C
D
f(z)
f : D C D ‚" C
z f w = f(z)
z = x + iy w = u + iv.
w = f(x + iy) = u(x, y) + iv(x, y)
Ref(z) := u(x, y) Imf(z) := v(x, y)
f x y
f(z) = iz2
f(z) = iz2 = i(x + iy)2 = i(x2 + 2ixy - y2) = ix2 - 2xy - iy2 = -2xy + i(x2 - y2)
Ref(z) = u(x, y) = -2xy Imf(z) = v(x, y) = x2 - y2
u(x, y) = x - y v(x, y) = 4xy f
z
z + z z - z
Å» Å»
z = x + iy z = x - iy Ò! x = y =
Å»
2 2i

z + z z - z z + z z - z
Å» Å» Å» Å»
f(z) = u(x, y) + iv(x, y) = (x - y) + i4xy = - + 4i · =
2 2i 2 2i


1 1 1 1
z - + z + - z2 - z2
Å» Å»
2 2i 2 2i
g f(z) z0
lim f(z) = g Ô! " > 0 "´ > 0 "z " D 0 < d(z, z0) < ´ Ò! d (f(z), g) <
zz0
d C

d(z1, z2) := (Rez1 - Rez2)2 + (Imz1 - Imz2)2 = |z1 - z2|
lim f(z) = g Ô! lim u(x, y) = Reg lim v(x, y) = Img
zz0
(x,y)(x0,y0) (x,y)(x0,y0)
z0 = x0 + iy0 f(z) = u(x, y) + iv(x, y)
http : //www.mini.pw.edu.pl/ <" eplonkow/
 f z0 Ô!
lim f(z) = f (z0)
zz0
f(z) = u(x, y) + iv(x, y) z0 Ô! u v
(x0, y0)

1
f " f
z
f (z + "z) - f(z)
lim
"z0 "z
f z f (z)
f (z + "z) - f(z)
f (z) := lim
"z0 "z
f (z) - f(z0)
f (z0) := lim .
zz0 - z0
z
f g z
(f + g) (z) = f (z) + g (z)
(f - g) (z) = f (z) - g (z)
(f · g) (z) = f (z) · g(z) + f(z) · g (z)

f f (z)·g(z)-f(z)·g (z)
(z) = z " g-1(0)
/
g
[g(z)]2
(f ć% g) (z) = f (g(z)) g (z)
f(z) = u(x, y) + iv(x, y) z0 = x0 + iy0 f (z0)
"u "u "v "v
(x0, y0) (x0, y0)
"x "y "x "y
"u "v "u "v
= = -
"x "y "y "x
u(x, y) v(x, y) (x0, y0)
f(z) = u(x, y) + iv(x, y) f (z0)
z " C f(z) = zz = |z|2 = x2 + y2
Å»
Ref(z) = u(x, y) = x2 + y2 Imf(z) = v(x, y) a" 0
u v "(x, y) " R2

u = 2x u = 2y vx = 0 vy = 0
x y
http : //www.mini.pw.edu.pl/ <" eplonkow/

u = vy Ô! x = 0 u = -vx Ô! y = 0
x y
z0 = 0
f
f(z) - f(0) zz
Å»
f (0) = lim = lim = lim z = 0
Å»
z0 - 0 z
z0 z0
z
f(z)

"f 1 "f "f "f 1 "f "f
:= - i := + i
"z 2 "x "y "z 2 "x "y
Å»
"f "u "v "f "u "v
= + i = + i
"x "x "x "y "y "y
f(z) z0 = x0 + iy0
"f
(z0) = 0
"z
Å»
f(z) z0
f(z)
D z " D f (z)
f " H(D)
f, g " H(D) (f Ä… g) " H(D) fg " H(D)
f
f, g " H(D) " H(D - (g-1(0))
g
f, g " H(D) (f ć% g)2H(D)
f(z) = |z|2 = zz
Å»
http : //www.mini.pw.edu.pl/ <" eplonkow/