Transcript Chapter 2 Complex numbers
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Chapter 2 Complex numbers (복소수)
Lecture 4 Introduction of complex numbers 고등수학 10-가 2장 실수와 복소수에 나옴
1
1. Introduction
az
2
bz
c
0
z
b
b
2 4
ac
, sometimes, 2
a
'
b
2 4
ac
' can be negative.
i
Let' s consider imaginary 2 1 ,
ex
) 2 8
i
number
i i
i
2 1 4 .
ex.
z
2 2
z
2 0
z
2 2 4 8 2 2 4 1
i
.
2
(READING) Once the new kind of number is admitted into our number system, fascinating possibilities open up. Can we attach any meaning to marks like sin
i
, e^
i
, ln (1+
i
)? We’ll see later that we can and that, in fact, such expressions may turn up in problems in physics, chemistry, and engineering, as well as, mathematics.
When people first considered taking square roots of negative numbers, they felt very uneasy about the problem. They thought that such numbers could not have any meaning or any connection with reality (hence the term “imaginary”). They certainly would not have believed that the new numbers could be of any practical use. Yet complex numbers are of good importance in a variety of applied fields; for example, the electrical engineer would, to say the least, be severely handicapped without them. The complex notation often simplifies setting up and solving vibration problems in either dynamical or electrical systems, and is useful in solving many differential equations which arise from problems in various branches of physics.
3
2. Real and imaginary parts of a complex number (
복소수의 실수와 허수 부분
)
z
x
iy x
: real part
y
: imaginary part (not imaginary!!)
4
3. Complex plane (
복소수 평면
)
- Complex plane: similar to the
xy
plane
5
-Rectangular form (
x,y
) vs. Polar form (
r,
) ( 직교형태 VS 극좌표 형태 )
x
r
cos ,
y
r
sin
r
z
x
iy
r
cos
ir
sin
x
2
y
2 , tan
y x
.
r
cos
i
sin
re i
(polar form).
6
Example) cf. : radian
7
4. Terminology and notation
real part imaginary : Re part : Im
x
y
Absolute angle of (or
z
: modulus) value
z
r
, ex)
z
1
i
principal angle
8
- Complex conjugate ( 켤레 복소수 ) complex conjugate
z z
r
cos
i
sin
x
iy
x
iy
.
r
cos
i
sin
re
i
.
9
5. Complex algebra (
복소수 연산
)
A. Simplifying to x+iy form
ex1.
1 2
i
i
2 1 2
i
1 2
i
.
ex2.
2 3
i i
2 3
i i
3 3
i i
6 3 9
i
2
i
2
i
i
2 5 5
i
10 1 2 1 2
i
ex3.
(polar form)
2
2
e i
/ 4
2 2
e i
/ 2 2
i
.
squaring ex4.
2
cos 20 1
i
sin 20
1 2
e i
20 1 2
e
i
20 1 2
cos 20
i
sin 20
10
B. Complex conjugate z
1
z
2
z
1
z
2 ,
z
f
ig
z
f
i g
( not
f
ig
) note) We can get the conjugate of an expression containing
i
’s by just changing the signs of all the
i
terms.
z
2
i
3
i
4
z
2
i
3
i
4 .
C. Absolute value z
re i
,
z
re
i
,
z z
r
2
e
i
i
r
2
z
2
z
z z
.
11
D. Complex equations
x
iy
2 2
i
,
x
2 2
ixy
y
2 2
i
x
2 2
xy
y
2 2 0
x
y
1 .
12
E. Graphs
ex.1
z
3 ,
z
x
iy
,
x
iy
3 ,
x
2
y
2 9 ex2.
z
1 9 , 2
y
2 9 2 ex3.
Angle of
z
: / 4 tan
y x
1
y
x
ex.4
Re
x
1 2 1 2
y
x
13
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Chapter 2 Complex numbers
Lecture 5 Euler formula & roots and powers
6. Complex infinite series (
복소수 무한 급수
)
S n
X n
iY n n
lim
S n
S
X
iY
, where lim
n
X n
X
,
n
lim
Y n
Y
.
In this case, we call the complex series convergent.
7. Complex power series; Disk of convergence (
복소수 멱급수
;
수렴 원판
)
a n z n
,
a n
: complex numbers ex.
1
z
z
2 2
z
3 3
z
4 4 For absolute convergenc e,
n
lim
n z
n
1
z
1 .
cf. real vs. complex
2.8 Elementary function of complex numbers (
복소수 기본함수
)
- elementary functions: powers, roots, trigonometric, inverse trigonometric, logarithmic, exponential, and combinations of these.
- Elementary functions of complex numbers behave just like those of real numbers
i
)
f
z
2 2
z
1 ,
f
2 2
1 1
ii
)
iii
)
e e z z
1 1
e z
2
z
z
2 2 !
e z
1
z
2
z
3 3 !
2.9 Euler’s formula
sin
e i
1
i
3 !
3 5 , cos 1 2 5 !
