Bessel, Neumann Function(properties,\(Y_\nu (x) \) ) CH5.5

이전글:

physical-world.tistory.com/41

Bessel's Equation, Bessel's Function Ch5.4

Bessel Function(properties) CH5.5

베셀 함수의 특징들을 알아보자.

$$J_{\nu }(x)\equiv (\frac{x}{2}{)}^{\nu }\sum _{j=0}^{\mathrm{\infty }}\frac{(-1{)}^j}{j!\mathrm{\Gamma }(j+\nu +1)}(\frac{x}{2}{)}^{2j}$$

 

정리1. 미분, 점화식

$$(a)[x^{\nu }{J}_{\nu }(x){]}^{\prime }=x^{\nu }{J}_{\nu -1}(x)$$

$$(b)[x^{-\nu }{J}_{\nu }(x){]}^{\prime }=-x^{-\nu }{J}_{\nu +1}(x)$$

$$(c)J_{\nu -1}(x)+J_{\nu +1}(x)=\frac{2\nu }{x}{J}_{\nu }(x)$$

$$(d)J_{\nu -1}(x)-J_{\nu +1}(x)=2[J_{\nu }(x){]}^{\prime }$$

 

베셀 방정식의 일반해:

$$x^2{y}^{″}+x{y}^{\prime }+(x^2-{\nu }^2)y=0$$

$$Indicialequation:r^2-{\nu }^2=0$$

 

If, $\nu$가 정수가 아니면,

$$J_{\nu }(x),J_{-\nu }(x)\text{are linearly independent}$$

따라서, 

일반해:

$$y=c_1{J}_{\nu }(x)+c_2{J}_{-\nu }(x)$$

$$forx\ne 0$$

 

If $\nu$가 정수라면, $\nu =n$

$J_{\nu }(x),J_{-\nu }(x)$은 선형 종속 관계.

$$\because J_{-n}(x)=(-1{)}^n{J}_n(x)$$

그렇다면, 또 다른 해는?

$$\Rightarrow Y_{\nu }(x)$$

Second kind of Bessel Function : $Y_{\nu }(x)$

= Neumann Function 

$$Y_{\nu }(x)\equiv \frac{J_{\nu }(x)\cos \nu \pi -J_{-\nu }(x)}{\sin \nu \pi }$$

 

따라서, 베셀 방정식의 일반해:

정리1.

$$y(x)=C_1{J}_{\nu }(x)+C_2{Y}_{\nu }(x)$$

$$forx>0,forall\nu$$

 

ㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡ

추가 설명, 그래프.

$\nu$가 $\nu =n=0,1,2\cdots$ 일 때, 그래프 개형을 그려보자.

First kind Bessel Function:$J_{\nu }(x)$

$$J_{\nu }(x)\equiv (\frac{x}{2}{)}^{\nu }\sum _{j=0}^{\mathrm{\infty }}\frac{(-1{)}^j}{j!\mathrm{\Gamma }(j+\nu +1)}(\frac{x}{2}{)}^{2j}$$

베셀 함수(n=0,1,2...)

 

Second kind Bessel Function:$Y_{\nu }(x)$

$$Y_{\nu }(x)\equiv \frac{J_{\nu }(x)\cos \nu \pi -J_{-\nu }(x)}{\sin \nu \pi }$$

 

추가 : Spherical Bessel function

$$x^2{y}^{″}+2x{y}^{\prime }+(x^2-l(l+1))y=0$$

sol:

$$\Rightarrow j_l(x),n_l(x)$$

$$\{\begin{array}{l}{j}_l(x)=(-x{)}^l(\frac{1}{x}\frac{d}{dx}{)}^l\frac{\sin x}{x}\\ n_l(x)=-(-x{)}^l(\frac{1}{x}\frac{d}{dx}{)}^l\frac{\cos x}{x}\end{array}$$

 

• $j_l(x)$

$\{\begin{array}{ll}\frac{x^l}{(2l+1)!!}& x\to 0\\ \frac{\sin (x-\pi l/2)}{x}& x\to \mathrm{\infty }\end{array}$

 

 

• $n_l(x)$

 $\{\begin{array}{ll}-\frac{(2l-1)!!}{x^{l+1}}& x\to 0\\ -\frac{\cos (x-\pi l/2)}{x}& x\to \mathrm{\infty }\end{array}$

여기서 보다시피,

$n_l(x)$은 x=0에서 irregular 하다.

 

※ Spherical Bessel function vs Bessel function

$$j_l(x)=\sqrt{\frac{\pi }{2x}}{J}_{l+\frac{1}{2}}(x)$$

$$n_l(x)=(-1{)}^{l+1}\sqrt{\frac{\pi }{2x}}{J}_{-l-\frac{1}{2}}(x)$$

 

그래프:

Spherical Bessel function

 

Spherical Neumann function