SEC. 25
the two functions
u(x, Y) =
x2  y2
2xy
and v ( x , y ) =
(x2 y2)2
(x2 y 2 p
+
+
are harmonic throughout any domain in the xy plane that does not contain the origin.
If two given functions u and v are harmonic in a domain L) and their firstorder
CHAP. 2
where grad u is the gradient vector
grad u = u,i
(2)
+ u, j.
Because u, and u, are zero everywhere in D, then, grad u is the zero vector at all
points on L. Hence it follows from equation (1) that the derivative d u l d s is zero along
L; and this
POLAR
COORDINATES
67
SEC. 22
and 0 exist everywhere in that neighborhood. Ifthose partial derivatives are continuous
at (ro, 6,) and satisfy the polar form
o the CauchyRiemann equations at (ro, go), then f'(zo) exists.
f
The derivative f '(zo) here can b
CHAP. 2
Evidently, then,
lim
( A X Ay)+(O,O)
,
Re
Aw
=
Az
lim
AY+O
v(x0. yo+ AY)  ~
AY
( ~ YO)
0 9
= vy (xo, YO)
and
Aw
lim
I m p (Ar,Ay)+(O,O) A2
lim
+A
~ ( ~ Yo 7
0
Ay+O
~ )U ( ~ 0YO)
,
= uy ( ~ 0 9
YO).
AY
Hence it follows from expression (3) that
respectively, it also shows that the real and imaginary components of a function of a
complex variable can have continuous partial derivatives of all orders at a point and
yet the function may not be differentiable there.
The function f ( 2 ) = 1z12 is co
SEC. 15
whenever
0 < I (x
+ i y )  (xo + YO) 1 < 6.
That is, limit (1) holds.
Let us now start with the assumption that limit (1) holds. With that assumption,
we know that, for each positive number E ,there is a positive number 6 such that
whenever
But
l
CHAP. 2
FIGURE 27
(See Fig. 27.) But the continuity of f at z0 ensures that the neighborhood lz  zol < 6
can be made small enough that the second of these inequalities holds. The continuity
of the composition g [ f (z)] is, therefore, established.
Theore
42
ANALYTIC
FUNCTIONS
CHAP. 2
Our final example here uses the images of horizontal lines to find the image of a
horizontal strip.
EXAMPLE 3. When w = eZ,the image of the infinite strip 0 5 y 5 rr is the upper
half v 2 0 of the w plane (Fig. 22). This is s
SEC. 12
MAPPINGS
37
EXAMPLE 1. According to Example 2 in Sec. 11, the mapping w = z2 can be
thought of as the transformation
from the xy plane to the u v plane. This form of the mapping is especially useful in
finding the images of certain hyperbolas.
It
32
COMPLEX
NUMBERS
CHAP. I
5. Let S be the open set consisting of all points z such that lzl < 1 or lz  21 < 1. State why
S is not connected.
6. Show that a set S is open if and only if each point in S is an interior point.
7. Determine the accumulation
EXAMPLES 27
SEC. g
EXAMPLE 3. The two values q (k = 0, 1) of
roots of & i, are found by writing
+
(J?+ i)1/2, which are the square
and (see Fig. 14)
I
FIGURE 14
Euler's formula (Sec. 6) tells us that
GO = h e x p ( i k )
= A (cos
1,
7r
 + i sin
"
12
12
SEC.
7
PRODUCTS QUOTIENTS EXPONENTIAL
AND
IN
FORM
17
geometrically obvious that
,n
i
and e i4n  1.

e in/2  i,
= 1,
Note, too, that the equation
is a parametric representation of the circle )zl = R, centered at the origin with radius
R. As the para
CHAP. I
and [see Exercise 2(b)]
10. Establish the identity
and then use it to derive Lagrange's trigonometric identity:
+
+
Suggestion: As for the first identity, write S = 1 + z z2 + .
zn and consider
the difference S  zS. To derive the second identity,
CHAP. I
0
X
FIGURE 5
So the conjugate of the sum is the sum of the conjugates:
In like manner, it is easy to show that
and
(5)
+
+r
The sum z
of a complex number z = x iy and its conjugate Z = x  iy is
the real number 2x, and the difference z  z is the
Group 1: Sasikanth Thota, Aravind
Title: Week 5: Activity 3: Assignment 2
Kumar
Ravi,
Sadhana
Subramanian
1. An overview of the category of software that your team is working with
Descriptive analytics provide insight from historical data. It helps an org
Group 1: Sasikanth Thota, Aravind Kumar Ravi, Sadhana Subramanian
Title: Week 3: Activity 3: Assignment 1
1. An overview of the category of software that your team is working with
Decision Support Software: It is a software which enables business managers
Group 1: Sasikanth Thota, Aravind
Title: Week 8: Activity 3: Assignment 3
Kumar
Ravi,
Sadhana
Subramanian
1.An overview of the category of software that your team is working with
Predictive analytics helps in predicting about the future events by analyzin
Group 1: Sasikanth Thota, Aravind
Title: Week 14: Activity 5: Assignment 5
Kumar
Ravi,
Sadhana
Subramanian
1.An overview of the category of software that your team is working with
A process where large sets or big data is collected, organized and then ana
Chapter 10
*
1. What is I/O Stream?
An I/O Stream represents an input source or an output destination.
A stream can represent many different kinds of sources and destinations, including disk files, devices, other programs, and memory arrays.
Streams s
*
*
CH 14
*
1. Define the components of RFID and how active and passive tags works.
Answer: In its simplest form, an RFID system consists of a tag attached to the product to be identified, a reader, one or more antennae attached to the reader, and a compu
MIS 513 DQ 6 Chapter 6
1. According to the textbook, what are the four pillars of Web 2.0?
The four pillars of Web 2.0 are discussed on pages 235 236.
The four pillars are:
o Utilizing the Web as a Platform.
o Harnessing Collective Intelligence.
o Levera
http:/www.teradata.com/td/pdf.aspx?a=134006&b=133896
3M Moves to a Customer Focus Using a Global Data Warehouse
Dale Goodhue, University of Georgia Barbara Wixom, University of Virginia Introduction In 1995, 3M Chairman and CEO L. D. DeSimone along with h
Question 1
1 out of 1 points
A data warehouse contains _ about how data are organized and how to use them effectively.
Answer
Selected Answer: metadata
Correct Answer: metadata
Question 2
1 out of 1 points
A star schema contains a central _ surr