讲授要点 O Taylor,展开 展开定理 讨论 基本函数展开式 Tavion展开举例 级数乘法与待定系数法 多值函数的 avlon展开 。在无穷远点的 Taylor展开 ③解析函数的唯一性 解析函数零点的孤立性 。解析函数的唯一性
Expansion in Taylor Series Taylor Expansion: Examples Identity Theorem for Analytic Functions Theorem (Taylor) Discussions & Remarks Illustrative Examples ùÇ: 1 TaylorÐm Ðm½n ?Ø Ä ¼êÐmª 2 TaylorÐmÞ~ ?ê¦{½Xê{ õ¼êTaylorÐm 3á:TaylorÐm 3 )Û¼ê5 )Û¼ê":á5 )Û¼ê5 C. S. Wu 1Ôù )Û¼êTaylorÐm
展开定理( Taylor) 设函数f(x)在以a为圆心的圆C内及C上解析,则 对于圆内的任何z点,∫(z)可用幂级数展开为(或 者说,f(z)可在a点展开为幂级数) an(z-a n=0 其中 an271 d )n n! C取逆时针方向 以后的围道积分,除特别说明的以外,均为逆时针方向
Expansion in Taylor Series Taylor Expansion: Examples Identity Theorem for Analytic Functions Theorem (Taylor) Discussions & Remarks Illustrative Examples Ðm½n(Taylor) ¼êf(z)3±a % CS9Cþ)Û§K éu S?Ûz:§ f(z)^?êÐm(½ ö`§f(z)3a:Ðm?ê) f(z) = X ∞ n=0 an(z − a) n Ù¥ an = 1 2π i I C f(ζ) (ζ − a) n+1dζ = f (n) (a) n! C_a a±È©§ØAO`²± §þ_ C. S. Wu 1Ôù )Û¼êTaylorÐm
展开定理( Taylor) (要点) f(2)=∑an(2-a)y ∫() f(n(a) 2i f i-antrds = n 根据 Cauchy积分公式,对于圆C内任意一点 <1的区域中一致收敛
Expansion in Taylor Series Taylor Expansion: Examples Identity Theorem for Analytic Functions Theorem (Taylor) Discussions & Remarks Illustrative Examples Ðm½n(Taylor) (:) f(z) = X∞ n=0 an(z − a) n an = 1 2π i I C f(ζ) (ζ − a) n+1 dζ = f (n) (a) n! âCauchyÈ©úª§éu CS?¿:z f(z) = 1 2π i I C f(ζ) ζ − z dζ ∵ 1 ζ−z = 1 (ζ−a)−(z−a) = 1 ζ−a X ∞ n=0 z−a ζ−a n 3 z − a ζ − a ≤ r < 1«¥Âñ C. S. Wu 1Ôù )Û¼êTaylorÐm
展开定理( Taylor) (要点) f(2)=∑an(2-a)y ∫() f(n(a) 2i f i-antrds = n 根据 Cauchy积分公式,对于圆C内任意一点z f(z)= f() d 2TiJC5-z <1的区域中一致收敛 C. S. Wu
Expansion in Taylor Series Taylor Expansion: Examples Identity Theorem for Analytic Functions Theorem (Taylor) Discussions & Remarks Illustrative Examples Ðm½n(Taylor) (:) f(z) = X∞ n=0 an(z − a) n an = 1 2π i I C f(ζ) (ζ − a) n+1 dζ = f (n) (a) n! âCauchyÈ©úª§éu CS?¿:z f(z) = 1 2π i I C f(ζ) ζ − z dζ ∵ 1 ζ−z = 1 (ζ−a)−(z−a) = 1 ζ−a X ∞ n=0 z−a ζ−a n 3 z − a ζ − a ≤ r < 1«¥Âñ C. S. Wu 1Ôù )Û¼êTaylorÐm
展开定理( Taylor) (要点) f(2)=∑an(2-a)y ∫() f(n(a) 2i f i-antrds = n 根据 Cauchy积分公式,对于圆C内任意一点z f(z)= f() d 2TiJC5-z c-2(-a)-(2-a)(-a2 在 ≤r<1的区域中一致收敛 C. S. Wu
Expansion in Taylor Series Taylor Expansion: Examples Identity Theorem for Analytic Functions Theorem (Taylor) Discussions & Remarks Illustrative Examples Ðm½n(Taylor) (:) f(z) = X∞ n=0 an(z − a) n an = 1 2π i I C f(ζ) (ζ − a) n+1 dζ = f (n) (a) n! âCauchyÈ©úª§éu CS?¿:z f(z) = 1 2π i I C f(ζ) ζ − z dζ ∵ 1 ζ−z = 1 (ζ−a)−(z−a) = 1 ζ−a X ∞ n=0 z−a ζ−a n 3 z − a ζ − a ≤ r < 1«¥Âñ C. S. Wu 1Ôù )Û¼êTaylorÐm