• ೾৔ॅ z = x + iy ʂও਻ખʦ r , ഫҺʦ θ ɼ ɭʟɼ

ۆ֩ 212 _ۆ׋ॅӁ IV

੓ 3 _Ѿ

೾৔ॅ ( _ਞɡ )

ঽѾʂ೹ࡁ (1)

• ^೾৔ॅ z = x + iy ʂও਻ખʦ r , ഫҺʦ θ ɼ ɭʟɼ

⊲ r = p

x ² + y ² ,

⊲ x 6 = 0 ɾʝ tan θ = y x

⊲ z = r (cos θ + i sin θ) ( ׎آߧ )

ঽѾʂ೹ࡁ (2)

• z ʂও਻ખʦ | z | , ഫҺʦ arg z ɼɘɚՒ܃ɻ

಺ɭ

• ^ഫҺ arg z ɿʃ 2π ʂय़ॅఝʂ೑଑ड़ɠɖʟ

• ഫҺʂ೑଑ʂೱഇʦ૛य़ɫɺҺ୞ɠ ( − π, π ]

ɿ࠴ʒʟʜɚɿɫɳʖʂʦഫҺʂࠞખɼɘɘ ,

ঽѾʂ೹ࡁ (3)

• e ^iθ = cos θ + i sin θ ( ʱʫ˰Ɵʂڷߧ )

• ^{މҺԟॅʂю൚଑๲} :

⊲ cos(θ + ϕ) = cos θ cos ϕ − sin θ sin ϕ

⊲ sin(θ + ϕ) = sin θ cos ϕ + cos θ sin ϕ

1/z _ʂ׎آߧ

z = re ^iθ = r (cos θ + i sin θ) ɿ਻ɫ , 1

z = 1

r e ^−iθ = 1

r (cos θ − i sin θ) ɼɾʟ .

๲ศ (re ^iθ ) × ( ¹ _r e ^−iθ ) = (r × ¹ _r )e ^{i(θ+(−θ))} = 1

z = re ^iθ _ɿ਻ɫ 1

z = 1

r e ^{− iθ}

ഫҺɼও਻ખʂड़߸ (1) (p.24)

଑๲ 1.3 (1) z = r (cos θ + i sin θ), w = ρ(cos ϕ + i sin ϕ) ɿ਻ɫ , Ιчɠढ़๼ɭʟ :

| zw | = | z || w | , arg(zw ) = arg z + arg w,

z w

= | z |

| w | , arg z w

= arg z − arg w

ഫҺɼও਻ખʂड़߸ (2) (p.24)

଑๲ 1.3 (2) z = r (cos θ + i sin θ), w = ρ(cos ϕ + i sin ϕ) ɿ਻ɫ , Ιчɠढ़๼ɭʟ :

(r (cos θ + i sin θ)) (ρ(cos ϕ + i sin ϕ))

= rρ (cos(θ + ϕ) + i sin(θ + ϕ))

|| z | − | w || ≤ | z + w | ≤ | z | + | w | ( މҺ೑୾ߧ )

˴Ɵ ρ

ʵ˱ʾʩഓߓ

ഫҺɼও਻ખʂड़߸ (3) (p.24)

଑๲ 1.3 ʂ঑෗ (1)

• ʒɮߘʂɧɼɿશΤɭʟ : z = r (cos θ + i sin θ ), w = ρ(cos ϕ + i sin ϕ) ɴɟʝ

r = | z | , θ = arg z, ρ = | w | , ϕ = arg w

• z = re ^iθ , w = ρe ^iϕ ɼࢆɡ૥ɭ

ഫҺɼও਻ખʂड़߸ (4) (p.24)

଑๲ 1.3 ʂ঑෗ (2)

• zw = (re ^iθ ) × (ρe ^iϕ ) = rρe ^i(θ+ϕ) ɴɟʝ

⊲ | zw | = rρ = | z || w |

⊲ arg(zw ) = θ + ϕ = arg z + arg w

ഫҺɼও਻ખʂड़߸ (5) (p.24)

