( ) i i i (Σe i i = e i + e i 2 + ), y n ( ) 2 y y = +iy 2 (, y) = + iy y > (upper hlf plne) y < (lower hlf plne) = + iy (rel prt) y (imginry prt) Re = Im = y = Re + iim = + iy = = ( + iy)( iy) = 2 + y 2 (r, θ) ( = r cos θ, y = r sin θ) = + iy = r cos θ + ir sin θ = r(cos θ + i sin θ) = re iθ = re iθ (polr form) r θ r = = 2 + y 2, tn θ = y
θ (rgument) θ = rg 2 ( ) n n n n = Re iθ = re iϕ n n = r n e inϕ Re iθ = r n e inϕ R = r n r = R /n, e inϕ = e iθ ϕ = θ/n e iθ = e iθ+2i = e iθ+4i = (e 2i = cos(2) + i sin(2) = ) 2 m m =,, 2,..., n ϕ = θ + 2m n n n = R /n e i(θ+2m)/n (de Moivre) (cos θ + i sin θ) n = cos(nθ) + i sin(nθ) 2
= re iϕ, n = r n e inϕ = r n (cos ϕ + i sin ϕ) n = r n (cos nϕ + i sin nϕ) r = R /n, ϕ = θ + 2m n (e iθ ) n = e inθ (cos θ + i sin θ) n = cos(nθ) + i sin(nθ) 2 = r e iθ r 2 e iθ2 = r (cos θ + i sin θ )r 2 (cos θ 2 + i sin θ 2 ) = r r 2 (cos θ cos θ 2 sin θ sin θ 2 + i cos θ sin θ 2 + i sin θ cos θ 2 ) = r r 2 (cos(θ + θ 2 ) + i sin(θ + θ 2 )) n = (re iθ ) n = r n (cos(nθ) + i sin(nθ)) i n = Re iθ = i = e i/2 (R = ) n = 2 = re iϕ = e i(/2+2m)/2 = e i(/4+m) = cos( 4 + m) + i sin( 4 + m) (r = R/2 = ) m = m = m = : cos 4 + i sin 4 = 2 ( + i) m = : cos( 4 + ) + i sin( 4 + ) = 2 ( + i) m = 2 m = m = 3 m = 2 + i 3 n = Re iθ = + i = 2e i3/4 (R = 2 /2 ) n = 3 = re iϕ = 2 /6 e i(3/4+2m)/3 3
3 m =,, 2 D (D ) (D ) D D (regulr) D df d = lim f( + ) D ( ) < < r (r ) ( < < r) ( ) r = = + iy =, y = (i), y = (ii) =, y 2 ( y = = y ) ( ) 2 D, y f( + ) f( +, y + y) f(, y) lim = lim + i y (i), y = f( +, y + y) f(, y) f( +, y) f(, y) f(, y) lim = lim = + i y (ii) =, y f( +, y + y) f(, y) f(, y + y) f(, y) f(, y) lim = lim = i + i y y i y y 2 f(, y) 4 f(, y) = i y
f(, y) f(, y) = u(, y) + iv(, y) u(, y) v(, y) u(, y) v(, y) + i = i + y y u(, y) = v(, y) y, u(, y) y v(, y) = (uchy-riemnn) = u(, y) + iv(, y) u(, y), v(, y) u(, y), v(, y) 2 ( u, v ) = = + iy = iy 2 + y 2 = u + iv (u = 2 + y 2, v = y 2 + y 2 ) u = 2 + y 2 2 2 ( 2 + y 2 ) 2 = 2 + y 2 ( 2 + y 2 ) 2 v y = 2 + y 2 u y = 2y ( 2 + y 2 ) 2 v = 2y ( 2 + y 2 ) 2 2y2 2 + y 2 = 2 + y 2 ( 2 + y 2 ) 2 = / = = / u y = / = = ( = ) = ( / ) D r = 2 + y 2 = r 2 r = = = iy = u + iv = (u =, v = y) 5
u =, v y =, u y =, v = d ( ) i d = lim N i= N f( i ) i d = d + i dy d = d + idy = u(, y) + iv(, y) d = (u(, y) + iv(, y))(d + idy) = (u(, y)d v(, y)dy) + i (v(, y)d + u(, y)dy), y t d = t2 t f((t)) d dt dt ( ) d 6
