Need information in deriving first integral of Bessel's function.

[yungman]

Can anyone give me the proof or point me to a link to derive this First Integral of Bessel's Function.

$$J_n(x)=\frac{j^{-n}}{\pi}\int_0^{\pi}\cos(n\phi)e^{jx\cos \phi}d\phi$$

I have been searching all over and can't find the derivation. In fact, other than some basic Bessel's identities that is shown in one PDE book, I have yet to find anything.

Thanks

 

[fzero]

I do not want to nitpick your English, but this expression is more properly called "an integral representation of the Bessel function of the first kind." I only point this out because "first integral" makes it seem as if you are integrating the Bessel function, which is another interesting thing that we could consider.

That said, the first thing that comes to mind when given any representation of a function is to check that the representation satisfies the definition of the function. So you could check that this integral expression actually satisfies Bessel's equation. It seems like you probably do not need to actually perform any of the $\phi$ integrals to check this.

Second, I have located a discussion of the http://www.math.psu.edu/papikian/Kreh.pdf. The text by Watson referred to there is a classic reference that you might also want to examine.

 

[yungman]

I do not want to nitpick your English, but this expression is more properly called "an integral representation of the Bessel function of the first kind." I only point this out because "first integral" makes it seem as if you are integrating the Bessel function, which is another interesting thing that we could consider.

That said, the first thing that comes to mind when given any representation of a function is to check that the representation satisfies the definition of the function. So you could check that this integral expression actually satisfies Bessel's equation. It seems like you probably do not need to actually perform any of the $\phi$ integrals to check this.

Second, I have located a discussion of the http://www.math.psu.edu/papikian/Kreh.pdf. The text by Watson referred to there is a classic reference that you might also want to examine.

thanks for the response. I downloaded the file and I need to spend some time reading it. So far I don't see the direct explanation of my question, can you give me some hint what equation to use in the article to derive the identity I asked?

I read the name from one of the article I read on line, they called first integration for whatever reason.

Thanks

 

[SteamKing]

Try this article on for size. It shows a derivation of the equation for Jn from the OP.

http://www.math.kau.se/mirevine/mf2bess.pdf

 

[fzero]

thanks for the response. I downloaded the file and I need to spend some time reading it. So far I don't see the direct explanation of my question, can you give me some hint what equation to use in the article to derive the identity I asked?

Sorry, I meant to point out what specific part to look at and neglected to do so. Theorem 2.11 on page 12 gives integral representations. The formulas is obtained from the contour around the real axis, while your formula probably comes from choosing the contour around the imaginary axis. SteamKing's link gives a derivation that doesn't use contour integration, so it's probably easier to follow. There is still a difference between these expressions and your formula, which is probably explained by applying trig identities.

 

[yungman]

Try this article on for size. It shows a derivation of the equation for Jn from the OP.

http://www.math.kau.se/mirevine/mf2bess.pdf

Thanks, this gives me a slightly different derivation that help me going through the first article by Martin Kreh.

Thanks

 

[yungman]

Sorry, I meant to point out what specific part to look at and neglected to do so. Theorem 2.11 on page 12 gives integral representations. The formulas is obtained from the contour around the real axis, while your formula probably comes from choosing the contour around the imaginary axis. SteamKing's link gives a derivation that doesn't use contour integration, so it's probably easier to follow. There is still a difference between these expressions and your formula, which is probably explained by applying trig identities.

Thanks. I am still on page 5 and 6 where the author using the Generating Function to derive all the identities. I am going to go through this a little as it's different from my PDE book. My PDE book does not get into integral representation of Bessel Function at all.

I am going to go through both articles as both seem to be very good in deriving the identities. I am sure I'll be back with more question.

Thanks