଑๲ 1.3 ʂ঑෗ (3)

• _w ¹ = ¹ _ρ e ^−iϕ ɴɟʝ

z

w = (re ^iθ )( ¹ _ρ e ^−iϕ ) = _ρ ^r e ^i(θ−ϕ) , ɫɳɠɷɺ

⊲ ^z

w

= _ρ ^r = _|w| ^|z|

⊲ arg ! _z

w

= θ − ϕ = arg z − arg w

ഫҺɼও਻ખʂड़߸ (6) (p.24)

଑๲ 1.3 ʂ঑෗ (4)

(r(cos θ + i sin θ)) (ρ(cos ϕ + i sin ϕ))

= rρ (cos(θ + ϕ) + i sin(θ + ϕ))

ɧʠʃ re ^iθ ρe ^iϕ = rρe ^i(θ+ϕ) ʦࢆɡ૥ɫɳɴɥ

ഫҺɼও਻ખʂड़߸ (7) (p.24)

• | z + w | ≤ | z | + | w | ( މҺ೑୾ߧ )

๲ศ މҺآɿԟɫ ,

ɖʟരʂૠɩ ≤ ^਩ʂ 2 രʂૠɩʂ໳

( ߘˡƟʿɿय );

ॅߧʦޡɷɳ঑෗ʃָѕࢆ 25 ˡƟʿ ( үߢɻ஬

ʧɻɞɣɧɼ )

֚߫

z w

z + w

| z |

| w |

| z + w |

ഫҺɼও਻ખʂड़߸ (8) (p.24)

• | z | − | w | ≤ | z + w | , | w | − | z | ≤ | w + z |

๲ศ z = (z + w ) − w ɴɟʝ

| z | ≤ | z + w | + | − w | = | z + w | + | w | , ʜɷɺ

| z | − | w | ≤ | z + w |

z ɼ w ʦ௡ʠԀɜʟɼ੓ 2 ʂ೑୾ߧ

ഫҺɼও਻ખʂड़߸ (9) (p.24)

ʒɼʕɺƧ

• || z | − | w || ≤ | z + w | ≤ | z | + | w |

( މҺ೑୾ߧ )

• ^଑๲ 1.3 ʂ࣍෗ʃʱʫ˰Ɵʂڷߧʦޡɷɳ൘

ɠɮɷɼԑ੺

ːƉ˩ʩ˝˲ʂڷߧ (1)

଑๲ 1.4 ( ːƉ˩ʩ˝˲ʂڷߧ )

(cos θ + i sin θ) ⁿ = cos nθ + i sin nθ

๲ศ !

e ^iθ n

= e ^inθ ɼɾʟɧɼʦɘɜʄʜɘ . ɧ ʠʦʱʫ˰Ɵʂڷߧɿɫɳɠɷɺࢆɡ૥ɭɼ଑

๲ 1.4 ɿɾʟ ( ࣏ݔʃॅӁଯՇ௽൚ , ߘˡƟʿ ).

ːƉ˩ʩ˝˲ʂڷߧ (2)

• n = 0 ʂɼɡ : ຑരɼʖɿ 1 ɴɟʝ୾ߧढ़๼

• n ≥ 0 ɿ਻ɫ ( e ^iθ ) ⁿ = e ^inθ _{ʦщ଑ɭʟɼ} ( e ^iθ ) ⁿ⁺¹ = ( e ^iθ ) ⁿ e ^iθ = e ^inθ e ^iθ = e ^inθ+iθ = e ^i(n+1)θ _ɼɾʞ , ( e ^iθ ) ⁿ⁺¹ = e ^i(n+1)θ _ɠɘɜʟ

