Isolated Singulari 讨论 6 Laurent展开既有正幂项,又有负幂项 ρ两部分合起来,就构成 Laurent级数,在环域 R1<|z-b<R2内绝对收敛,在环域内的任 意一个闭区域中一致收敛 当R1=0时, Laurent级数的主要部分就完全 反映了f(x)在z=b点的奇异性
Expansion in Laurent Series Isolated Singularities of Uniform Function Analytic Continuation Theorem (Laurent) Illustrative Examples Laurent Expansion: Multivalued Functions ?Ø ❺ LaurentÐmQk§qkK üÜ©Üå5§Ò¤Laurent?ê§3 R1 < |z − b| < R2SýéÂñ§3S? ¿4«¥Âñ R1 = 0§Laurent?êÌÜ©Ò N f(z)3z = b:ÛÉ5 C. S. Wu 1lù )Û¼êLaurentÐm
Isolated Singulari 讨论 Taylor展开的唯一性 设f()在R1<2一<R2内有两个aent级数 两端同乘以(2 沿环域内绕内圆一周的 任一围道C积分(这两个级数在围道上显然一致收 敛,因而可以逐项积 则由于 2元10,故有a1=
Expansion in Laurent Series Isolated Singularities of Uniform Function Analytic Continuation Theorem (Laurent) Illustrative Examples Laurent Expansion: Multivalued Functions ?Ø ❻ TaylorÐm5 f(z)3R1<|z−b|<R2SküLaurent?ê f(z) = P ∞ n=−∞ an(z − b) n = P ∞ n=−∞ a 0 n (z − b) n üàÓ¦±(z − b) −k−1 §÷S7S ± ?CÈ©(ùü?ê3þw, ñ§ Ï ±ÅÈ©) KduI C (z − b) n−k−1 dz = 2π iδnk§kak = a 0 k Ïk?¿§k ak =a 0 k k=0, ±1, ±2, · · · C. S. Wu 1lù )Û¼êLaurentÐm
Isolated Singulari 讨论 Taylor展开的唯一性 设f()在R1<|z-b<R2内有两个 Laurent级数 f(z)=∑an(z-b)=∑an(z-b)2 n==00 两端同乘以(2-b) 沿环域内绕内圆一周的 任一围道C积分(这两个级数在围道上显然一致收 因而可以逐项积分 则由于
Expansion in Laurent Series Isolated Singularities of Uniform Function Analytic Continuation Theorem (Laurent) Illustrative Examples Laurent Expansion: Multivalued Functions ?Ø ❻ TaylorÐm5 f(z)3R1<|z−b|<R2SküLaurent?ê f(z) = P ∞ n=−∞ an(z − b) n = P ∞ n=−∞ a 0 n (z − b) n üàÓ¦±(z − b) −k−1 §÷S7S ± ?CÈ©(ùü?ê3þw, ñ§ Ï ±ÅÈ©) KduI C (z − b) n−k−1 dz = 2π iδnk§kak = a 0 k Ïk?¿§k ak =a 0 k k=0, ±1, ±2, · · · C. S. Wu 1lù )Û¼êLaurentÐm
Isolated Singulari 讨论 Taylor展开的唯一性 设f()在R1<|z-b<R2内有两个 Laurent级数 f(x)=∑an(z-b)2=∑an(z-b) 两端同乘以(z-b)-k-1,沿环域内绕内圆一周的 任一围道C积分(这两个级数在围道上显然一致收 敛,因而可以逐项积分) 则由于 b)--d2=2元i5k,故有a=a
Expansion in Laurent Series Isolated Singularities of Uniform Function Analytic Continuation Theorem (Laurent) Illustrative Examples Laurent Expansion: Multivalued Functions ?Ø ❻ TaylorÐm5 f(z)3R1<|z−b|<R2SküLaurent?ê f(z) = P ∞ n=−∞ an(z − b) n = P ∞ n=−∞ a 0 n (z − b) n üàÓ¦±(z − b) −k−1 §÷S7S ± ?CÈ©(ùü?ê3þw, ñ§ Ï ±ÅÈ©) KduI C (z − b) n−k−1 dz = 2π iδnk§kak = a 0 k Ïk?¿§k ak =a 0 k k=0, ±1, ±2, · · · C. S. Wu 1lù )Û¼êLaurentÐm
Isolated Singulari 讨论 Taylor展开的唯一性 设f()在R1<|z-b<R2内有两个 Laurent级数 f(x)=∑an(z-b)2=∑an(z-b) 两端同乘以(z-b)-k-1,沿环域内绕内圆一周的 任一围道C积分(这两个级数在围道上显然一致收 敛,因而可以逐项积分) 则由于∮(=-b)--2=2k,故有ak=a 因为k任意,故有=01k=0.±1,+2
Expansion in Laurent Series Isolated Singularities of Uniform Function Analytic Continuation Theorem (Laurent) Illustrative Examples Laurent Expansion: Multivalued Functions ?Ø ❻ TaylorÐm5 f(z)3R1<|z−b|<R2SküLaurent?ê f(z) = P ∞ n=−∞ an(z − b) n = P ∞ n=−∞ a 0 n (z − b) n üàÓ¦±(z − b) −k−1 §÷S7S ± ?CÈ©(ùü?ê3þw, ñ§ Ï ±ÅÈ©) KduI C (z − b) n−k−1 dz = 2π iδnk§kak = a 0 k Ïk?¿§k ak =a 0 k k=0, ±1, ±2, · · · C. S. Wu 1lù )Û¼êLaurentÐm