z1 = x1 + iy1 z2 = x2 + iy2
z1z2
x1 y1 - z1z2
= .
x2 y2
2i
z1, z2, z3
1 1 1
D = i z1 z2 z3
z1 z2 z3
V, W K
T : V W K
K
T (v1 + v2) = T (v1) + T (v2) v1, v2 " V
T (av) = aT (v) a " K, v " V
C z z R C
T : C C R
z " C
T (z) = z + µz,
 = (T (1) - iT (i))/2 µ = (T (1) + iT (i))/2
T : C C C
R µ = 0
a b
A =
c d
A
R T : C C
x
T (x + iy) = A .
y
T C a = d
-b = c
w, z " C
w, z := Re wz.
 w, z w z
R2
aw, az = |a|2 w, z , a " C
w, z = w, z
2 2
w, z + iw, z = |w|2|z|2
| w, z | |w||z|
|w + z|2 = |w|2 + |z|2 + 2 w, z
|w + z| |w| + |z|
w, z " C |w|, |z| < 1
w - z |w| - |z| w - z
< 1
1 - wz 1 - |w||z| 1 - wz
a, b " C.
z2 + az + b = 0
C
(1+i)4m (1+i)4m+2
Õ " R \ {Ä„/2 + kÄ„ : k " Z} m " Z,
m
1 + i tan Õ 1 + i tan mÕ
= .
1 - i tan Õ 1 - i tan mÕ
sin 3¸ = 3 sin ¸ - 4 sin3 ¸
cos 4¸ = 8 cos4 ¸ - 8 cos2 ¸ + 1
sin 5¸ = 5 sin ¸ - 20 sin3 ¸ + 16 sin5 ¸
sin6 ¸ + cos6 ¸ = (3 cos 4¸ + 5)/8
zn + z-n = 2 cos n¸, z = ei¸
z " C \ {1}
1 - zn+1
1 + z + · · · + zn = .
1 - z
n
1
sin n + ¸
1
2
cos k¸ = +
1
2 2 sin ¸
2
k=0
n
1
cos n + ¸
1 1
2
sin k¸ = cot ¸ -
1
2 2 2 sin ¸
2
k=0
0 < ¸ < 2Ä„.
"
-1 + i
"
1, i, , 1 + 3i, . . .
2
1 1
d(z1, z2) = d(z1, z2) = d ,
z1 z2
z1 z2 1 + z1z2 = 0
B ‚" C |z| <
M z " B z1, z2 " B z1 = z2

d(z1, z2) < |z1 - z2| < (1 + M2)d(z1, z2)
N = (0, 0, 1)
N
N
N
Bz + Bz + C = 0, B " C \ {0}, C " R
zz + Bz + Bz + C = 0, B " C, C " R, |B|2 - C > 0
 (zn)
limn" |zn| = +" (zn) "
z2 z
lim , lim .
z0 z0
z z
Õ Õ
lim n cos + i sin - 1 = iÕ, Õ " R,
n"
n n
1 Ä„
lim n Arg i + - = -1
n"
n 2
lim nArg (n + i) = 1
n"
n
z
lim 1 + = eRe z.
n"
n
z
Õn := Arg 1 + , n " N.
n
Õn z/n = -1

n limn" nÕn = Im z.
n
z
lim 1 + = eRe z(cos Im z + i sin Im z).
n"
n
"
zn
n
n=1
|z| < 1 |z| > 1 z = 1
|z| = 1 z = 1

