2

p1 i 2 = 1 i 2 x, y x + iy 2 (x + iy) + (γ + iδ) = (x + γ) + i(y + δ) (x + iy)(γ + iδ) = (xγ yδ) + i(xδ + yγ) i 2 = 1 γ + iδ 0 x + iy γ + iδ xγ + yδ xδ = γ 2 + iyγ + δ2 γ 2 + δ 2 p7 = x 2 +y 2 z z p13 (x, y) (r, θ) x = rcosθ y = rsinθ z == r(cosθ + isinθ) r r z θ z argz 1

2

p1 i 2 = 1 i 2 α, β α + iβ 2 (α + iβ) + (γ + iδ) = (α + γ) + i(β + δ) (α + iβ)(γ + iδ) = (αγ βδ) + i(αδ + βγ) i 2 = 1 γ + iδ 0 α + iβ γ + iδ = αγ + βδ γ 2 + δ 2 αδ + iβγ γ 2 + δ 2 p7 = α 2 +β 2 a a p13 (α, β) (r, ϕ) α = rcosϕ β = rsinϕ a == r(cosϕ + isinϕ) r r a ϕ a arga p1 i 2 := 1 2 3

(x 1 + iy 1 ) + (x 2 + iy 2 ) := (x 1 + x 2 ) + i(y 1 + y 2 ) (x 1 + iy 1 )(x 2 + iy 2 ) := (x 1 x 2 y 1 y 2 ) + i(y 1 x 2 + x 1 y 2 ) p1 2 x,y (x,y) x + iy +i 1 2 0+i0,1+i0 0,1 4

5

p25 f f(z + h) f(z) (z) = lim h 0 h [ ε > 0, δ > 0, h + ik{0 < h + ik < δ}, {u(x + h, y + k) + iv(x + h, y + k)} {u(x, y) + iv(x, y)} {a + ib} < ε] h + ik D = {x + iy x + iy < 10} u(x, y) + iv(x, y) = x 2 y 2 + i2xy 1 + i1 2 + i2 ε > 0 δ > 0 0 < h + ik < δ h + ik {{1 + h}2 {1 + k} 2 + i2{1 + h}{1 + k}} {1 2 1 2 + i2 1 1} {2 + i2} < ε h + ik {{1 + h}2 {1 + k} 2 + i2{1 + h}{1 + k}} {1 2 1 2 + i2 1 1} {2 + i2} < 100 h + ik 0 < h + ik < 1 {0 < 0.5 + i{ 0.5} < 1 {{1 + 0.5}2 {1 0.5} 2 + i2{1 + 0.5}{1 0.5}} {1 2 1 2 + i2 1 1} {2 + i2} < 100 0.5 + i{ 0.5} 0 < 0.1 + i0.2 < 1 {{1 + 0.1}2 {1 + 0.2} 2 + i2{1 + 0.1}{1 + 0.2}} {1 2 1 2 + i2 1 1} {2 + i2} < 100} 0.1 + i0.2 {{1 + h}2 {1 + k} 2 + i2{1 + h}{1 + k}} {1 2 1 2 + i2 1 1} {2 + i2} < 200 h + ik [ ] 6

h 0 h 0 y x f (z) = f x = u x + i v x {u(x + h, y) + iv(x + h, y)} {u(x, y) + iv(x, y)} [ lim =] h 0 h + i0 h ik 0 f f(z + ik) f(z) (z) = lim = i f k 0 ik y = i u y + v y {u(x, y + k) + iv(x, y + k)} {u(x, y) + iv(x, y)} [lim k 0 0 + ik u(x, y + k) u(x, y) i{v(x, y + k) v(x, y)} = lim + lim ] k 0 ik k 0 ik f(z) f x = i f y [ u x u x = v y, u y = v x (6) (x, y) = v y (x, y), u y v (x, y) = (x, y)] x [ ] 7

