複素関数・同演習第 2 -...

http://nalab.mind.meiji.ac.jp/~mk/complex/
2020 9 23

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+α 3
3 Hamilton ( Q, R, . . . ) (), Gauss
√

[1] §1.7 ( PDF):
C : C := {a+ bi | a, b ∈ R}. Cardano 3
> 0 → 3 () p, q → ()
1 9 29 13:30.

1
x − yi
x2 + y 2
x + yi 1
x + yi a, b, c (b, c = 0)
a b = ac
x − yi
(x + yi)

(1) Hamilton R2
(a, b) + (c , d) = (a+ c , b + d),
(a, b) · (c , d) = (ac − bd , ad + bc)
(R2; +, ·) (0, 0). (x , y) −(x , y) = (−x ,−y). (1, 0). (x , y)
(x , y)−1 =
x2 + y2
) R2 C R2 (a, b) a+ bi

K , 0K (1) (∀a, b, c ∈ K) (a+ b) + c = a+ (b + c)
(2) (∃0K ∈ K) (∀a ∈ K) a+ 0K = 0K + a = a
(3) (∀a ∈ K) (∃a′ ∈ K) a+ a′ = a′ + a = 0K
(4) (∀a, b ∈ K) a+ b = b + a
(5) (∀a, b, c ∈ K) (ab)c = a(bc)
(6) (∃1K ∈ K) (∀a ∈ K) a1K = 1Ka = a
(7) (∀a ∈ K \ {0K}) (∃a′′ ∈ K) aa′′ = a′′a = 1K
(8) (∀a, b, c ∈ K) (a+ b)c = ac + bc, a(b + c) = ab + ac
(9) (∀a, b ∈ K) ab = ba
K = Q,R,C K = H () (1)–(8) (9)

C :=
a −b b a
) = aI + bJ, J2 = −I .
( e iθ ( cos θ − sin θ sin θ cos θ
) (
) )
()
(3) C R[x ] (x2 + 1) :
C = R[x ]/(x2 + 1).
(x2 + 1) R[x ] C ……
[2] ( )

Hamilton (1805–1865)
(quaternion) H = {a+ bi + cj + dk | a, b, c , d ∈ R} ,
i2 = j2 = k2 = ijk = −1.
( ij = k , jk = i , ki = j )
H (the skew field of Hamilton quaternions) [3], [4]

C (): (1 )
Q, R, C Q R (2 ) : C
() Hamilton H


1.5 2.1 ()
c z2 = c z ∈ C c (square root of c) :
√ R (
) C √
c

1.5 2.2 ()
c c z c = 0 z = 0 c = 0 c 2 ( −1)c = a+ bi (a, b ∈ R)
(1) z =
± (√√
|b|

1.5 2.3
z2 = 1− i z (1− i ) () z = x + yi (x , y ∈ R) z2 = x2 − y2 + 2xyi
z2 = 1− i ⇔ x2 − y2 = 1 ∧ 2xy = −1.
2xy = −1 y = − 1 2x . x2 − y2 = 1
4x4 − 4x2 − 1 = 0.
x2 = 2 +
√ 22 + 4
()

z = x + yi = ±
(√ 2 √ 2 + 2

1.5 √
(1) c ∈ R x2 = c x ∈ R c ≥ 0
(2) c ≥ 0 x x x ≥ 0 1 √
c (3) c = 0 c 0 √
c = 0. (4) c > 0 c √
c − √ c 2
(5) c1, c2 ≥ 0 √ c1 √ c2 =
√ c1c2.
√ −c i ( √
−1 = i ,√ −3 =
√ 3i).
√
√

c1 √ c2 =
√ c1c2
√ −c i √

() √
√ c c 1 ()
(b) √ c c ( )
(c) √ c c
√ −3 √
√ 3i

1.5 2 2.4 ( 2)
a, b, c ∈ C, a = 0 2 az2 + bz + c = 0
(2) z = −b ±
√ b2 − 4ac b2 − 4ac (2
z ) D := b2 − 4ac D = 0 2, D = 0 1 ()
()
b
2a
)2
4a2
± √ b2 − 4ac

1.6 (3) z = x + yi (x , y ∈ R)
(the complex conjugate of z) z (4) z := x − yi
(5) z = z .
(6) z + w = z + w , z − w = z − w , zw = z w , ( z
w
(3 z = x + yi , w = u + iv 4 3)
(7) x = z + z
2i .
x , y z , z (8) z = z ⇔ z ∈ R.

1.6
1 + 2i ax2 + bx + c = 0 (a, b, c ∈ R) 1− 2i 1 + 2i ax3 + bx2 + cx + d = 0 (a, b, c, d ∈ R) 1− 2i
2.5 ()
n ∈ N, a0, a1, . . . , an ∈ R f (z) = a0z n + a1z
n−1 + · · ·+ an−1z + an c ∈ C f (c) = 0 f (c) = 0.
f (z) = a0zn + an−1zn−1 + · · ·+ an−1z + an
= a0zn + a1zn−1 + · · ·+ an−1 z + an
= a0 (z) n + a1 (z)
n−1 + · · ·+ an−1 z + an
= a0 (z) n + a1 (z)
n−1 + · · ·+ an−1z + an
x= f (z)
f (c) = 0 ⇔ f (c) = 0 ⇔ f (c) = 0.

√ x2 + y2.
R2 () (x , y) (, )
(10) |−z | = |z | , |z | = |z | (),
(11) zz = |z |2
zz = (x + yi)(x − yi) = x2 − (yi)2 = x2 + y2 =
(√ x2 + y2

(1) |z | ≥ 0. ⇔ z = 0.
(2) |z + w | ≤ |z |+ |w |
(3) |zw | = |z | |w |. ( z w
= |z| |w | )
(4) |z − w | ≥ |z | − |w |
. (1) R2 x ≥ 0. ⇔ x = 0 (2) R2 x + y ≤ x+ y (3) z = a+ bi , w = c + di
|zw |2 = zwzw = zwz w = zzww = |z |2|w |2 = (|z | |w |)2 √
(4) |z | = |z − w + w | ≤ |z − w |+ |w | |z | − |w | ≤ |z − w |. z w |w | − |z | ≤ |w − z | = |z − w |.
|z | − |w | = max{|z | − |w | , |w | − |z |}
≤ |z − w |.
1 PDF Oh-o! Meiji
, A4 PDF, 2 PDFLATEX Word PDF, 1 21 PDF 1 Mac 1

c1 √ c2 =
√ c1c2
√ −c i √
c1 √ c2 =
√ c1c2.
(2) c1 = −1, c2 = −1 √ c1 √ c2 = i · i = −1,
√ c1c2 =

http://nalab.mind.meiji.ac.jp/~mk/lecture/complex-function-2020/
complex2020.pdf (2014).
[3] , (2007).
[4] , (2016/12/1).

(), Gauss

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複素関数・同演習第 2 -...

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