Why bessel function

This, in outermost quotes is a paragraph taken from Cajori's Book "A history of Mathematical Symbols" , clause , pg , vol-II explaining the symbol chosen for Bessel Function of the first kind. Hope it helps. Cajori, F. II: Notations mainly in higher mathematics. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Ask Question.

Asked 10 years, 6 months ago. Active 11 months ago. Viewed times. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. I came across an integral that is the Fourier transform of the nth power of a surface roughness correlation function with correlation length zeta.

I am wondering why does this integral contains a Bessel function and what is the intuitive meaning of this? Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more.

At the same time, the point is a branch point except in the case of integer for the two functions. For fixed integer , the functions and are entire functions of. For fixed , the functions , , , and are entire functions of and have only one essential singular point at.

For fixed noninteger , the functions , , , and have two branch points: , , and one straight line branch cut between them. For fixed integer , only the functions and have two branch points: , , and one straight line branch cut between them.

All Bessel functions , , , and do not have periodicity. The two Bessel functions of the first kind have special parity either odd or even in each variable:. The two Bessel functions of the second kind have special parity either odd or even only in their parameter:.

The last four formulas have restrictions that do not allow their right sides to become indeterminate expressions for integer. In such cases, evaluation of the limit from the right sides leads to much more complicated representations, for example:. The asymptotic behavior of the Bessel functions , , , and can be described by the following formulas which show only the main terms :.

In particular cases, when or , the second and fourth formulas can be simplified to the following forms:. The Bessel functions , , , and have simple integral representations through the cosine or the hyperbolic cosine or exponential function and power functions in the integrand:.

The argument of the Bessel functions , , , and sometimes can be simplified through formulas that remove square roots from the arguments. For the Bessel functions of the second kind and with integer index , this operation is realized by special formulas that include logarithms:.

If the argument of a Bessel function includes an explicit minus sign, the following formulas produce Bessel functions without the minus sign argument:. The Bessel functions , , , and satisfy the following recurrence identities:. The last eight identities can be generalized to the following recurrence identities with jump length :. The derivatives of all the four Bessel functions , , , and have rather simple and symmetrical representations that can be expressed through other Bessel functions with different indices:.

The symbolic -order derivatives have more complicated representations through the regularized hypergeometric function or generalized Meijer G function:. When is real, the functions and each have an infinite number of real zeros, all of which are simple with the possible exception of the zero :. When , the zeros of are all real.