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 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 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_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $|nl| = 〈value〉$ .
Constraint: $|nl| \leq 101$ .

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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
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 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 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 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_complex_bessel_j_seq (s18gkc)