• n < 0 ʂɼɡʃ n = − m _{ɼɫɺࣘՒʦޡɚɼ}

( e ^iθ ) ⁿ = ( e ^iθ ) ^(−m) = (( e ^iθ ) ⁻¹ ) ^m = ( e ^imθ ) ⁻¹ =

e ^−imθ = e ^inθ ( ɳɴɫߘˡƟʿɻࡧʍʟड़߸ʦ

೦ʂʍɡʂड़߸ n ≥ 0 ɿ਻ɫ , z ⁻ⁿ = ( z ⁻¹ ) ⁿ

๲ศ ָѕࢆ 15 ˡƟʿɻʃ , z ⁻ⁿ = ( z ⁿ ) ⁻¹ ɼ଑

զɫɳ . ɧʠɠ ( z ⁻¹ ) ⁿ ɼλઠɭʟɧɼʦ٪ɳɘʂ ɻɖʟɠ , ( z ⁻¹ ) ⁿ z ⁿ = ( zz ⁻¹ ) ⁿ = 1 ɼɾʟɧɼɟʝ ( z ⁻¹ ) ⁿ = z ⁻ⁿ _{ɻɖʟɧɼɠʣɟʟ} ( ɳɴɫߘˡƟʿ ɻࡧʍʟड़߸ʦޡɷɺɘʟ )

ɧʂˡƟʿʂ௅฻ʃݔɟɘ໴ɾʂɻ๲ѽɫɾɣɺʜɘ

ঀʂʍɡ α, β ∈ C , n ≥ 0 ɿ਻ɫ , ( αβ ) ⁿ = α ⁿ β ⁿ

๲ศ ॅӁଯՇ௽൚ɿʜʟ . n = 0 ʂɼɡ ʃຑരɼʖɿ 1. n ɻ୾ߧɠढ़๼ɭʟɼщ଑ɭ ʠʄ , ( αβ ) ⁿ⁺¹ = ( αβ ) ⁿ ( αβ ) = ( α ⁿ β ⁿ )( αβ ) = ( α ⁿ α )( β ⁿ β ) = α ⁿ⁺¹ β ⁿ⁺¹

ɧʂˡƟʿʂ௅฻ʃݔɟɘ໴ɾʂɻ๲ѽɫɾɣɺʜɘ

• ^ຶ 1.7 ʃүߢɻ஬ʧɻɞɣɧɼ

֜๸ɼयآ (1) (p.34)

଑զ 1.13 ೾৔ॅ z ɼ w ʂ֜๸ʦ | w − z | ^ɿ ʜʞ଑զɫ , ɧʠʦ d(z, w ) ɼࢆɣ :

d(z, w) = | w − z |

֜๸ɼयآ (2) (p.35)

଑๲ 1.5 z, w, u ʦ೾৔ॅɼɫɳɼɡ

(1) d(z, w) ≥ 0, ୾܃ढ़๼ʃ z = w ʂɼɡʂʓ (2) d(z, w) = d(w, z )

(3) d(z, w) ≤ d(z, u) + d(u, w )

๲ศ (1)(2) ʃ଑զɟʝʣɟʟ ; (3) ʃމҺ೑୾

ߧɱʂʖʂ

֜๸ɼयآ (3) (p.35)

α ʦ೾৔ॅ , k ʦ।ʂ߹ॅɼɫɳɼɡ ,

• | z − α | = k ʦභɳɭ೾৔ॅʦࡍʕɳʖʂʃ α ʦયअɼɭʟౝأ k ʂЌɿλઠɭʟ

• | z − α | < k ʦභɳɭ೾৔ॅʦࡍʕɳʖʂʃ α

ʦયअɼɭʟౝأ k ʂҔЌౣɿλઠɭʟ ( Ҕ

ЌౣɼʃЌ࠵ʦ࢑ɘɳЌౣʂɧɼ )

֜๸ɼयآ (4) (p.35)

α ʦ೾৔ॅ , k ʦ।ʂ߹ॅɼɫɳɼɡ ,

• | z − α | ≤ k ʦභɳɭ೾৔ॅʦࡍʕɳʖʂʃ α

ʦયअɼɭʟౝأ k ʂടЌౣɿλઠɭʟ ( ട

ЌౣɼʃЌ࠵ʦԥʔЌౣʂɧɼ )

• z = x + iy , α = a + ib ɼɭʟ

• ^ച෣ɿ P (x, y ) ɼ Q(a, b) ʦࠟʟ .