u v 1 1 (6) u(x + h, y + k) u(x, y) = u x h + u y k + ε 1 v(x + h, y + k) v(x, y) = v x h + v y k + ε 2 [u(x + h, y + k) u(x, y) = u u (x, y)h + x y (x, y)k + ε 1(h, k) ε > 0, δ > 0, h + ik{0 < h + ik < δ}, ε 1(h, k) h2 + k 2 < ε] h + ik 0 ε 1 /(h + ik) 0 ε 2 /(h + ik) 0 ε 1, ε 2 h + ik 0 f(z) = u(x, y) + iv(x, y) (6) f(z + h + ik) f(z) = ( u x + i v x )(h + ik) + ε 1 + iε 2 lim h+ik 0 f(z + h + ik) f(z) h + ik f(z) = u x + i v x u(x, y), v(x, y) f(z) = u(z) + iv(z) f (z) [v(x 0 + h, y 0 + k) v(x 0, y 0 ) = v x (x 0, y 0 )h + v y (x 0, y 0 )k + ε 2 (h, k) [ ] [ ] h h ε > 0, δ > 0, {0 < k k < δ}, ε 2 (h, k) [ ] h < ε] k 8

p24 The class of analytic functions is formed by the complex functions of a complex variable which possess a derivative wherever the function is defined. The term holomorphic function is used with identical meaning. The definition of the derivative can be rewritten in the form f f(z + h) f(z) (z) = lim h 0 h [ ε > 0, δ > 0, h + ik{0 < h + ik < δ}, {u(x + h, y + k) + iv(x + h, y + k)} {u(x, y) + iv(x, y)} {a + ib} < ε] h + ik The limit of the difference quotient must be the same regardless of the way in which h approaches zero. If we choose real values for h, then the imaginary part y is kept constant, and the derivative becomes a partial derivative with respect to x. We have thus 9

Ip162 1 C D f[= u(x, y) + iv(x, y)] a[= a + ib] D lim h 0,h 0 f(a + h) f(a) h = b C [ ε > 0 δ > 0 0 < h + ik < δ h + ik {u(a + k, b + l) + iv(a + k, b + l)} {u(a, b) + iv(a, b)} {α + iβ} < ε ] b f a f (a)[=] D f D D = {x + iy x + iy < 10} u(x, y) + iv(x, y) = x 2 y 2 + i2xy 1 + i1 2 + i2 ε > 0 δ > 0 0 < < δ {{1 + k}2 {1 + l} 2 + i2{1 + k}{1 + l}} {1 2 1 2 + i2 1 1} {2 + i2} < ε {{1 + k}2 {1 + l} 2 + i2{1 + k}{1 + l}} {1 2 1 2 + i2 1 1} {2+i2} < 100 0 < < 1 {0 < 0.5 + i{ 0.5} < 1 {{1 + 0.5}2 {1 0.5} 2 + i2{1 + 0.5}{1 0.5}} {1 2 1 2 + i2 1 1} {2 + i2} < 100 0.5 + i{ 0.5} 0 < 0.1 + i0.2 < 1 {{1 + 0.1}2 {1 + 0.2} 2 + i2{1 + 0.1}{1 + 0.2}} {1 2 1 2 + i2 1 1} {2 + i2} < 100} 0.1 + i0.2 {{1 + k}2 {1 + l} 2 + i2{1 + k}{1 + l}} {1 2 1 2 + i2 1 1} {2+i2} < 200 Ip162 Ip163 164 - p44 p169 f (a)[= [ ε > 0, δ > 0, {0 < < δ}, {u(a + k, b + l) + iv(a + k, b + l)} {u(a, b) + iv(a, b)} d dx+idy {u(a, b) + iv(a, b)}] {α + iβ} < ε] 10

11

p26 = lim h 0 f(z + h) f(z) h f (z) p51 h 0 0 p52 u(x, y) = u(x 0, y 0 ) + a(x x 0 ) + b(y y 0 ) + o() a = u x (x 0, y 0 ), b = u y (x 0, y 0 ) α = a + ib f(z) = f(z 0 ) + α(z z 0 ) + o( z z 0 ), α = df dz (z 0) u(x, y) = u(x 0, y 0 ) + a(x x 0 ) b(y y 0 ) + o( x x 0 2 + y y 0 2 ), v(x, y) = v(x 0, y 0 ) + b(x x 0 ) + a(y y 0 ) + o( x x 0 2 + y y 0 2 ) p54 f(z) 12

