K = AB
:< ą, � > t = [x(t), y(t)] " K �" R2
r r(t)

D �" C f : D C K �" D f(z)dz
K
" < ą, � > ą = t0 < t1 < t2 < ... < tn = �
" zk-1zk k = 1, 2, ..., n, zk = z(tk)
" �k " zk-1zk"zk = zk - zk-1
n
" Sn = f(�k)"zk
k=1
< ą, � >
�k

f K f(z)dz
K

f g K = AB a, b " C
K
af + bg
K af(z) + bg(z)dz = a K f(z) + b K g(z)dz.

f(z)dz = f(z)dz
AB BA


C " AB f(z)dz = f(z)dz + f(z)dz
AB AC CB

f AB z = z(t) t "< ą, � >

�
f(z)dz = f(z(t)) � z (t)dt
AB ą

zdz
Ż
AB
z(t) = x(t) + iy(t) = t + it t "< 0, 1 >
z (t) 1 + i f(z(t)) z(t) = t - it
= =
1 1 1
zdz = f(z(t)) � z (t)dt = (t - it)(1 + i)dt = (1 + i)(1 - i) tdt = 2 � 0, 5 = 1
Ż
AB 0 0 0
D �" C f : D C
F : D C f D

"z " D : F (z) = f(z).

F " H(D) f = F (z) " C(D) K �" D
z1 z2

f(z)dz = F (z2) - F (z1)
K
"
"
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 D �" C f " H(D) K " D

f(z)dz = 0
K

f D K �" D f(z)dz = 0
K

"z " D F (z) = f(z)
n-1

D = (D0 *" K0) - Di
i=1
"i = j, Di )" Dj = 0 "i : Di �" D0 "Di = Ki, i = 0, 1, ..., n - 1 Ki

Di f " H(D)

n-1

f(z)dz = f(z)dz.
K0 Ki
i=1
f D "D
z " D

1 f(�)
f(z) = d�
2Ąi � - z
"D

z - 2
dz
z + 1
|z|=2

f(�)
d� = 2Ąif(z)
"D �-z
D = {z : |z| < 2} z = -1 " D f(�) = � - 2

z - 2 f(�) � - 2
dz = d� = d� = 2Ąi � f(-1) = 2Ąi � (-1 - 2) = -6Ąi
z + 1 � - z � - (-1)
|z|=2 "D "D
f D "D
z " D

n! f(�)
f(n)(z) = d�
2Ąi (� - z)n+1
"D

dz
dz
(z2 + 9)2
|z-2i|=2

f(�)
2Ąi
d� = f(z)
"D (�-z)n+1 n!
1
D = {z : |z - 2i| < 2} z = 3i " D f(�) =
(�+3i)2

1
2
dz f(�) 2Ąi Ą
(�+3i)2
dz = d� = d� = �f(1)(3i) = 2Ąi�(-2)(3i+3i)-3 =
z2 + 9 (� - z)n+1 (� - 3i)2 (2 - 1)! 54
|z-2i|=2 "D "D

f (�) = (� + 3i)-2 = -2(� + 3i)-3
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 D �" C

f " C(D) K �" D f(z)dz = 0 f " H(D)
K
ex
"z " C ez := ex(cos y + i sin y)
" ez "z " C
" |ez| = ex
" (ez) = ez
1 1 2
" ez +z2 = ez � ez
" "z " C ez = 0

" ez+2kĄi = ez k " Z T = 2Ąi
n
z
" "z " C ez = limn" 1 +
n
eiz + e-iz eiz - e-iz
"z " C cos(z) := sin(z) :=
2 2i
" "z " C
" (cos z) = - sin z (sin z) = cos z
" cos2 z + sin2 z = 1
" T = 2Ą
ez + e-z ez - e-z
"z " C ch(z) := sh(z) :=
2 2
" "z " C
" cos(iz) = chz sin(iz) = ish(z)
" (chz) = shz (shz) = chz
" ch2z - sh2z = 1
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