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I am currently submitting a paper in a distantly related field (experimental psychology) in which we are using a von Mises distribution to model certain aspects of perceptually-driven behavior. One of our reviewers has requested that we write out the full equation to the Bessel function of zero order, which is a component of the former. I'm happy to oblige, but I've been unable to find a clear example of this equation. Would any of you folks be so kind as to point me in the right direction?

Thanks!

Louis Thibault
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    What do you mean by equation? Differential equation? Or just a formula (e.g. Taylor series) for $J_0(x)$ or even $Y_0(x)$? – gammatester Dec 14 '15 at 09:12
  • @gammatester, I have to admit I'm in over my head. The context is that we lifted the definition of a von Mises distribution from wikipedia, which includes "the modified Bessel function of order 0". A reviewer then asked us to "write out the full equation for the Bessel function", so I'm at a complete loss. What do you think is most appropriate? – Louis Thibault Dec 14 '15 at 09:16
  • Can't you refer to the NIST page ? http://dlmf.nist.gov/10.25 –  Dec 14 '15 at 09:33
  • @YvesDaoust I was not aware such a thing existed. Thanks! – Louis Thibault Dec 14 '15 at 09:42
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    There's a paper version: http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521192255 Also famous and "citeable": Handbook of mathematical functions http://people.math.sfu.ca/~cbm/aands/intro.htm#006 –  Dec 14 '15 at 09:54

2 Answers2

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Your question should be more specific. First, don't confuse "Bessel functions" and "Modified Bessel functions": they are different. In one of each of whose two sets of functions, they can be of the "first kind" or of the "second kind" : again different sub-sets of functions. And in each one of these sub-sets, they are different Bessel functions of various order.

For examples:

The modified Bessel function of the first kind and order $0$ is $I_0(x)$. One integral definition is : $$I_0(x)=\frac{1}{\pi}\int_0^\pi \exp\left(x\cos(t)\right)dt$$

The modified Bessel function of the second kind and order $0$ is $K_0(x)$. One integral definition is : $$K_0(x)=\int_0^\infty \cos\left(x \sinh(t) \right)dt$$

Series expressions can be found in : http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html

and related differential equation : http://mathworld.wolfram.com/ModifiedBesselDifferentialEquation.html

JJacquelin
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Perhaps is is best to give both informations. The modified Bessel functions $I_n$ of the first kind of order $n$ are solutions of the modified Bessel differential equation: $$x^2y''(x) + xy'(x)-(x^2+n^2) y(x)=0$$ Their Taylor series are $$I_n(x) = (\tfrac{1}{2}x)^n \sum\limits_{k=0}^{\infty} \frac{(\tfrac{1}{4}x^2)^k}{k!(n+k)!}$$ Note that in https://en.wikipedia.org/wiki/Von_Mises_distribution#Definition not only $I_0$ appears but all functions $I_n$.

But this is only the formal part, I have no clue why the $I_n$ appear in the von Mises distribution.

gammatester
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    Many thanks! We'll be sure to toast to you when we publish :) – Louis Thibault Dec 14 '15 at 09:29
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    I happen to have been browsing though my stack-exchange history and stumbled on my previous comment. The paper was published a few years ago, so thank you once again for your help. Link if you're interested: https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0148504 – Louis Thibault Feb 10 '21 at 21:59