p201 f(z) z f(z + h) f(z) lim = f (z) h 0 h f(z + h) = f(z) + hf (z) + o(h) h 0 f (z) K K f(z) = u + vi z = x + yi f (z) = p + qi h = dz = dx + idy (1), ( h = dx 2 + dy 2 ) u,v x,y u x = v y = p, u y = v x = q f (z) = u x + iv x = i() f(z) u v u x = v y, u y = v x (2) (2) Cauchy-Riemann u,v x,y (2) h = dx + idy (2) du = u x dx + u y dy + o( h ), dv = v x dx + v y dy + o( h ), du + idv = (u x + iv x )(dx + idy) + o( h ) 13

p33 30 p85 u(x, y) = x 2 y 2, v(x, y) = 2xy 30 p89 f(z) f(z 0 ) lim z z 0 z z 0 f z 0 f (z 0 ) A 30 p91 {u(x, y)+iv(x, y)} {u(x 0, y 0 )+iv(x 0, y 0 )} = (a+ib){(x x 0 )+i(y y 0 )}+(){(x x 0 )+i(y y 0 )} u(x, y) u(x 0, y 0 ) = a(x x 0 ) b(y y 0 ) + z z 0 ɛ 0 x x 0 y y 0 0 ɛ 1, ɛ 2 0 30 p92 {u(x, y 0 ) + iv(x, y 0 )} {u(x 0, y 0 ) + iv(x 0, y 0 )} lim = lim x x 0 (x + iy 0 ) (x 0 + iy 0 ) u(x, y 0 ) u(x 0, y 0 ) x x 0 x x 0 +i lim x x 0 v(x, y 0 ) v(x 0, y 0 ) x x 0 14

30 p97 D D f D f(z) f (z) = u x + i v x f (z) p23 1 f(x) x a A lim f(x) = A x a ε > 0 δ > 0 x a < δ x a x f(x) A < ε p25 f (z) = lim h 0 f(z + h) f(z) h [ ε > 0 δ > 0 h 0 < δ h 0 h f(z + h) f(z) f (z) < ε ] h 15

f(z + δ) f(z) δ = f f(z + h) f(z) (z) = lim ( ) h 0 h {u(x + h, y + k) + iv(x + h, y + k)} {u(x, y) + iv(x, y)} ( ) h + ik [f {u(x + h, y + k) + iv(x + h, y + k)} {u(x, y) + iv(x, y)} (z) = lim ] h 0 h + ik q : lim f(x) = b ( ) x a q : ɛ > 0, δ > 0, x(0 < x a < δ), f(x) b < ɛ ( ) [ ɛ > 0, δ > 0, h(0 < h < δ), {u(x + h, y + k) + iv(x + h, y + k)} {u(x, y) + iv(x, y)} f (z) < ɛ] h + ik 1 f(z + δ) f(z) δ P51 52 = [ lim h 0,h 0 f(a + h) f(a) lim = b C ( 1.1) h 0,h 0 h {u(x + h, y + k) + iv(x + h, y + k)} {u(x, y) + iv(x, y)} ( ) h + ik {u(a + k, b + l) + iv(a + k, b + l)} {u(a, b) + iv(a, b)} ] 2 f R n A R m a Ā, b Rm x a f(x) b ε > 0 δ > 0 x a < δ x A f(x) b < ε lim f(x) = b x a [ ε > 0 δ > 0 0 < h + ik < δ h + ik {u(a + k, b + l) + iv(a + k, b + l)} {u(a, b) + iv(a, b)} {α + iβ} < ε ] 16

