Solutions in vicinit 方程奇点邻域内的解 1(2)=(2-20)∑ck(2-20) k=-∞o ul2(2)=gu1()n(2-x0)+(x-20)2∑dk(z-0)
Solutions in Vicinity of Regular Singularity Outlines & Conclusions) Example: Bessel Equation Solutions in Vicinity of Singularity Regular Singularity §Û:S) w1(z) = (z−z0) ρ1 P ∞ k=−∞ ck(z−z0) k w2(z) = gw1(z) ln(z−z0)+(z−z0) ρ2 P ∞ k=−∞ dk(z−z0) k XJρ1½ρ2´ê§ g = 0§Kz0:§ )4:½ 5Û: XJρ1½ρ2شꧽg 6= 0§K§) õ¼ê§z0:Ù{: C. S. Wu 1ù ~©§?ê){()
Solutions in vicinit 方程奇点邻域内的解 1(2)=(2-20)∑ck(2-20) k=-∞o l2(2)=gu(2)n(2-20)+(x-20)2∑d(2-x0) 如果1或p2是整数,且g=0,则20点为方程 解的极点或本性奇点 如果P1或P2不是整数,或9≠0,则方程的解 为多值函数,0点为其枝点
Solutions in Vicinity of Regular Singularity Outlines & Conclusions) Example: Bessel Equation Solutions in Vicinity of Singularity Regular Singularity §Û:S) w1(z) = (z−z0) ρ1 P ∞ k=−∞ ck(z−z0) k w2(z) = gw1(z) ln(z−z0)+(z−z0) ρ2 P ∞ k=−∞ dk(z−z0) k XJρ1½ρ2´ê§ g = 0§Kz0:§ )4:½ 5Û: XJρ1½ρ2شꧽg 6= 0§K§) õ¼ê§z0:Ù{: C. S. Wu 1ù ~©§?ê){()
Solutions in vicinit 方程奇点邻域内的解 1(2)=(2-20)∑ck(2-20) k=-∞o l2(2)=gu(2)n(2-20)+(x-20)2∑d(2-x0) 如果1或P是整数,且g=0,则0点为方程 解的极点或本性奇点 ●如果1或P不是整数,或g≠0,则方程的解 为多值函数,2点为其枝点
Solutions in Vicinity of Regular Singularity Outlines & Conclusions) Example: Bessel Equation Solutions in Vicinity of Singularity Regular Singularity §Û:S) w1(z) = (z−z0) ρ1 P ∞ k=−∞ ck(z−z0) k w2(z) = gw1(z) ln(z−z0)+(z−z0) ρ2 P ∞ k=−∞ dk(z−z0) k XJρ1½ρ2´ê§ g = 0§Kz0:§ )4:½ 5Û: XJρ1½ρ2شꧽg 6= 0§K§) õ¼ê§z0:Ù{: C. S. Wu 1ù ~©§?ê){()
Solutions in vicinit 方程奇点邻域内的解 u1(2)=(2-x0)∑ck(z-20) u2(z)=gn1(z)n(z-x0)+(x-30)∑dk(z-0) k= 现在如果我们把上面的解式代入方程,尽管仍然 能得到系数之间的递推关系,但却无法求出系数 的普遍表达式,因为这时的级数解中,一般说 来,都有无穷多个正幂项和负幂项,我们无法设 定系数的“初值 尜
Solutions in Vicinity of Regular Singularity Outlines & Conclusions) Example: Bessel Equation Solutions in Vicinity of Singularity Regular Singularity §Û:S) w1(z) = (z−z0) ρ1 P ∞ k=−∞ ck(z−z0) k w2(z) = gw1(z) ln(z−z0)+(z−z0) ρ2 P ∞ k=−∞ dk(z−z0) k y3XJ·rþ¡)ª\§§¦+E, UXêm4í'X§%Ã{¦ÑXê ÊHLª© Ïù?ê)¥§` 5§ÑkáõÚK§·Ã{ ½Xê/Ð0 C. S. Wu 1ù ~©§?ê){()
方程奇点邻域内的解 1(2)=(z-20)2 Ck k u2(2)=gu1(2)n(2-0)+(x-20)2∑d(z-x0) k==oo 如果级数解中只有有限个负幂项,这时总可以调 整相应的ρ值,使得级数解中没有负幂项 1(2)=(x-20)∑ck(z-20) (z)=91(2) )+(2-2x0)∑dk(2-0)
Solutions in Vicinity of Regular Singularity Outlines & Conclusions) Example: Bessel Equation Solutions in Vicinity of Singularity Regular Singularity §Û:S) w1(z) = (z−z0) ρ1 P ∞ k=−∞ ck(z−z0) k w2(z) = gw1(z) ln(z−z0)+(z−z0) ρ2 P ∞ k=−∞ dk(z−z0) k XJ?ê)¥kkK§ùo±N Aρ§¦?ê)¥vkK w1(z) = (z − z0) ρ1 P ∞ k=0 ck(z − z0) k w2(z) = gw1(z) ln(z−z0)+(z−z0) ρ2 P ∞ k=0 dk(z−z0) k C. S. Wu 1ù ~©§?ê){()