NAG Library Function Document

nag_complex_bessel_j_seq (s18gkc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_complex_bessel_j_seq (s18gkc) returns a sequence of values for the Bessel functions

J_{α + n - 1} (z)

J_{α - n + 1} (z)

for complex

z

, non-negative

α < 1

and

n = 1, 2, \dots, |N| + 1

2 Specification

#include <nag.h>

#include <nags.h>

void	nag_complex_bessel_j_seq (Complex z, double a, Integer nl, Complex b[], NagError *fail)

3 Description

nag_complex_bessel_j_seq (s18gkc) evaluates a sequence of values for the Bessel function of the first kind

J_{α} (z)

, where

z

is complex and nonzero and

α

is the order with

0 \leq α < 1

. The

(|N| + 1)

-member sequence is generated for orders

α, α + 1, \dots, α + |N|

when

N \geq 0

. Note that

+

is replaced by

-

when

N < 0

. For positive orders the function may also be called with

z = 0

, since

J_{q} (0) = 0

when

q > 0

. For negative orders the formula

J_{- q} (z) = \cos (π q) J_{q} (z) - \sin (π q) Y_{q} (z)

is used to generate the required sequence. The appropriate values of

J_{q} (z)

and

Y_{q} (z)

are obtained by calls to nag_complex_bessel_y (s17dcc) and nag_complex_bessel_j (s17dec).

4 References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

5 Arguments

1: z – ComplexInput: On entry: the argument $z$ of the function.
Constraint: $z \neq (0.0, 0.0)$ when $nl < 0$ .
2: a – doubleInput: On entry: the order $α$ of the first member in the required sequence of function values.
Constraint: $0.0 \leq a < 1.0$ .
3: nl – IntegerInput: On entry: the value of $N$ .
Constraint: $abs (nl) \leq 101$ .
4: b[ $abs nl + 1$ ] – ComplexOutput: On exit: with $fail . code =$ NE_NOERROR or NW_SOME_PRECISION_LOSS, the required sequence of function values: $b [n - 1]$ contains $J_{α + n - 1} (z)$ if $nl \geq 0$ and $J_{α - n + 1} (z)$ otherwise, for $n = 1, 2, \dots, abs (nl) + 1$ .
5: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $abs (nl) = 〈value〉$ .
Constraint: $abs (nl) \leq 101$ .; On entry, $nl = 〈value〉$ .
Constraint: when $nl < 0$ , $z \neq (0.0, 0.0)$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_OVERFLOW_LIKELY: Computation abandoned due to the likelihood of overflow.
NE_REAL: On entry, $a = 〈value〉$ .
Constraint: $a < 1.0$ .; On entry, $a = 〈value〉$ .
Constraint: $a \geq 0.0$ .
NE_TERMINATION_FAILURE: Computation abandoned due to failure to satisfy the termination condition.
NE_TOTAL_PRECISION_LOSS: Computation abandoned due to total loss of precision.
NW_SOME_PRECISION_LOSS: Computation completed but some precision has been lost.

7 Accuracy

All constants in nag_complex_bessel_y (s17dcc) and nag_complex_bessel_j (s17dec) are specified to approximately

18

digits of precision. If

t

denotes the number of digits of precision in the floating point arithmetic being used, then clearly the maximum number of correct digits in the results obtained is limited by

p = \min (t, 18)

. Because of errors in argument reduction when computing elementary functions inside nag_complex_bessel_y (s17dcc) and nag_complex_bessel_j (s17dec), the actual number of correct digits is limited, in general, by

p - s

, where

s \approx \max (1, |\log_{10} |z||, |\log_{10} |α||)

represents the number of digits lost due to the argument reduction. Thus the larger the values of

|z|

and

|α|

, the less the precision in the result.

8 Further Comments

None.

9 Example

This example evaluates

J_{0} (z), J_{1} (z), J_{2} (z)

and

J_{3} (z)

z = 0.6 - 0.8 i

, and prints the results.

NAG Library Function Documentnag_complex_bessel_j_seq (s18gkc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_complex_bessel_j_seq (s18gkc)