• | z − α | ^ʃ P (x, y ) ɼ Q(a, b) ɼʂ֜๸ .

• ^ɧʂɳʕ , Ιࣘɻࡧʍɳɧɼʃച෣௅ʂЌ , Ҕ

Ќౣ , ടЌౣʂड़߸ɼஔλ .

• ^ຶ 1.8 ʃүߢɻ஬ʧɻɞɣɧɼ .

n _ࣛܧ (1) (p.40)

଑զ 1.14 z ∈ C , n ∈ N _ɿ਻ɫ , w ⁿ = z ʦභ ɳɭ೾৔ॅ w ʦ z ʂ n ࡚ܧɼɘɘ , z ^1/n ɖʟɘʃ

√

z ɻɖʝʣɭ .

ߢীॅুਹ N

n _ࣛܧ (2)

n ࣛܧʂ঑෗ʂࡵಖʂɳʕɿ n ࣛʂड़߸ʦ೹ࡁɭʟ . z = re ^iθ ɼɭʟɼ z ⁿ = r ⁿ e ^inθ ɴɟʝ

| z ⁿ | = | z | ⁿ , arg (z ⁿ ) = n arg z

n _ࣛܧ (3)

ɳɼɜʄ z = 1.2e ^iπ/3 ɾʝ

z ² = 1.44e ^i2π/3 , z ³ = 1.728 e ^iπ

|{z}

3π/3=π

, z ⁴ = 2.0736e ^i4π/3

0 _߹߫

֚߫

z ² z z ³

z ⁴

଑๲ 1.9 z ∈ C , n ∈ N _ɿ਻ɫ , z 6 = 0 ɻɖʠ ʄ , z ʃ n ڃʂ n ࣛܧʦߖɸ . z = re ^iθ ɼɫɳɼ ɡ , w ₀ , w ₁ , . . . , w _n−1 ʦ n ڃʂܧɼɭʟɼ ,

w _k = √

re ⁱ⁽

⁺

⁾

ɼࢆɥʟ ( √

r ʃ r > 0 ʂ߹ॅʂ౮Ξɻʂಒ೦ʂ

n ࣛܧ ).

n _ࣛܧ (4) (p.41)

଑๲ 1.9 ʂ๲ศ w _k ⁿ =

√

re ⁱ⁽

⁺

⁾ n

=

re ^{i(θ+2kπ )} = re ^iθ ɴɟʝ , w ₀ , . . . , w _n−1 ʃ z ʂ

n ࣛܧɻɖʟ . ʒɳ , e ⁱ

, e ⁱ

, e ⁱ

,

. . . , e ⁱ

ʃ੺ΛЌʦ n ୾ഇɭʟɟʝ ,

w ₀ , . . . , w _n−1 ʃɭʍɺΪɾʟ . w ₀ , . . . , w _n−1 Ιҙ

ɿܧɠɾɘɧɼʃ੐ॅӁʂԷඕ଑๲ ( ָѕࢆ 213

ˡƟʿ ʂՇٌɻɖʟ

଑๲ 1.9 ʂߧ

w _k = √

re ⁱ⁽

⁺

⁾

ʦʱʫ˰Ɵʂڷߧʦޡɷɺࢆɡ૥ɭɼ w _k = √

r

cos

θ

n + 2kπ n

+ i sin

θ

n + 2kπ n

ɼɾʟʂɻ , ܞࡧʍɳɧɼʃָѕࢆ 40 ˡƟʿɼஔ

n _ࣛܧ (5)

ຶ : z = 2i = 2e ^iπ/2 ʂ 3 ࣛܧ w ₀ , w ₁ , w ₂ ʦظޔɭ ʟɼ

w ₀ = √

2e ⁱ

, w ₁ = √

2e ⁱ⁽

⁺

⁾ , w ₂ = √

2e ⁱ⁽

⁺

⁾ = √

2e ⁱ⁽

⁺

⁾

ഫҺʦࠞખɿ૥ɫɺɘɾɘʂɻશΤ

0 _߹߫

֚߫

π/6 2π/3

2π/3

w ₀ w ₁

w

1

n _ࣛܧ (6)