{u(x, y 0 ) + iv(x, y 0 )} {u(x 0, y 0 ) + iv(x 0, y 0 )} u(x, y 0 ) u(x 0, y 0 ) v(x, y 0 ) v(x 0, y 0 ) lim = lim +i lim ( ) x x 0 (x + iy 0 ) (x 0 + iy 0 ) x x 0 x x 0 x x 0 x x 0 {u(x 0 + h, y 0 ) + iv(x 0 + h, y 0 )} {u(x 0, y 0 ) + iv(x 0, y 0 )} [ lim =] h 0 (x 0 + h + iy 0 ) (x 0 + iy 0 ) R n x f {u(x + h, y) + iv(x + h, y)} {u(x, y) + iv(x, y)} [ lim =] h 0 h + i0 x = x 1 x 2.. x n ( ) f(x) lim x a,x a g(x) = 0 ( ) f o(g){x a} h = dx + idy ( h = dx 2 + dy 2 ) ( ) du = u x dx + u y dy + o( h ) ( ) f(h) [ lim h 0,h 0 h (h) = 0] f(h) [ lim dx+idy 0,dx+idy 0 dx2 + dy 2 (h) = 0] q : lim f(x) = b ( ) x a q : ɛ > 0, δ > 0, x(0 < x a < δ), f(x) b < ɛ ( ) [ ɛ > 0, δ > 0, h(0 < h 0 < δ), f(h) 0 < ɛ] h (h) f(h) [ ε > 0, δ > 0, dx + idy{0 < dx + idy 0 < δ}, 0 < ε] dx2 + dy 2 (h) [ ] [ ] h h [ ε > 0, δ > 0, {0 < k k < δ}, ε 1 (h, k) < ε] h2 + k 2 f(z + h) f(z) = (a + bi)h + o(h) (h 0) ( 1.8) u(x + k, y + l) u(x, y) = ak bl + o(h) (h 0) ( 1.9) f(h) [ lim h 0,h 0 h(h) = 0] 17

q : lim f(x) = b ( ) x a q : ɛ > 0, δ > 0, x(0 < x a < δ), f(x) b < ɛ ( ) [ ɛ > 0, δ > 0, h(0 < h 0 < δ), f(h) 0 < ɛ] h(h) 1 R n U f au n c = (c 1, c 2,, c n ) f(a + h) f(a) = ch + o( h ) {h 0} ( 5.3) f(h) [ lim h 0,h 0 h(h) = 0] [ ] [ ] h h [ ε > 0, δ > 0, {0 < k k < δ}, ε 1 (h, k) [ ] h < ε] k Ip162 1 C D f a D f(a + h) f(a) lim = b C h 0,h 0 h {u(a + k, b + l) + iv(a + k, b + l)} {u(a, b) + iv(a, b)} [ lim = b C] h 0,h 0 b f a f (a) D f D P163 164 30P91 f(z) = u(x, y) + iv(x, y)(z = x + yi) h = k + li u(x + k, y + l) u(x, y) f (z) = a + ib = - p44 q : lim f(x) = b x a q: 0 < x a < δ x f(x) b < ɛ 18

ɛ > 0, δ > 0 : 0 < x a < δ = f(x) b < ɛ q q : ɛ > 0, δ > 0, x(0 < x a < δ), f(x) b < ɛ f(a + h) f(a) [ lim = b C (1.1)] h 0,h 0 h f(a + h) f(a) [q : ɛ > 0, δ > 0, x(0 < h 0 < δ), b < ɛ] h {u(a + k, b + l) + iv(a + k, b + l)} {u(a, b) + iv(a, b)} [ lim = b C] h 0,h 0 [q : ɛ > 0, δ > 0, h(0 < h < δ), {u(a + k, b + l) + iv(a + k, b + l)} {u(a, b) + iv(a, b)} b < ɛ] p169 f(z + δ) f(z) δ = {u(x + h, y + k) + iv(x + h, y + k)} {u(x, y) + iv(x, y)} h + ik 19

- p44 q : lim f(x) = b x a q: 0 < x a < δ x f(x) b < ɛ ɛ > 0, δ > 0 : 0 < x a < δ = f(x) b < ɛ q q : ɛ > 0, δ > 0, x(0 < x a < δ), f(x) b < ɛ [f f(z + h) f(z) (z) = lim ] h 0 h f(z + h) f(z) [q : ɛ > 0, δ > 0, h(0 < h 0 < δ), f (z) < ɛ] h {u(x + h, y + k) + iv(x + h, y + k)} {u(x, y) + iv(x, y)} [ lim = f (z) + ] h 0 h + ik [q : ɛ > 0, δ > 0, h(0 < h < δ), {u(x + k, y + l) + iv(x + k, y + l)} {u(x, y) + iv(x, y)} f (z) < ɛ] p169 f(z + δ) f(z) δ = {u(x + h, y + k) + iv(x + h, y + k)} {u(x, y) + iv(x, y)} h + ik 20

p28 p26 d ( dz f(z)) d [ {u(x, y) + iv(x, y)}] dx + idy d [ {u(a, b) + iv(a, b)}] dx + idy p50 d dz (f(z)g(z)) z (f(z)) p4 dz = dx + idy 21

