11
.. Chapter d
t
5
• The power series L::'=o
anx
n
converges absolutely within the interval of
convergence
Ixl <
R,
where
I
an
I
R
=
lim
 .
ntoo
a
n
+l
Jr'
COMPLEX NUMBERS
• Relations between functions will be satisfied order by order when they are
replaced by their power series. You must know how to expand functions of
functions, out to some desired order within the common interval of conver
gence.
jlr.
• The power series representing a function may be integrated term by term and
differentiated term by term within the interval of convergence to obtain the
series for the integral or derivative of the function in question.
;jii.
ili
Pr!
,lh·
5.1.
Introduction
Let us consider the quadratic function
f {x)
=
x
2
 5x+6
and ask where it vanishes.
If we plot it against
x ,
we will find that it vanishes at
x
=
2 and
x
=
3. This is
also clear if we write
f
in factorized form as
f{x)
=
(x

2){x

3). We could
equivalently use the wellknown formula for the roots
x
±
of a quadratic equation:
89
A plot will show that this function is always positive and does not vanish for any
point on the z axis. We are then led to conclude that this quadratic equation has
no roots. Let us pass from the graphical procedure which gives no solution to the
algebraic one which does give some form of answer even now. It says
The problemof course is that we do not know what to make of
H
since there is
no real number whose square is
~3.
Thus if we take the stand that a number is not
a number unless it is a real number, we will have to conclude that some quadratic
equations have roots and other do not.
.
"'~
~
n
~ ~0
;..J

~
i!
~~
o
~
(])
~~
~
00",
~
~~
~ 5~
~~
,...l
>,
~
r.Q:i
§B
~
,....
,
.

~
O
~ [~
<>.~
QQ
UU
>0
='"
~
~
8"~
~~
g$
o"ffi
'EQ
~
~
~
~
~8
'25
.5 2
g~
1l;;
~~
~~
'i~
~~
~g
:/J
=
§~
u
e."
(5.1.4)
(5.1.3)
(5.1.2)
(5.1.1)
x±
=
 l ± H
2
x
2
+ x
+
1
=
O.
b
±
,jb'1

4ac
x ±
=
2a
to find the roots
x ±
=
2,3. Suppose instead we consider
ax
2
+
bx
+c
=
0
namely
~t.
:>;'t f.
:si)'
,}
~:
~.
f)I"
'.
~.;
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Chapter5
Complex Numbers
91
and refer to
x
and
v
as its
real and imaginary parts
and denote them by the symbols
Re
z
and 1m
z: A number withjust Y
'#
0
is called a pure imaginary number.
The
solution to our quadratic equation has a real part
x
=
!
and an imaginary part
'J
:l.J
'J)
"~
h
~~
o~
~§
§8
H
~0

~
ti
(1)
~S
~ ~;;
~
~
~~
:5;:
"8
"'
~
>[email protected]
f....
4
. .
::i
(1)
§~
r"'I __
~
O
~ [~
"'~
""
uu
>0
§C"'J
8'~
:"3
=Ol
gf;i
c'
o
~
'EQ
gg
a
~
.g8
::2~
s~
;s.Q
'g'5
~~
«
(5.2.11)
(5.2.8)
(5.2.9)
(5.2.10)
(5.2.4)
(5.2.5)
rule
(5.2.6)
multiplication rule (5.2.7)
Z2
=
x 2
+
iY2
implies
X2
Y2 ·
Z·
=
x

iy
Xl
+
iYl
X2
+
iY2,
we define.
(Xl
+
X2)
+
i(Yl
+
Y2)
addition
(X1X2

Y1Y2)
+
i(X1Y2
+
X2Ytl
Zl
=
+
Yl
Z2
Zl
+
Z2
ZlZ2
Suppose this were not true. This would imply
Xl 
X2
=
i(Y2

Yl). without
both of them vanishing separately. Squaring both sides. we would find a positive
definite lefthand side and a negative definite righthand side. The only way to
avoid a contradiction is for both sides to vanish, giving us 0
=
0, which is
something we can live with.
.
Now, given any real number z, we can associate with it a unique number
x,
called its negative. We can do that with a complex number
Z
=
x
+
too, by
negating
x
and
y.
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 Fall '08
 RAMAMURTISHANKAR
 Physics, Complex Numbers, Power, Complex number, imaginary parts

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