t b (), (b) () = (b) t t (t) (t ) ( ) (Jordn) () (b) 2 (simply connected) (multiply connected) D ( ) () (b) d = F ((b)) F (()) df () ( =, F () = f(ξ)dξ) d 2 2 d = F ( 2 ) F ( ) 2 ( + g())d = d = d 2 d = d = + 2 d = 2 d = d + g()d d + d 2 d + 2 d 3 2 + 2 (2 ) 4, 2 = + i = + iy ( ) d = ( + iy)(d + idy) = (d ydy + iyd + idy) y (, ) y = y (y y ) 7
(d ydy + iyd + idy) = (d d + id + id) = 2i d = i = = = + i 2 ( y ) 2 + i 2 d = (d ydy + iyd + idy) = 2 2 = 2 (d ydy + iyd + idy) + d + ( y + i)dy 2 (d ydy + iyd + idy) = i 2 2 y =, dy = 2 =, d = = + i t ( )(t) (t) = ( + i)t ( t ) d = (t) d(t) dt = dt ( + i)t( + i)dt = 2i 2 = i (t) ( + i)t y (, ) (, ) (t) = (t, t) = (, )t = = + i ( =, y = ) = + iy (t) = ( + i)t t ( + i)t = = + i 2 2 t ) (t) = t 2 (t) = + it ( d = 2 2 d + d = 2 tdt + ( + it)idt = 2 + i 2 = i 2 (t) = + it = y ( ) / r = r, y = = r, y = d = = = i re iθ ireiθ dθ ( = re iθ ) idθ 8
= + i = + i (y = ) d == ( iy)(d+idy) = (d+ydy iyd+idy) = (d+d id+id) = 2 d = =, y = = + i ( =, y = ) 2 d = (d + ydy iyd + idy) = (d + ydy iyd + idy) = 2 2 2 d + (ydy + idy) = + i y = = y =, y = = + i 3 d = (d + ydy iyd + idy) = 3 3 ydy + (d id) = i (uchy) D D d =, 2,..., n,..., n (,..., n ), 2,..., n (,..., n ) d = n i= i d D D f() = 2i d = sin r > sin sin /( ) 9
sin d = 2i sin f (n) () = n! d 2i ( ) n+ (Gourst) f() = 2i d, f( + δ) = 2i δ d (f( + δ) f())/δ = = n= c n ( ) n = c n ( ) n + c n ( ) n n= n= c n = f(ξ) dξ (n =, ±, ±2,...) 2i (ξ ) n+ n= c n ( ) n (principl prt) ( ) = = e / ep e = + + 2 2 + = n= n n!
e / = n! ( )n n= = ( ) singulrity / = ( R < < R ) = = (isolted singulrity) removble singulrity n n (pole) (essentil singulrity) < < R ( < δ δ < R) = g() = g() ( ) n ( < < δ) n n n lim ( ) n n (residue) = Res(f, ) = d 2i Res(f, ) Res n n d = 2i Res(f, i ) i= n n d = 2i Res(f, i ) i=
d = c n ( ) n d n= r ( ) n d (n =, 2,...) n = d = 2 re iθ ireiθ dθ ( = re iθ ) = 2 idθ = 2i () = r = re iθ n ( ) n d = 2 = i r n r n e inθ ireiθ dθ 2 e i(n )θ dθ = i [ r n i(n ) e i(n )θ] 2 = i ( r n i(n ) ) i(n ) = e i2n n n =, 2,... ( ) n (n ) ( ) d = = c n ( ) n d n= c ( ) d = 2ic 2
Res(f, ) = c n c = c + c ( ) + c 2 ( ) 2 + c + c 2 ( ) 2 2 c d d (( ) 2 ) = = d d (c ( ) 2 + c ( ) 3 + c 2 ( ) 4 + c ( ) + c 2 ) = = c n ( c = ( (n )! d n ) n ) = d n c ( ) n = n d n ( Res(f, ) = c = ( (n )! d n ) n ) = ( ) ( ) lim e i d = ( > ) R R R R +R +R (ir) R R R = Re iθ lim e i d = R R R R ( R R ( ir) +R ) 3
R R ( ) R R R R + = R ( R R R R + R = ) ( ) ( ) I = e i d, I = R R e i d R I I I = e i d = i R e ireiθ f(re iθ )Re iθ dθ ( = Re iθ ) b f() d b f()d b f() f() ( ) f() f() b f() d b b b f() d f()d f()d 2 b f() d b f()d b f() d f() f() b f()d 4 b f() d
