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yungman

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I am trying to evaluate[tex]\int J_{2}(x)dx[/tex]

I have been trying to use all the identities involving Bessel function to no prevail. The ones I used are:

[tex]\frac{d}{dx}[x^{-p}J_{p}(x)]=-x^{-p}J_{p+1}(x)[/tex] (1)

[tex]\frac{d}{dx}[x^{p}J_{p}(x)]=-x^{p}J_{p-1}(x)[/tex] (2)

[tex]J_{p-1}(x)]+J_{p+1}(x)=\frac{2p}{x}J_{p}(x)[/tex] (3)

[tex]J'_{0}(x)=-J_{1}(x)[/tex] (4)

So far this is the closest:

[tex]\int J_{2}(x)dx=\int x[x^{-1}J_{2}(x)]dx[/tex] Using (1) with p=1 [tex]=-\int x \frac{d}{dx}[x^{-1}J_{1}](x)]dx[/tex]

[tex]u=x,du=dx, v=x^{-1}J_{1}(x)[/tex] using intergate by parts:

[tex]\int J_{2}(x)dx=-J_{1}+\int x^{-1}J_{1}(x)dx[/tex] Using (3) p=1 [tex]\int J_{2}(x)dx=-J_{1}(x)+\frac{1}{2}\int J_{0}(x)dx+\frac{1}{2}\int J_{2}(x)dx[/tex]

[tex]\Rightarrow \frac{1}{2}\int J_{2}(x)dx=-J_{1}(x)+\frac{1}{2}\int J_{0}(x)dx \Rightarrow \int J_{2}(x)dx=-2J_{1}(x)+\int J_{0}(x)dx[/tex]

I have not been able to go any further, can anyone help or at least give me a hint. Is there a general approach of solving these problems of integration. I worked on integrating of Bessel function of order 3, order 1. The methods are very different.

Thanks

Alan

I have been trying to use all the identities involving Bessel function to no prevail. The ones I used are:

[tex]\frac{d}{dx}[x^{-p}J_{p}(x)]=-x^{-p}J_{p+1}(x)[/tex] (1)

[tex]\frac{d}{dx}[x^{p}J_{p}(x)]=-x^{p}J_{p-1}(x)[/tex] (2)

[tex]J_{p-1}(x)]+J_{p+1}(x)=\frac{2p}{x}J_{p}(x)[/tex] (3)

[tex]J'_{0}(x)=-J_{1}(x)[/tex] (4)

So far this is the closest:

[tex]\int J_{2}(x)dx=\int x[x^{-1}J_{2}(x)]dx[/tex] Using (1) with p=1 [tex]=-\int x \frac{d}{dx}[x^{-1}J_{1}](x)]dx[/tex]

[tex]u=x,du=dx, v=x^{-1}J_{1}(x)[/tex] using intergate by parts:

[tex]\int J_{2}(x)dx=-J_{1}+\int x^{-1}J_{1}(x)dx[/tex] Using (3) p=1 [tex]\int J_{2}(x)dx=-J_{1}(x)+\frac{1}{2}\int J_{0}(x)dx+\frac{1}{2}\int J_{2}(x)dx[/tex]

[tex]\Rightarrow \frac{1}{2}\int J_{2}(x)dx=-J_{1}(x)+\frac{1}{2}\int J_{0}(x)dx \Rightarrow \int J_{2}(x)dx=-2J_{1}(x)+\int J_{0}(x)dx[/tex]

I have not been able to go any further, can anyone help or at least give me a hint. Is there a general approach of solving these problems of integration. I worked on integrating of Bessel function of order 3, order 1. The methods are very different.

Thanks

Alan

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