s17af returns the value of the Bessel function J1x.


public static double s17af(
	double x,
	out int ifail
Visual Basic
Public Shared Function s17af ( _
	x As Double, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
static double s17af(
	double x, 
	[OutAttribute] int% ifail
static member s17af : 
        x : float * 
        ifail : int byref -> float 


Type: System..::..Double
On entry: the argument x of the function.
Type: System..::..Int32%
On exit: ifail=0 unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

s17af returns the value of the Bessel function J1x.


s17af evaluates an approximation to the Bessel function of the first kind J1x.
Note:  J1-x=-J1x, so the approximation need only consider x0.
The method is based on three Chebyshev expansions:
For 0<x8,
J1x=x8r=0arTrt,   with ​t=2x82-1.
For x>8,
where P1x=r=0brTrt,
and Q1x=8xr=0crTrt,
with t=28x2-1.
For x near zero, J1xx2. This approximation is used when x is sufficiently small for the result to be correct to machine precision.
For very large x, it becomes impossible to provide results with any reasonable accuracy (see [Accuracy]), hence the method fails. Such arguments contain insufficient information to determine the phase of oscillation of J1x; only the amplitude, 2πx, can be determined and this is returned on failure. The range for which this occurs is roughly related to machine precision; the method will fail if x1/machine precision (see the Users' Note for your implementation for details).


Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Clenshaw C W (1962) Chebyshev Series for Mathematical Functions Mathematical tables HMSO

Error Indicators and Warnings

Errors or warnings detected by the method:
x is too large. On failure the method returns the amplitude of the J1 oscillation, 2πx.
An error occured, see message report.


Let δ be the relative error in the argument and E be the absolute error in the result. (Since J1x oscillates about zero, absolute error and not relative error is significant.)
If δ is somewhat larger than machine precision (e.g., if δ is due to data errors etc.), then E and δ are approximately related by:
(provided E is also within machine bounds). Figure 1 displays the behaviour of the amplification factor xJ0x-J1x.
However, if δ is of the same order as machine precision, then rounding errors could make E slightly larger than the above relation predicts.
For very large x, the above relation ceases to apply. In this region, J1x2πxcosx-3π4. The amplitude 2πx can be calculated with reasonable accuracy for all x, but cosx-3π4 cannot. If x-3π4 is written as 2Nπ+θ where N is an integer and 0θ<2π, then cosx-3π4 is determined by θ only. If xδ-1, θ cannot be determined with any accuracy at all. Thus if x is greater than, or of the order of, the reciprocal of machine precision, it is impossible to calculate the phase of J1x and the method must fail.
Figure 1
Figure 1

Parallelism and Performance


Further Comments



This example reads values of the argument x from a file, evaluates the function at each value of x and prints the results.

Example program (C#): s17afe.cs

Example program data: s17afe.d

Example program results: s17afe.r

See Also