2 !
5 4 !
4 2 !
3 !
4 !
5 !
1
i
2 2 !
i
3 !
3 4 4 !
i
5 5 !
1 2 2 !
4 4 !
i
3 3 !
5 5 !
cos
i
sin
e i
cos
i
sin Euler' s fomula
z
x
iy
r
cos
i
sin
re i
Ex. Find the graph expressing a given
z
. 2
e i
/ 6 ,
e i
, 3
e
i
/ 2 ,
e
2
n
i
- Multiplication, division z
1
z
2
r
1
e i
1
r
2
e i
2
r
1
r
2
e i
1 2 ,
z
1
z
2
r
1
r
2
e i
1 2 .
ex.
2
2 2
e i
2
e
i
/ / 4 4 2 2
e i
/ 2 2
e
i
/ 4 2
e
3
i
/ 4 .
10. Powers and roots of complex numbers (
복소수의 멱수와 근
)
n
cos
i
sin
n
cos
n
i
sin
n
z n i
1 /
n
cos
n
i
sin
r n e in
,
1 /
n z
1 /
n
cos
n
1 /
n
i
sin
n
.
r
1 /
n e i
/
n
ex.1
cos
/ 10
i
sin
/ 10
25
e i
/ 10 25
e
2
i e i
/ 2
i
.
ex.2 Cube roots of 8?
8 8
i
0 8
e
2
k i
3 8 8
e
2
k i
1 / 3 8 1 / 3
e i
2
k
3 .
k k
0 , 1 ,
k
2 ,
z
2
z
2
e i
2 3 2 1 2
i
2 3 .
z
2
e i
4 3 2 1 2
i
2 3
ex.3 Find the plot all values of 4 64
r
1 / 4 64 1 / 4 2 4 2
k
4 2 4 , 3 4 , 5 4 , 7 4 .
ex. 4 6 8
i r
1 / 6 8 1 / 6 2 6 2
k
3 / 2 6 4 3
k
( 0 , 1 , 2 , 3 , 4 , 5 ).
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Chapter 2 Complex numbers
Lecture 6 application
11. The exponential and trigonometric functions (
지수함수와 삼각함수
)
- exponential function e z
e x
iy
e x e iy
e x
cos
y
i
sin
y
ex.
e
2
i
e
2
e
i
e
2
e
2 .
- trigonometric function e i
cos
i
sin , sin
e i
e
i
, 2
i e
i
cos
i
sin cos
e i
e
i
.
2 Similarly, sin
z
e iz
e
iz
, 2
i
cos
z
e iz
e
iz
.
2
12. Hyperbolic functions (
쌍곡함수
)
sinh
z
e z
e
z
, 2 cosh
z
e z
e
z
2 - The other hyperbolic functions are named and defined in a similar way to parallel the trigonometric functions: tanh
z
sinh
z
cosh
z
, sec h sin
iy z
1 cosh
z
,
i
sinh
y
, coth
z
1 tanh
z
csc h
z
cos
iy
1 sinh cosh
z y
.
.
cosh 2
z
sinh 2
z
1 ,
d dz
cosh
z
sinh
z
,
cf
.
cf
.
sin 2
z
cos 2
z
1
d dz
cos
z
sin
z
.
13. Logarithms (
로그함수
)
w
ln
z
ln
Ln
i
.
- Since has an infinite values (all differing by multiples of 2 ), a complex number has infinitely many logarithms. (principal value) ex. ln
Ln
2
n
i
,
i
, 3
i
, .
14. Complex roots and powers (
복소수 근과 멱수
)
- For complex
a
and
b
,
a b
e b
ln
a
- Since ln
a
is multiple values, powers a^b are usually multiple values (cf. principal value).
ex. 1 Find all values of i^(-2i) ln
i
ln 1
i
( / 2 2
n
)
i
( / 2 2
n
)
i
2
i
e
2
i
ln
i
e
2
i
i
/ 2 2
n
e
4
n
e
,
e
,
e
9 , .
15. Inverse trigonometric and hyperbolic functions (
역삼각함수와 역쌍곡함수
)
e
iz w
cos
z
e iz
2
z
arccos
w
16. Some applications (
응용
)
- Electricity V R
IR
,
V L
L dI dt
,
V
Q
dV C dt
I C
.
(method 1)
I
I
0 sin
t V R
RI
0 sin
t
,
V L
LI
0 cos
t
,
V C
1
C I
0 cos
t
.
Total voltage
V
V R
V L
V C
‘complicated function’
(method 2)
I
0 sin
t
Im Im
I
0
e i
t
.
After describing with a complex
I
, we can take the imaginary part of the solution.
V R
RI
0
e i
t
RI
,
V L
V C
i
LI
0
e i
t
1
i
C
I
0
e i
t i
LI
, 1
i
C I
.
V
V R
V L
V C
R
i
L
1
C
I
ZI
Impedence
Z
R
i
L
1
C
.
cf.
Resonance :
L
1
C
0