자코비 다항식

개요

• 직교다항식

 

 

정의

• 초기하급수(Hypergeometric series)를 통해 정의된다\[P_n^{(\alpha,\beta)}(z)=\frac{(\alpha+1)_n}{n!} \,_2F_1\left(-n,1+\alpha+\beta+n;\alpha+1;\frac{1-z}{2}\right)\]
  
• 다항식표현\[P_n^{(\alpha,\beta)} (z) = \frac{\Gamma (\alpha+n+1)}{n!\Gamma (\alpha+\beta+n+1)} \sum_{m=0}^n {n\choose m} \frac{\Gamma (\alpha + \beta + n + m + 1)}{\Gamma (\alpha + m + 1)} \left(\frac{z-1}{2}\right)^m\]
  

 

 

 

 

 

 

미분방정식

• 자 코비 다항식은 다음을 만족시킨다\[(1-x^2)y'' + ( \beta-\alpha - (\alpha + \beta + 2)x )y'+ n(n+\alpha+\beta+1) y = 0\]
  

 

 

직교성

• weight함수와 구간\[w(x) = (1-x)^{\alpha} (1+x)^{\beta}\]\[[-1,1]\]
  
• \(m\neq n\) 일 때\[\int_{-1}^1 (1-x)^{\alpha} (1+x)^{\beta} P_m^{(\alpha,\beta)} (x)P_n^{(\alpha,\beta)} (x) \; dx= 0\]
  
• \(m=n\) 일 때\[\int_{-1}^1 (1-x)^{\alpha} (1+x)^{\beta} P_m^{(\alpha,\beta)} (x)P_n^{(\alpha,\beta)} (x) \; dx= \frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1} \frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)n!} \delta_{nm}\]
  예 \(\alpha=2,\beta=2,m=n=2\)\[\int_{-1}^1 (1-x)^{\frac{1}{2}} (1+x)^{\frac{1}{2}} P_2^{(\frac{1}{2},\frac{1}{2})} (x)P_2^{(\frac{1}{2},\frac{1}{2})} (x) \; dx= \frac{4}{6} \frac{\Gamma(3+\frac{1}{2})\Gamma(3+\frac{1}{2})}{\Gamma(4)2!}=\frac{4(\frac{15\sqrt{\pi}}{8})^2}{12\cdot 3!}=\frac{25\pi}{128}\]
  

 

 

목록

• 매쓰매티카 코드
  
  1. Do[Print["P_",i,"(z)=",JacobiP[i,a,b,z]],{i,0,4}]

 

P_0(z)=1
P_1(z)=1/2 (a-b+(2+a+b) z)
P_2(z)=1/2 (1+a) (2+a)+1/2 (2+a) (3+a+b) (-1+z)+1/8 (3+a+b) (4+a+b) (-1+z)^2
P_3(z)=1/6 (1+a) (2+a) (3+a)+1/4 (2+a) (3+a) (4+a+b) (-1+z)+1/8 (3+a) (4+a+b) (5+a+b) (-1+z)^2+1/48 (4+a+b) (5+a+b) (6+a+b) (-1+z)^3
P_4(z)=1/24 (1+a) (2+a) (3+a) (4+a)+1/12 (2+a) (3+a) (4+a) (5+a+b) (-1+z)+1/16 (3+a) (4+a) (5+a+b) (6+a+b) (-1+z)^2+1/48 (4+a) (5+a+b) (6+a+b) (7+a+b) (-1+z)^3+1/384 (5+a+b) (6+a+b) (7+a+b) (8+a+b) (-1+z)^4

 

 

역사

 

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사 연표
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사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Jacobi_polynomials
• http://www.wolframalpha.com/input/?i=
• NIST Digital Library of Mathematical Functions
• The On-Line Encyclopedia of Integer Sequences
  
  • http://www.research.att.com/~njas/sequences/?q=