f(x) f(z) z = x + i f(z). x f(x) + R f(x)dx = lim f(x)dx. R + f(x)dx = = lim R f(x)dx + f(x)dx f(x)dx + lim R R f(x)dx Im z R Re z.: +R. R f(z) = R f(x)dx + f(z) 3
4 R f(x)dx = f(z) f(z) R f(z) = lim R f(x) p(x) q(x) f(x) = p(x) q(x) = [ q(x) [ p(x) + p(x) [ q(x) dx =πi Res(z ) + Res(z )+ + Res(z n ) Res(z k ) k =,,,n z k f(z) z k q(z) f(x) f(x) x 4 + dx = 4 π R x 4 + dx = R x 4 + dx R f(z) = z 4 +. R f(z) x 4 + dx = [ z 4 + z 4 + z = +i, z = +i, z 3 = i, z 4 = i
. 5 R z z [ Res(z ) = lim (z z ) z z z 4 + [ Res(z ) = lim (z z ) z z z 4 + = i 4 = i 4 ( ) z 4 + = πi Res(z ) + Res(z ) = πi 4 ( i + i ) = π (.) z 4 + = z 4 + z 4 = R 4 < R 4 R lim R (.) (.) z 4 = (.) + x 4 + dx = π = 4 π
6. sin θ cos θ F (sin θ, cos θ) z =e iθ θ π z : z =. Im z Re z.: sin θ cos θ = iz = iz sin θ = z z, cos θ = z + z i π ( z z F (sinθ, cos θ) = F, z + ) z (.3) i iz z,z,,z n π [ F (sinθ, cos θ) =πi Res(z ) + Res(z )+ + Res(z n ) (.4) π z =e iθ π F (sinθ, cos θ) = =iz = iz F ( z z i, z + ) z. (.5) iz
. 7 π + sin θ = π θ π z =e iθ. sin θ = cos θ + sin θ = 3 cos θ Euler cos θ = z + z z + sin θ = 3 z + z = 4 6 z z = 4z z 6z + =iz = iz π + sin θ = 4z z 6z + iz =i z 6z + (.6) f(z) = z 6z + f(z) z = z =3 z = z =3+ z Res(3 ) = = z z 4 ( z 6z + =πi ) 4 = πi. (.7) (.6) (.7) π + sin θ = π
8.3 sin mx cos mx x F (x) e ix = cos x + i sin x F (x) f(x) sin x dx { sin mx cos mx f(x)e ix dx } dx f(x) cos x dx.3 R f(z)e iz = f(x)e ix dx + f(z)e iz R f(x)e ix dx = f(z)e iz f(z)e iz Im z R Re z.3: R lim R f(z)e iz = f(x) p(x) q(x) f(x) = p(x) q(x)
.3 9 [ q(x) [ p(x) + p(z) = lim R q(z) eiz = p(x) [ ( ) q(x) cos x dx =Re πi Res(z ) + Res(z )+ + Res(z n ) p(x) [ ( ) q(x) sin x dx =Im πi Res(z ) + Res(z )+ + Res(z n ) Res(z k ) k =,,,n z k f(z)e iz z k q(z) cos ax x + b dx = π b e ab a, b > eiaz f(z) = z.3 z = ±ib + b z = ib z = ib [ e iaz Res(ib) = lim (z ib) = e ab z ib (z ib)(z + ib) ib e iaz e ab z =πi + b ib = π b e ab R cos ax x + b dx + i R sin ax x + b dx + e iaz z + b = π b e ab R R e iaz z (R ) + b
3.4.3. Jordan R> π/ e sin θ < π R θ π/ sin θ π/ e sin θ sin θ π θ sin θ R π θ π/ e (R/π) θ = π ( e ) < π R R. M z = R e iθ F (z) M, k > = lim Rk R F (z) M, k > = lim Rk R F (z) = e imz F (z) = R x m k> πr F (z) M πm πr = Rk lim R F (z) R k = lim F (z) = R
.4 3 k> z = R e iθ e imπ F (z) = π e imr eiθ F (R e iθ ) ir e iθ π e imr eiθ F (R e iθ )ir e iθ = = π π π M R k = M R k e imr eiθ F (R e iθ )ir e iθ e imr cos θ mr sin θ F (R e iθ )ir e iθ e mr sin θ F (R e iθ ) R π π/ e mr sin θ e mr sin θ Jordan M π/ R k e mrθ/π < πm mr k m k R
3.5 auchy F (x) a<x <b x a x b ε ε [ b x ε b F (x)dx = lim F (x)dx + F (x)dx a ε,ε a x +ε ε ε ε = ε = ε [ b x ε b F (x)dx = lim F (x)dx + F (x)dx a ε a x +ε auchy sin x x dx = π f(z) = eiz z z =.4 ε R Im z ε ε R Re z.4: z = = e iz z = ε e ix x dx + e iz R z + ε e ix x dx + e iz z
.5 33 x x R ε e ix e ix x e dx + iz z + ε R i R ε sin x x dx = e iz z e iz z z = ε e iθ = e iz z e iεeiθ lim ε π εe iθ iε eiθ = lim i e iεeiθ = πi ε π R R lim i sin x dx = πi R,ε ε x sin x x dx = π.