I = e i d R = i = = = e ireiθ f(re iθ )Re iθ dθ e ireiθ f(re iθ ) Re iθ dθ ep[ir(cos θ + i sin θ) ] f(re iθ ) Rdθ e R sin θ f(re iθ ) Rdθ ( e i = e i e i = ) e R sin θ f(re iθ ) Rdθ ep = f(re iθ ) R θ ϵ I ϵr e R sin θ dθ ϵr e R sin θ dθ = ϵr = ϵr = ϵr = ϵr = 2ϵR /2 /2 /2 /2 /2 e R sin θ dθ + ϵr e R sin θ dθ /2 e R sin θ dθ + ϵr e R sin(θ +) dθ (θ = θ ) /2 e R sin θ dθ + ϵr e R sin θ dθ (sin(θ + ) = sin θ) /2 e R sin θ dθ + ϵr e R sin θ dθ /2 e R sin θ dθ (sin( θ) = sin θ) sin θ θ /2 2θ sin θ ( sin θ sin θ 2θ/) 5
I 2ϵR /2 e R sin θ dθ 2ϵR /2 e 2Rθ/ dθ = 2ϵR [ 2R e 2Rθ/] /2 = ϵ ( e R ) R I ϵ ϵ lim I = R I I = e i d = i R 2 e ireiθ f(re iθ )Re iθ dθ ( = Re iθ ) I 2 e ireiθ f(re iθ ) Rdθ = 2 ep[ ir(cos θ + i sin θ)] f(re iθ ) Rdθ = R ϵr = ϵr = ϵr = 2ϵR 2 2 /2 e R sin θ f(re iθ ) dθ e R sin θ dθ e R sin(θ +) dθ (θ = θ ) e R sin θ dθ e R sin θ dθ θ 3/2 2θ/ sin θ sin d = sin 6
( ) n sin = (2n + )! 2n+ = 3! 3 + n= 2, 4,... = = R r sin d = R 2i r = 2i = 2i = 2i = 2i J ( R r ( R r ( R r e i e i d e i R d e i ) r d e i R d e i ) r d e i r d + e i ) R d e = + + = 2i lim lim J = R r sin d (2) r R r R d + d + R r d + d 2 ( ) = r ( ) 2 R ( ) R r +r +r +R +R R J = e i r d = e i R d + e i R d + e i r 2 d + e i d = J = J + e i d + 2 e i d = (3) J r =, R = (3) R 7
2 e i d = e ir cos θ e R sin θ dθ e ireiθ Re iθ ireiθ dθ = i e ireiθ dθ = i e ir cos θ e R sin θ dθ = e ir cos θ e R sin θ dθ e R sin θ dθ (3) e i d = i = i re iθ eir cos θ e r sin θ re iθ dθ ( = re iθ ) e ir cos θ e r sin θ dθ r i lim r e ir cos θ e r sin θ dθ = i dθ = i (3) J (2) sin / sin d = 2i lim lim J = R r 2i lim e r i d 2i e lim R 2 i d = 2 sin / sin d = 2 ( sin + sin ) d = ( sin 2 = 2 = 2 = 2 ( ( sin sin + sin d sin( ) ) d sin ) d sin ) d sin d r R sin d + sin R d + r sin d 8
r sin / R, r ( ) R lim R R sin d = lim lim R r ( r R sin d + sin R d + sin ) r d = lim lim R r sin d R +R 2 d, 2 d 2 2 ( R) R 2 e i d =, 2 e i d = sin d = = 2i = 2i sin d + e 2i 2 i d 2i 2 ( ei + e i e i )d + 2i d 2i e i 2 e i d e i d 2i d 2 e i d + = + 2 = + 2 + R +R R +R + + = ( ) + e i d = 2ie = 2i sin d = ( 2i ei e i )d = 2i + e i d 2i e i d = sin d = 2 lim lim sin R r d = 2 9
+ + ( ) f() lim ϵ ( ϵ b ) f()d + f()d +ϵ (principl vlue) pv b ( ϵ f()d = lim ϵ b ) f()d + f()d +ϵ PV p.v. vp P ( ) d ( ϵ) R R R R ( R, ϵ ) ϵ d = R R d + +ϵ d + d + 2 d (R, ϵ ) /( ) Res(f/( ), i ) n d = 2i f Res(, i) i= = ϵ R d = pv R d + lim ϵ d + 2 d ϵ 2 R /( ) R 2 () lim ϵ d = i lim ϵ f( + ϵe iθ ) ϵe iθ ϵe iθ dθ = i lim 2 ϵ f( + ϵe iθ )dθ = if()
R, ϵ pv n d = if() + 2i f Res(, i) i= = f() = e i e i d ( = ) pv e i d = i f( = ) = e = = e i 2