ຶ : z = i = e ^iπ/2 ʂ 2 ࣛܧ w ₀ , w ₁ ʦظޔɭʟɼ w ₀ = e ⁱ

,

w ₁ = e ⁱ⁽

⁺

⁾ = e ⁱ⁽

^+π)

( ഫҺʦࠞખɿ૥ɫɺɘɾɘʂɻશΤ )

0 _߹߫

֚߫

1 w ₁

w ₂

π/4

π

n _ࣛܧ (7) (p.41 ∼ 42)

• ೾৔ॅʂ౮Ξɻ۪ɜʟɼɡɿʃ i ^1/2 ʃ 2 ڃʂ

೾৔ॅ e ^iπ/4 , e ^i(π/4+π) ʦʒɼʕɳ಺Ւɻ੺λ ʂ೾৔ॅɻʃɾɘ

• ^{ɧʠʦ൳ʠʟɼָѕࢆ} 41 ˡƟʿຶ 1.9 ʂʜɚ

ɾܩ๥ɠ౎१ɭʟ ( үߢɻɧʂຶʦ஬ʧɻɞ

ɣɧɼ )

n _ࣛܧ (8) (p.42)

Ւ܃ʂޡɘഇɥ

√

x ಒ೦ʂ߹ॅ x ʂ , ಒ೦ʂ߹ n ࣛܧ

z

೾৔ॅ z ʂ೾৔ॅʂ౮Ξɿɞɥʟ n ڃ

ʂ n ࣛܧ , ɖʟɘʃɱʂયʂʆɼɸ

• ^{ಒ೦ʂ߹ॅ} x ɿ਻ɫɺ , ɱʂಒ೦ʂ n ࣛܧ √

x ɠɳɴλɸਣݚɭʟ ( ɧʠʃಘഇঀഇӁʂ౮ Ξɻ঑෗ɩʠʟߏ߹ )

• ^ָѕࢆશΤ 1.16 ʃƹಒ೦ʂ߹ॅ x ʂಒ೦ʂ n

ࣛܧƺɼࢆɟɾɘɼ೑।ҷ

• √

z ɼ z

ʂޡɘഇɥʦɫɾɘָѕࢆʖɖʟ

ʂɻ , ਩ʂָѕࢆʦ஬ʔɼɡɿʃܩ๥ɫɾɘ

• ^ຶ 1.10 ʃүߢɻ஬ʧɻɞɣɧɼ

Ѿ໚๲໱ɼʂԟؙ (1)

೾৔ѽॾ Ѿ໚๲໱

֚ॅ੺Λ i j

೾৔ॅ z Z ˙

Ѿ໚๲໱ɼʂԟؙ (2)

• ^{Ѿ໚๲໱ɿʃ} i ɠ୊ກʦ಺ɭɼɘɚӾຶɠɖ ʟɳʕ , ܩ๥ɫɾɘʜɚ֚ॅ੺Λɼɫɺ j ʦ ๆɘʟ

• ^ɧʂ۳զɻʃ , ωɡਞɡ , ֚ॅ੺Λɼɫɺ i ʦ

ޡɘਞɥʟ

Ѿ໚๲໱ɼʂԟؙ (3)

ߗ܌ʦ t ɼɭʟɼ , ।ٸఉڱກɿɞɥʟڱກ୊΄

v(t) ʃߘʂʜɚɿࢆɥʟ :

v (t) = V _m sin(ωt + φ)

Ւ܃ ෕ࣁ ੺Λ

V _{m ୊΄ʂ݂੒इ೺} V

ω _Һ࠵ఉॅ rad/s

φ _Λ৷Һ rad

Ѿ໚๲໱ɼʂԟؙ (4)

v(t) = V _m e ^i(ωt+φ) = V _m (cos(ωt + φ) + i sin(ωt + φ))

