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Chapter 2 Complex numbers ( 복소수 )
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Lecture 4 Introduction of complex numbers
고등수학 10- 가 2 장 실수와 복소수에 나옴
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1. Introduction
.4168282) ,1
1number imaginary consider sLet'
22
iiiexi
i
ex.
.12
42
2
842
0222
iz
zz
negative. becan '4' sometimes, ,2
4
0
22
2
acba
acbbz
cbzaz
3
(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.
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2. Real and imaginary parts of a complex number ( 복소수의 실수와 허수 부분 )
iyxz
x: real party: imaginary part (not imaginary!!)
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3. Complex plane ( 복소수 평면 )
- Complex plane: similar to the xy plane
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form).(polar sincossincos
.tan,sin,cos 22
ireirirriyxz
x
yyxrryrx
-Rectangular form (x,y) vs. Polar form (r,) ( 직교형태 VS 극좌표 형태 )
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Example)
cf. : radian
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4. Terminology and notation
: of angle
, valuemodulus)(or Absolute
Im :part imaginary
Re :part real
z
rz
yz
xz
ex) iz 1
principal angle
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.sincossincos
. conjugatecomplex *
ireirirz
iyxiyxzz
- Complex conjugate ( 켤레 복소수 )
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5. Complex algebra ( 복소수 연산 )
A. Simplifying to x+iy form
.212121111 ex1. 22 iiiiiii
squaring
i
i
i
iii
ii
ii
i
i
2
1
2
1
10
55
9
236
33
32
3
2 ex2.
2
2
20sin20cos
2
1
2
1
2
1
20sin20cos2
1 ex4. 20
20ie
eii
i
.2221 form)(polar ex3. 2/24/2 ieei ii
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B. Complex conjugate
)not ( ,2121 igfgifzigfzzzzz
C. Absolute value
.,,222 zzzzrerzzrezrez iiii
note) We can get the conjugate of an expression containing i’s by just
changing the signs of all the i terms.
.4
32
4
32
i
iz
i
iz
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D. Complex equations
.122
022
,2
22
22
2
yxxy
yxiyixyx
iiyx
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E. Graphs
ex.1
9,3,
,3
22
yxiyxiyxz
z
ex2. 222 91,91 yxz
ex3.
xyx
y
z
1tan
4/: of Angle
ex.4 2
1Re xz
xy
2
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Chapter 2 Complex numbers
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Lecture 5 Euler formula & roots and powers
6. Complex infinite series ( 복소수 무한 급수 )
.lim,lim where,lim YYXXiYXSS
iYXS
nn
nn
nn
nnn
In this case, we call the complex series convergent.
7. Complex power series; Disk of convergence ( 복소수 멱급수 ; 수렴 원판 )
numberscomplex :, nn
n aza
.11
lim e,convergenc absoluteFor
4321
432
z
n
nz
zzzz
n
ex.
cf. real vs. complex
2.8 Elementary function of complex numbers ( 복소수 기본함수 )
2121)
!3!21)
111211,12)32
22
zzzz
z
eeeiii
zzzeii
iiifzzzfi
- 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
2.9 Euler’s formula
sincos!5!3!4!2
1
!5!4!3!21
!5!4!3!21
!4!21cos,
!5!3sin
5342
5432
5432
4253
ii
iii
iiiiiei
fomula sEuler' sincos iei
ireiriyxz sincos
Ex. Find the graph expressing a given z.
iniii eeee 22/6/ ,3,,2
.
,
21
2121
2
121
212121
i
iii
er
rzz
errererzz
- Multiplication, division
ex. i
i
1
1 2
.2
2
2
2
2
1
1 4/3
4/
2/
4/
24/2
i
i
i
i
i
ee
e
e
e
i
i
10. Powers and roots of complex numbers ( 복소수의 멱수와 근 )
ninnininnnin
nni
nni
errezerrez
ni
nie
ninie
//1/1/1
/1/1
,
.sincossincos
sincossincos
ex.1 .10/sin10/cos 2/22510/25 ieeei iii
ex.2 Cube roots of 8?
.8888088 3
23/13/1232
ki
ikik eeei
2
3
2
122,2
.2
3
2
122,1
2,0
3
4
3
2
iezk
iezk
zk
i
i
ex.3 Find the plot all values of 4 64
.4
7,
4
5,
4
3,
44
2
4
2264 4/14/1
k
r
ex. 4 6 8i
).5,4,3,2,1,0(346
2/32
6
28 6/16/1
kk
r
Chapter 2 Complex numbers
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Lecture 6 application
11. The exponential and trigonometric functions ( 지수함수와 삼각함수 )
yiyeeeee xiyxiyxz sincos
.2
cos,2
sin Similarly,
.2
cos,2
sin
sincos,sincos
iziziziz
iiii
ii
eez
i
eez
ee
i
ee
ieie
.1 ex. 2222 eeeee ii
- exponential function
- trigonometric function
12. Hyperbolic functions ( 쌍곡함수 )
2cosh,
2sinh
zzzz eez
eez
- The other hyperbolic functions are named and defined in a similar way to
parallel the trigonometric functions:
.sincos.,sinhcosh
1cossin.,1sinhcosh
.coshcos,sinhsin
.sinh
1h csc,
cosh
1hsec
tanh
1coth,
cosh
sinhtanh
2222
zzdz
dcfzz
dz
d
zzcfzz
yiyyiiy
zz
zz
zz
z
zz
13. Logarithms ( 로그함수 )
.lnln irLnrezw i
- Since has an infinite values (all differing by multiples of 2), a complex
number has infinitely many logarithms. (principal value)
ex. .,3,,211ln iiiniLn
14. Complex roots and powers ( 복소수 근과 멱수 )
abb ea ln
ex. 1 Find all values of i^(-2i)
- For complex a and b,
- Since ln a is multiple values, powers a^b are usually multiple values (cf. principal value).
.,,,
)22/()22/(1lnln9422/2ln22
eeeeeei
niniinniiiii
15. Inverse trigonometric and hyperbolic functions ( 역삼각함수와 역쌍곡함수 )
wzee
zwiziz
arccos2
cos
16. Some applications ( 응용 )
- Electricity
.,,C
I
dt
dV
C
QV
dt
dILVIRV LR
CLR
CLR
VVVV
tIC
VtLIVtRIV
tII
voltageTotal
.cos1
,cos,sin
sin
000
0
(method 1)
‘complicated function’
(method 2)
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,
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