# Bessel Functions

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## Bessel Functions

cylinder functions of the first kind; they occur in the consideration of physical processes (heat conduction, diffusion, oscillations, and so on) in areas with circular and cylindrical symmetry. They are solutions of the Bessel equation.

The Bessel function J_{p} of the order (index) *p*,- ∞ < *p* < ∞, is represented by the series

which converges for all *x.* Its graph for x > 0 has the form of a damped oscillation; J_{p}(*x*) has an infinite number of zeros. The behavior of J_{p} (*x*) for small | *x* | is given by the first term of the series (1); for large x *>* 0, the following asymptotic representation holds:

Here the oscillatory character of the function is clearly shown. The Bessel functions of the “half-integer” order *p = n +* ½ are expressed in terms of elementary functions—in particular,

The Bessel functions , where are the positive zeros of J_{p}*(x)* and where *p* > - ½, form an orthogonal system with weight *×* in the interval (0, l).

The function J_{0} was first considered by D. Bernoulli in a paper devoted to the oscillation of heavy chains (1732). L. Euler, considering the problem of the oscillations of a circular membrane (1738), arrived at the Bessel equation with whole values *p = n* and found an expression for J_{n}(x) in the form of a series of the powers of *x.* In later works he extended this expression to the case of arbitrary *ρ* values. F. Bessel investigated the functions J_{p} (*x*) in connection with the study of the motion of the planets around the sun (1824). He compiled the first tables for J_{0} (x), J_{1} (*x*), and J_{2} (x).

### REFERENCES

Watson, G. N.*Teoriia besselevykh funktsii,*parts 1–2. Moscow, 1949. (Translated from English.)

Lebedev, N. N.

*Spetsial’nye funktsii i ikh prilozheniia,*2nd ed. Moscow-Leningrad, 1963.

Bateman, H., and A. Erdély.

*Vysshie transtsendentnye funktsii: Funktsii Besselia, funktsii parabolicheskogo tsilindra, orto-gonal’nye mnogochleny.*Moscow, 1966. (Translated from English.) P. I. Lizorkin