"자코비 다항식"의 두 판 사이의 차이

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<h5>목록</h5>
 
<h5>목록</h5>
  
−
1<br> 1/2 (a-b+(2+a+b) z)<br> 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<br> 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_0(z)=1<br> P_1(z)=1/2 (\[Alpha]-\[Beta]+z (2+\[Alpha]+\[Beta]))<br> P_2(z)=1/2 (1+\[Alpha]) (2+\[Alpha])+1/2 (-1+z) (2+\[Alpha]) (3+\[Alpha]+\[Beta])+1/8 (-1+z)^2 (3+\[Alpha]+\[Beta]) (4+\[Alpha]+\[Beta])<br> P_3(z)=1/6 (1+\[Alpha]) (2+\[Alpha]) (3+\[Alpha])+1/4 (-1+z) (2+\[Alpha]) (3+\[Alpha]) (4+\[Alpha]+\[Beta])+1/8 (-1+z)^2 (3+\[Alpha]) (4+\[Alpha]+\[Beta]) (5+\[Alpha]+\[Beta])+1/48 (-1+z)^3 (4+\[Alpha]+\[Beta]) (5+\[Alpha]+\[Beta]) (6+\[Alpha]+\[Beta])<br> P_4(z)=1/24 (1+\[Alpha]) (2+\[Alpha]) (3+\[Alpha]) (4+\[Alpha])+1/12 (-1+z) (2+\[Alpha]) (3+\[Alpha]) (4+\[Alpha]) (5+\[Alpha]+\[Beta])+1/16 (-1+z)^2 (3+\[Alpha]) (4+\[Alpha]) (5+\[Alpha]+\[Beta]) (6+\[Alpha]+\[Beta])+1/48 (-1+z)^3 (4+\[Alpha]) (5+\[Alpha]+\[Beta]) (6+\[Alpha]+\[Beta]) (7+\[Alpha]+\[Beta])+1/384 (-1+z)^4 (5+\[Alpha]+\[Beta]) (6+\[Alpha]+\[Beta]) (7+\[Alpha]+\[Beta]) (8+\[Alpha]+\[Beta])
  
 
 
 
 

2010년 5월 2일 (일) 09:20 판

이 항목의 스프링노트 원문주소

 

 

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
    \(P_n^{(\alpha,\beta)} (z) = \frac{\Gamma (\alpha+n+1)}{n!\Gamma (\alpha+\beta+n+1)} \sum_{m=0}^n {n\choose m} \frac{\Gamma (\alpha + \beta + n + m + 1)}{\Gamma (\alpha + m + 1)} \left(\frac{z-1}{2}\right)^m\)
     

 

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
    \(\int_{-1}^1 (1-x)^{\alpha} (1+x)^{\beta} P_m^{(\alpha,\beta)} (x)P_n^{(\alpha,\beta)} (x) \; dx= \frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1} \frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)n!} \delta_{nm}\)

 

 

3항 점화식

 

 

 

미분방정식
  • 자코비 다항식은 다음을 만족시킨다
    \((1-x^2)y'' + ( \beta-\alpha - (\alpha + \beta + 2)x )y'+ n(n+\alpha+\beta+1) y = 0\)

 

 

목록

P_0(z)=1
P_1(z)=1/2 (\[Alpha]-\[Beta]+z (2+\[Alpha]+\[Beta]))
P_2(z)=1/2 (1+\[Alpha]) (2+\[Alpha])+1/2 (-1+z) (2+\[Alpha]) (3+\[Alpha]+\[Beta])+1/8 (-1+z)^2 (3+\[Alpha]+\[Beta]) (4+\[Alpha]+\[Beta])
P_3(z)=1/6 (1+\[Alpha]) (2+\[Alpha]) (3+\[Alpha])+1/4 (-1+z) (2+\[Alpha]) (3+\[Alpha]) (4+\[Alpha]+\[Beta])+1/8 (-1+z)^2 (3+\[Alpha]) (4+\[Alpha]+\[Beta]) (5+\[Alpha]+\[Beta])+1/48 (-1+z)^3 (4+\[Alpha]+\[Beta]) (5+\[Alpha]+\[Beta]) (6+\[Alpha]+\[Beta])
P_4(z)=1/24 (1+\[Alpha]) (2+\[Alpha]) (3+\[Alpha]) (4+\[Alpha])+1/12 (-1+z) (2+\[Alpha]) (3+\[Alpha]) (4+\[Alpha]) (5+\[Alpha]+\[Beta])+1/16 (-1+z)^2 (3+\[Alpha]) (4+\[Alpha]) (5+\[Alpha]+\[Beta]) (6+\[Alpha]+\[Beta])+1/48 (-1+z)^3 (4+\[Alpha]) (5+\[Alpha]+\[Beta]) (6+\[Alpha]+\[Beta]) (7+\[Alpha]+\[Beta])+1/384 (-1+z)^4 (5+\[Alpha]+\[Beta]) (6+\[Alpha]+\[Beta]) (7+\[Alpha]+\[Beta]) (8+\[Alpha]+\[Beta])

 

 

재미있는 사실

 

 

 

역 사

 

 

 

메 모

 

 

관련된 항목들

 

 

수학용어번역

 

 

사전 형태의 자료

 

 

관 련논문

 

 

관련도서

 

 

관 련기사

 

 

블 로그
원본 주소 "https://wiki.mathnt.net/index.php?title=자코비_다항식&oldid=17395"