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Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_specfun_bessel_j_seq_complex (s18gk)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_specfun_bessel_j_seq_complex (s18gk) 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

Syntax

[b, ifail] = s18gk(z, a, nl)

[b, ifail] = nag_specfun_bessel_j_seq_complex(z, a, nl)

Description

nag_specfun_bessel_j_seq_complex (s18gk) 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_specfun_bessel_y_complex (s17dc) and nag_specfun_bessel_j_complex (s17de).

References

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

Parameters

Compulsory Input Parameters

1: $z$ – complex scalar: The argument $z$ of the function.

Constraint: $z \neq (0.0, 0.0)$ when $nl < 0$ .
2: $a$ – double scalar: The order $α$ of the first member in the required sequence of function values.

Constraint: $0.0 \leq a < 1.0$ .
3: $nl$ – int64int32nag_int scalar: The value of $N$ .

Constraint: $abs (nl) \leq 101$ .

Optional Input Parameters

None.

Output Parameters

1: $b (abs (nl) + 1)$ – complex array: With $ifail = 0$ or $3$ , the required sequence of function values: $b (n)$ contains $J_{α + n - 1} (z)$ if $nl \geq 0$ and $J_{α - n + 1} (z)$ otherwise, for $n = 1, 2, \dots, abs (nl) + 1$ .
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail = 1$

On entry,	$z = (0.0, 0.0)$ when $nl < 0$ ,
or	$a < 0.0$ ,
or	$a \geq 1.0$ ,
or	$abs (nl) > 101$ .

$ifail = 2$: The computation has been abandoned due to the likelihood of overflow.

W $ifail = 3$: The computation has been completed but some precision has been lost.

$ifail = 4$: The computation has been abandoned due to total loss of precision.

$ifail = 5$: The computation has been abandoned due to failure to satisfy the termination condition.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

All constants in nag_specfun_bessel_y_complex (s17dc) and nag_specfun_bessel_j_complex (s17de) 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_specfun_bessel_y_complex (s17dc) and nag_specfun_bessel_j_complex (s17de), 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.

Further Comments

None.

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.

Open in the MATLAB editor: s18gk_example

function s18gk_example


fprintf('s18gk example results\n\n');

z =  0.6 - 0.8i;
a = 0;
nl = int64(3);

[b, ifail] = s18gk(z, a, nl);

fprintf('   alpha        J_alpha(%5.1f%+5.1fi)\n',real(z), imag(z));
for j=1:nl+1
  fprintf('%10.2e   %12.4e%+12.4ei\n', a+double(j-1), real(b(j)), imag(b(j)));
end

s18gk example results

   alpha        J_alpha(  0.6 -0.8i)
  0.00e+00     1.0565e+00 +2.4811e-01i
  1.00e+00     3.5825e-01 -3.7539e-01i
  2.00e+00    -2.5974e-02 -1.2538e-01i
  3.00e+00    -1.9369e-02 -8.6380e-03i

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Chapter Contents

Chapter Introduction

NAG Toolbox