22

P51 52 2 f R n A R m a Ā, b Rm x a f(x) b ɛ > 0 δ > 0 x a < δ x A f(x) b < ɛ lim f(x) = b x a ( ɛ > 0)( δ > 0)( x A)( x a < δ = f(x) b < ɛ) [ ɛ > 0, δ > 0, x{0 < x a < δ}, f(x) b < ɛ] [ ɛ > 0 δ > 0 0 < h 0 < δ {a + h} D f(a+h) f(a) h b < ɛ ] P120 n c=[,,,] 30p34 ɛ δ 0 < x a < δ = f(x) A < ɛ lim f(x) = A x a 23

Ip162 1 C D u(x, y) + iv(x, y) a + ib D α + iβ ε > 0, δ > 0, {0 < < δ}, {u(a + k, b + l) + iv(a + k, b + l)} {u(a, b) + iv(a, b)} α + iβ = {u + iv} (a + ib) {α + iβ} < ε D u(x, y) + iv(x, y) D D = {x + iy x + iy < 10} u(x, y) + iv(x, y) = x 2 y 2 + i2xy 1 + i1 2 + i2 ε > 0 δ > 0 0 < < δ {{1 + k}2 {1 + l} 2 + i2{1 + k}{1 + l}} {1 2 1 2 + i2 1 1} {2 + i2} < ε ε = 100 δ = 1 0 < 0.5 + i{ 0.5} < 1 {{1 + 0.5}2 {1 0.5} 2 + i2{1 + 0.5}{1 0.5}} {1 2 1 2 + i2 1 1} {2 + i2} < 100 0.5 + i{ 0.5} ε = P162 P51 52 120 163 30 P89 1 C D u(x, y) + iv(x, y) {a + ib} D α + iβ ɛ > 0 δ > 0 0 < k+il < δ {{a+ib}+{k+il}} {u(a + k, b + l) + iv(a + k, b + l)} {u(a, b) + iv(a, b)} D {α + iβ} < ɛ α + iβ = {u + iv} (a + ib) D u(x, y) + iv(x, y) D D = {x + iy x + iy < 10} u(x, y) + iv(x, y) = x 2 y 2 + i2xy 1 + i1 ɛ > 0 δ > 0 0 < k+il < δ {{1+i1}+{k+il}} D {{1 + k}2 {1 + l} 2 + i2{1 + k}{1 + l}} {1 2 1 2 + i2 1 1} {2+i2} < ɛ ɛ = 100 δ = 1 OK 0 < 0.5 + i{ 0.5} < 1 {{1 + 0.5}2 {1 0.5} 2 + i2{1 + 0.5}{1 0.5}} {1 2 1 2 + i2 1 1} {2 + i2} < 100 0.5 + i{ 0.5} ɛ = 24

25

P15 [6] 26

P162 1 C D u(x, y) + iv(x, y) {a + ib} D α + iβ { ɛ > 0}{ δ > 0}{ {{a + ib} + {}} D} {u(a + k, b + l) + iv(a + k, b + l)} {u(a, b) + iv(a, b)} {0 < < δ = α + iβ = {u + iv} (a + ib) {α + iβ} < ɛ} D u(x, y) + iv(x, y) D D = {x + iy x + iy < 10} u(x, y) + iv(x, y) = x 2 y 2 + i2xy 1 + i1 ɛ > 0 0 < < δ {{1 + k}2 {1 + l} 2 + i2{1 + k}{1 + l}} {1 2 1 2 + i2 1 1} {2 + i2} < ɛ δ ɛ = 100 δ = 1 OK 0 < 0.5 + i{ 0.5} < 1 {{1 + 0.5}2 {1 0.5} 2 + i2{1 + 0.5}{1 0.5}} {1 2 1 2 + i2 1 1} {2 + i2} < 100 0.5 + i{ 0.5} ɛ = 27