ɼɘɚߧʦ۪ɜʟ

Ѿ໚๲໱ɼʂԟؙ (5)

।ٸఉڱກɼ V _m e ^i(ωt+φ) ʦಅҼɭʟɼƧ

v (t) = V _m sin(ωt + φ)

v (t) = V _m cos(ωt + φ)+ iV _m sin(ωt + φ)

֚ॅढ़ഇɠλઠɫɺɘʟ

Ѿ໚๲໱ɼʂԟؙ (6)

଑զ

v (t) = V _m e ^i(ωt+φ) = V _m (cos(ωt+φ)+i sin(ωt+φ)) ɿʜɷɺ଑զɩʠʟ୊΄ʦ೾৔୊΄ɼɘɚ

• ^{֚ॅढ़ഇʦڱກ୊΄} (sin(ωt + φ)) ɼ܄ʣɯɺ

ɘʟʂʃ໅ޤଯرέɼްʣʠʟ

߹ڹખɿԟɭʟ೹ࡁ (1)

ڱກ୊΄ v (t) ɿʜɷɺଘې R ɻࢶಐɩʠʟ୊ຢɼ , ஔɬଘې R ɿ૥ກ୊΄ʦτюɫɳɼɡɿࢶಐɩʠ ʟ୊ຢɼɠ୾ɫɣɾʟʜɚɾ૥ກ୊ກ ( ɖʟɘʃ૥

ກ୊΄ ) ʂ੒ɡɩʦ߹ڹખɼڅʊ ( ਘ๼ , ऊ , ୊ՉѾ

໚ʂԷ৐ , ୱ֣୊Ն੒Ӂࡥ౦׌ , 2007).

• ^࠵Ճɠ T ɾʝ I =

q 1 T

R _T

0 i ² (t)dt

߹ڹખɿԟɭʟ೹ࡁ (2)

• ^{।ٸఉڱກɻʃ} , i(t) = I _m sin ωt ɼɭʟɼ , T = 2π/ω ɻ , މҺԟॅʂड़߸ʦޡɚɼ ,

I =

s

I _m ² T

Z _T

0

( 1

2 (1 − cos 2ωt))dt = I _m

√ 2 ɴɟʝ , ߹ڹખ I ʃ I _m ʦ √

2 ɻӍɷɳʖʂ ( ୊

΄ɻʖஔɬ

Ѿ໚๲໱ɼʂԟؙ (7)

୊΄ʂ߹ڹખʦ V ɼɫɺ (V = V _m / √

2), ೾৔୊΄

ʂߧʦࢆɡԀɜʟɼ , v (t) = √

2V e ^i(ωt+φ) = √

2V e ^iφ e ^iωt

ɼʖࢆɟʠʟ .

Ѿ໚๲໱ɼʂԟؙ (8)

଑զ ೾৔୊΄ʦ߹ڹખɿમෳɫɺࢆɡ૥ɫ ( √

2 ʦ๿ɭ ), ɩʝɿߗԝɿΜਣɭʟ۽ e ^iωt ʦࢽ

๿ɫɳʖʂ :

V = V e ^iφ

ʦ , ˜ʮƟʽ୊΄ɼڅʊ .

Ѿ໚๲໱ɼʂԟؙ (9)

• ^{˜ʮƟʽ୊΄} V = V e ^iφ ʦ V = V ∠ φ ɼʖࢆɣ

• ˜ʮƟʽ൚ɿԟɭʟՒࡧʃ୊ՉѾ໚ʂָѕࢆ

ɿʜɷɺʒɵʒɵɾʂɻશΤɭʟɧɼ

Ѿ໚๲໱ɼʂԟؙ (10)

ʒɼʕ

ڱກ୊΄ v (t) = √

2V sin(ωt + φ)

೾৔୊΄ v (t) = √

2V e ^i(ωt+φ)

˜ʮƟʽ୊΄ V = V e ^iφ