!"#$%&"'()$"*

!"#"$%&'%()'%("$%&"""*!"#+,-./

01223&045647428&'9#:9*;4564742;/

?@AB@%'(C0"&!"#"$%!%$CD(%#&'%()'%("$%&E&"D'C&?)FB)(%GC%F&%D%'

CF"H)@#"(%#&F)B)@"&IABI%$%@(%

z = x + iy x yi i =

1

i2 = 1

f(z) = u(x, y) + iv(x, y)

u(x, y) v(x, y)x y

f(z) = z2 f(z) = z2 = (x+ iy)2 = x2y2 + i2xyu(x, y) = x2 y2 v(x, y) = 2xy

f(z)f(z)

f (z) = limz0

f(z + z) f(z)z

z = x + iy z

f(z)

f(z) z = a f(z) z =a f(z)

f(z)

f(z) = u(x, y) + iv(x, y)

u

x=

v

yu

y= v

x

u(x, y) v(x, y) x yf(x)

f(z) = z3

f(z) = z3 = (x + iy)3 = x3 + 3x2iy + 3x(iy)2 + (iy)3 = x3 3xy2 + i(3x2y y3)

u(x, y) = x3 3xy2 v(x, y) = 3x2y y3

u

x= 3x2 3y2 ; u

y= 6xy ; v

x= 6xy ;

v

y= 3x2 3y2

u

x=

v

y

u

y= v

x

f(z) = z3

u v x y

2u

x2+

2u

y2= 0

2v

x2+

2v

y2= 0

2u = 0

f(z) = u + iv u vu

v

f(z)

z

f(z)

u(x, y) = x2 y2

2u = 2u

x2+

2u

y2= 2 2 = 0

v(x, y) u + iv z

v

y=

u

x= 2x

y

v(x, y) = 2xy + g(x)

g(x) x x

v

x= 2y +

dg

dx= u

y= 2y

dg

dx= 0, g =

f(z) = u + iv = x2 y2 + i(2xy + )

f(z)

Cf(z)dz = 0

z2z1

f(z)dz z2z1

f(z)dz = F (z2) F (z1) F (z) = f(z) f(z)

f(z) = 2z

C2z dz = 0

1+i

2i2z dz = z2|1+i2i = (1 + i)

2 (2i)2 = 2i + 4

f(z) a

C

f(z)z a dz = 2i f(a)

a

a

C

f(z)(z a)n+1 dz =

2in!

f (n)(a)

f (n)(z) n f(z)

C

sin z2z dz

|z| = 1 |z| = 2

C

sin z2z dz =

C

sin z2(z 2 )

dz =12

C

sin zz 2

dz

a = 2 = 1, 57 a

a

12

C

sin zz 2

dz =122i sin

2= i

f(z) z = a

a

f(z) = f(a) + f (a)(z a) + f(a)2!

(z a)2 + f(a)3!

(z a)3 +

f(z) f(z)f(z) f(z)

f(z) z = a

C1 C2 z0 f(z)R f(z)

f(z) = a0 + a1(z z0) + a2(z z0)2 + +b1

z z0+

b2(z z0)2

+

an =1

2i

C

f(z) dz(z z0)n+1

; bn =1

2i

C

f(z) dz(z z0)n+1

z0

R

R.

b

b f(z) z = z0 z0

b = 0 b bn f(z)z = z0 n = 1 f(z)

b f(z)z = z0

b1 1zz0 f(z) z = z0

z0 f(z)C f(z)dz

C z0 z0

Cf(z)dz = 2i f(z) C

C z0, z1,z2, . . . C0, C1C2 . . . f(z) C

Cf(z)dz = 2i f(z) C

C

f(z)

f(z) z = z0R(z0) b1 1(zz0)

f(z) = ez

z1 z = 1

ez

z 1 =e ez1

z 1 =e

z 1

1 + (z 1) + (z 1)2

2!+ . . .

=e

z 1 + e +e

2(z 1) + . . .

1z1 R(1) = e

f(z) z = z0f(z) (z z0) z = z0

f(z) = z(2z+1)(5z) R(12) R(5)

R(12) f(z) (z +12) (2z + 1)

z = 12

(z +12)f(z) = (z +

12)

z

(2z + 1)(5 z) =z

2(5 z)

R(12) =

122(5 + 12)

= 122

R(5)

(z 5)f(z) = (z 5) z(2z + 1)(5 z) =

z

2z + 1

R(5) = 511

f(z) g(z)h(z) g(z) z = z0h(z0) = 0 h(z0) = 0

R(z0) =g(z0)h(z0)

h(z0) h(z) z = z0.

f(z) = z(2z+1)(5z) g(z) = z h(z) = (2z + 1)(5 z)h(z) = 2(5 z) + (2z + 1)(1) = 4z + 9

R(z0) =z0

4z0 + 9

R(12) =

122 + 9

= 122

R(5) =5

20 + 9 = 511

f(z) nf(z) (z z0)m m

n (m 1)(m 1)! z = z0

R(z0) =1

(m 1)!dm1

dzm1(z z0)mf(z)

z=z0

f(z) = z sin z(z)2 z =

m = 3 (m 1) = 2

R() =12!

d2

dz2z sin z

z=

=12(z sin z + 2 cos z)

z=

= 1

2

0

d

5 + 4 cos

z = ei 0 2z |z| = 1

z = ei dz = ieid = izd d = 1izdzcos = 12(e

i + ei) = 12(z +1z )

I = 2

0

d

5 + 4 cos =

C

1izdz

5 + 2(z + 1z )=

1i

C

dz

5z + 2z2 + 2=

1i

C

dz

(2z + 1)(z + 2)

C f(z) z = 12z = 2 z = 12 C f(z)

z = 12

R(12) =

1(2z + 1) + 2(z + 2)

z= 12

=1

4z + 5

z= 12

=13

I = 1i 2iR(12) =

23

0 13