" " " "
in (3 + 4i)n cos n sin2 n
, " , , .
ln n 5n n n n
n=2 n=1 n=1 n=1
 f(z) = 1/z
y = x + 1
y = x
x2 + y2 = r2 r > 0
(x - 1)2 + y2 = 4
(x - 1)2 + y2 = 1
f
z - i
f(z) =
z + i
f y = c c > 0
t2 + c2 - 1 2it
R t -
t2 + (c + 1)2 t2 + (c + 1)2
c 1
z " C : z - =
c + 1 c + 1
f H = {z " C :
Im z > 0} K = {z " C : |z| < 1}
f
1 + z
g(z) = i
1 - z
g {z " C : |z| = r} 0 < r < 1
-2r sin t 1 - r2
0, 2Ä„ t + i
r2 - 2r cos t + 1 r2 - 2r cos t + 1
1 + r2 2r
z " C : z - i =
1 - r2 1 - r2
g K = {z " C : |z| < 1}
H = {z " C : Im z > 0}
f g
 exp
{x + it : t " (-Ä„, Ä„ } x " R {z " C : |z| = ex}
{t + iy : t " R} y " R {reiy : r > 0}
exp
A = {z " C : -Ä„ < Im z Ä„}
A C \ {0}
sin cos C.
C sin z = a cos z = a a
-1, 0, 1, i, -i, . . .
ex - e-x ex + e-x
sinh x := , cosh x := , x " R.
2 2
ez - e-z ez + e-z
sinh z := , cosh z := , z " C.
2 2
z = x + iy
sin iz = i sinh z cos iz = i cosh z
Re sin z = sin x cosh y Im sin z = cos x sinh y
Re cos z = cos x cosh y Im cos z = - sin x sinh y
| sin z|2 = sin2 x + sinh2 y = cosh2 y - cos2 x
| cos z|2 = cos2 x + sinh2 y = cosh2 y - sin2 x
C
A = {z " C :
-Ä„/2 < Re z < Ä„/2}
Iy = {t + iy : -Ä„/2 t Ä„/2} y " R
 -1, 1 y = 0
u2 v2
Ey = (u, v) " R2 : + = 1, v 0 ,
cosh2 y sinh2 y
y > 0
u2 v2
Ey = (u, v) " R2 : + = 1, v 0 ,
cosh2 y sinh2 y
y < 0
Lx = {x + it : t " R} -Ä„/2 x Ä„/2
1, +") x = Ä„/2
(-", -1 x = -Ä„/2
{it : t " R} x = 0
u2 v2
Hx = (u, v) " R2 : - = 1, u < 0 ,
cos2 x
sin2 x
-Ä„/2 < x < 0
u2 v2
Hx = (u, v) " R2 : - = 1, u > 0 ,
cos2 x
sin2 x
0 < x < Ä„/2
sin(A) = C A = {z " C : -Ä„/2 Re z Ä„/2}
sin(B) = {z " C : Im z = 0 (" (Im z = 0 '" - 1 < Re z < 1)}

