Solutions of the Bessel differential equation
where is an arbitrary real or complex number (see Bessel equation).
- 1 Cylinder functions of arbitrary order.
- 2 Cylinder functions of integral and half-integral orders.
- 3 Integral representations of cylinder functions.
- 4 Asymptotic behaviour of cylinder functions.
- 5 Zeros of cylinder functions.
- 6 Addition theorems and series expansions of cylinder functions.
- 7 Calculation of values of cylinder functions on a computer.
- 8 Tables.
Cylinder functions of arbitrary order.
If is not an integer, then the general solution of equation (1) has the form
where and are constants and are the so-called cylinder functions of the first kind or Bessel functions. They have the expansion
The series for on the right-hand side converges absolutely and uniformly for all , , where and are arbitrary positive numbers. The functions and are analytic with singular points and ; the derivatives of and satisfy the following identity:
But if is an integer, then and are linearly independent and the linear combinations of them no longer yield the general solutions of (1). Therefore, apart from cylinder functions of the first kind one introduces cylinder functions of the second kind (or Neumann functions, Weber functions, cf. Neumann functions):
(another notation is ). By means of these functions the general solution of equation (1) can be written in the form
For applications, cylinder functions of the third kind (or Hankel functions) are also important, being solutions of (1). They are denoted by and , where by definition
and the relations
hold. For real and the Hankel functions are complex-conjugate solutions of (1). The functions give the real part and the functions give the imaginary part of the Hankel functions.
The cylinder functions of the first, second and third kind satisfy the recurrence formulas
Every pair of functions
forms (when is not an integer) a fundamental system of solutions of (1).
Modified cylinder functions are cylinder functions of an imaginary argument:
and the Macdonald functions (cf. Macdonald function)
These functions are solutions of the differential equation
and satisfy the recurrence formulas
Cylinder functions of integral and half-integral orders.
If is an integer, can be defined by means of the Jacobi–Anger formulas
hold. The function is an entire transcendental function of the argument ; when , , is algebraic, is a transcendental number, and for . As a second solution of (1), linearly independent of , one takes, as a rule, the function
where is Euler's constant. If in one of the finite sums the upper summation index is less than the lower one, the corresponding sum has the value 0. The equality
Cylinder functions turn into elementary functions if and only if the index takes the values , (spherical Bessel functions or cylinder functions of half-integral order). The following formulas hold :
in particular, ;
in particular, ;
Integral representations of cylinder functions.
When there are Bessel's integral representations
For and there are Poisson's integral representations
Asymptotic behaviour of cylinder functions.
For , , , one has
For real ,
For , there is the following estimate
For the series (9) and (10) terminate. The Hankel functions are the only cylinder functions that tend to 0 for complex values of the variable as (and this is their merit in applications):
Zeros of cylinder functions.
The zeros of an arbitrary cylinder function are simple except for . If and are real, then between two real zeros of lies one real zero of . For real , has infinitely many real zeros; for all zeros of are real; if are the positive zeros of , then
For , for the smallest positive zero of . The pairs of functions ; , have no common zeros except . If
then has exactly complex zeros, two of which are pure imaginary; if , , then has exactly complex zeros with non-zero real part.
Addition theorems and series expansions of cylinder functions.
The following addition theorems hold:
Connected with spherical functions are the Anger functions, the Struve functions, the Lommel functions (cf. Anger function; Struve function; Lommel function), as well as the Kelvin functions and the Airy functions.
Cylinder functions can be defined as limit functions of spherical functions in the following way:
Here, asymptotic representations of spherical functions are connected with cylinder functions and vice versa, for example, as in Hilb's formula
Calculation of values of cylinder functions on a computer.
For the numerical evaluation of the functions , , , , , , , , approximations by polynomials and rational functions are convenient (see ). For expansions with respect to Chebyshev polynomials see . For the calculation of functions of large integral order, especially on a computer, one uses the recurrence relations (5)–(7) (see ).
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Cylinder functions. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Cylinder_functions&oldid=12530