
Riemann sphere, 51 Taylor’s theorem
Riemann’s theorem, 258 explanation of, 189

Brown-bindex-v2 11/27/07 4:50pm 468

468 Index

Taylor’s theorem (continued) Schwarz–Christoffel, Unity
to find Maclaurin series 405–425 (See also nth roots of, 28, 30
expansions, 192–195 Schwarz–Christoffel radius, 17
proof of, 190–192, 201 transformation)
Unstable component, 309
Temperatures table of, 452–460
in half plane, 375–377 w = sine z, 330–334 Value
in quadrant, 379–381 w = 1/z, 313–317 absolute, 10
steady, 373–377 Transforms maximum and minimum,
in thin plate, 377–379 inverse, 301n 176–178
inverse z-, 207
Thermal conductivity, 373 Laplace, 298–301, 309 Vector field, 45
Thron, W. J., 125n Poisson integral, 432, 437 Vectors, 9–12
Transformations. See also Schwarz integral, 442 Velocity potential, 392–393
z-, 207 Viscosity, 392
Mappings Translation, 38
argument principle and, 291 Triangle inequality, 11–12, Winding number, 292
bilinear, 319
of boundary conditions, 174 Zero of order m, 249–250, 252,
Triangles, 409–410 253, 256
367–370 Trigonometric functions
of circles, 314–317, 400 Zeros
conformal, 355–370, 418 definite integrals involving, of analytic functions,
288–290 249–252, 294
(See also Conformal isolated, 251
mapping) explanation of, 104–107 poles and, 249, 252–255
critical point of, 357–358 identities for, 105–107 of polynomials, 173, 174,
explanation of, 38 inverse of, 112–114 268, 296
fixed point of, 324 mapping by, 330–331 pure imaginary, 289
of harmonic functions, periodicity of, 107, 111 in trigonometric functions,
365–367, 438 zeros of, 106–107 106–107
integral, 432, 437, 442 Two-dimensional fluid flow,
inverse of, 320, 347, 354, z-transform
387, 401, 418 391–393 explanation of, 207
kernel of, 431 inverse, 207
linear, 311–313 Unbounded sets, 32
linear fractional, 319–323, Uniform convergence, 210–213
325–327, 341, 416