B = {z " C : -Ä„/2 < Re z < Ä„/2}
f : C z z + µz, , µ " C.
f z0 " C
f z " C
µ = 0
f : C C
f(x + iy) = x3y2 + ix2y3
f(x + iy) = x4y5 + ixy3
f(x + iy) = y2 sin x + iy
f(x + iy) = sin2(x + y) + i cos2(x + y)
u: C R
v : C R u + iv C
u(x + iy) = 2x3 - 6xy2 + x2 - y2 - y
u(x + iy) = x2 - y2 + e-y sin x - ey cos x
u(x + iy) = x2 + y2
1 y
L(z) = ln(x2 + y2) + iarctg
2 x
C \ {z " C : Re z = 0} 1/z
" " "
zn
nnzn, , 2nzn,
nn
n=1 n=1 n=1
" " "
(ln n)2zn, 2-nzn, n2zn,
n=1 n=1 n=1
" " "
n! (n!)3 z2n
zn, zn, , c " C \ {0},
nn (3n)! cn
n=1 n=1 n=1
" " "
n!
(z - 1)n, (n2 + an)zn, a " C, (sin n)zn.
2n(2n)!
n=1 n=0 n=1
"
anzn
n=0
r, M > 0
|anrn| M, n = 0, 1, 2, . . .
{z " C : |z| < r}
R r > 0
"
(|anrn|) R anzn
n=0
"
zn
n=0
K = {z " C : |z| < 1}
1
K z .
1 - z
k = 0, 1, 2, . . .
"
1 n
= zn-k, z " C, |z| < 1
(1 - z)k+1 k
n=k
c " C d " C \ {c} k = 0, 1, 2, . . .
"
n-k
1 1 n z - d
= , z " C, |z - d| < |c - d|
(c - z)k+1 (c - d)k+1 k c - d
n=k
"
zn
n2
n=1
1
z0
1/z z02 + 3i
(z + i)/(z - i) z0 = 0, z0 = -i z0 = i
1/(z2 + 1) z0 = x " R z0 = i
(2z - i)/(z2 + iz + 2) z0 = 1 z0 = -i
z/(z2 - (1 + 2i)z - 1 + i) z0 = 0
t " 0, 1
Å‚(t) = 1 + it
Å‚(t) = e-Ä„it
ł(t) = eĄit
Å‚(t) = 1 + it + t2.
f(z) = z3,
f(z) = z
1
f(z) =
z
“
z0 " C r > 0
Ć
(z - z0)kdz
“
k " Z.
 f
"
f(z) = an(z - z0)n
n=-"
P = {z " C : r < |z - z0| < R} r < Á < R “
z0 Á k " Z
Ć
1 f(z)
ak = dz.
2Ä„i (z - z0)k+1
“
x " R \ {0}
Ć
1
Ix := dz.
z - x
[-i,i]
limx0- Ix limx0+ Ix
x > 0
Ć Ć
1 1
- +
IR := dz, IR := dz.
z + ix z - ix
[-R,R] [-R,R]
- +
limR+" IR limR+" IR
Ć
eśdś = ez - 1, z " C.
[0,z]
|ez - 1| < |z|, Re z < 0.
PR -R, R, R+
iR, -R + iR R > 1
“R := [-R, R] + "R, "R := [R, R + iR, -R + iR, -R],
“R = "PR.
Ć
+"
cos t Ä„
dt = .
1 + t2 e
-"
eiz
f(z) = , z " C \ {-i, i}.
1 + z2
Ć
Ä„
f(z)dz = .
e
“R
|eiz| 1 Im z 0
Ć
lim f(z)dz = 0.
R+"
"R
a > 0
Ć
+"
t sin t
dt = Ä„e-a.
t2 + a2
-"
zeiz
f(z) = , z " C \ {-ia, ia}.
z2 + a2
R > a
Ć
f(z)dz = iĄe-a.
“R
Ć
lim f(z)dz = 0.
R+"
"R
n
Ć
+"
dt 1 · 3 · 5 · · · (2n - 1)
= Ä„.
(1 + t2)n+1 2 · 4 · 6 · · · (2n)
-"
n
Ć
d¾ 1 · 3 · 5 · · · (2n - 1)
= Ä„.
(1 + ¾2)n+1 2 · 4 · 6 · · · (2n)
“R
Ć
d¾
lim = 0
R+" (1 + ¾2)n+1
"R
Ć
+"
cos nt
dt = Ä„e-n.
1 + t2
-"
 f k z0
f(z0) = f (z0) = · · · = f(k-1)(z0) = 0, f(k)(z0) = 0.

sin z, exp z - 1, tg z, cos z - 1.
z0 k f
g(z) = (z - z0)kf(z) z0
g
1
res f = lim g(k-1)(z).
z0
zz0
(k - 1)!
z2 1
, , tg z,
(z2 - 1)2 sin z
z2 - Ä„2 1 - cos z 1 1
, , - .
sin z sin z ez - 1 z - 2Ä„i
cos t = (eit + e-it)/2
Ć
2Ä„
dt
I :=
5 + 4 cos t
0
Ć
dz
-i
2z2 + 5z + 2
“
I = 2Ä„/3
a > 1
Ć
2Ä„
dt 2Ä„
= " ,
a + sin t - 1
a2
0
Ć
2Ä„
dt 2Ä„a
= .
(a + cos t)2 (a2 - 1)3/2
0
Ć
"
x2
I := dx.
1 + x4
-"
“R := [-R, R] + CR R > 1 CR
0, Ä„ t exp (it)
z2
f (z) := .
1 + z4
Ć
exp (-Ä„i/4) i exp (-Ä„i/4) Ä„
"
f (z) dz = 2Ä„i - = .
4 4
2
“R
Ć
lim f (z) dz = 0.
R"
CR
"
I = Ä„/ 2