s Chapter Contents
s Chapter Introduction
NAG C Library Manual

NAG Library Function Documentnag_complex_bessel_j_seq (s18gkc)

1  Purpose

nag_complex_bessel_j_seq (s18gkc) returns a sequence of values for the Bessel functions ${J}_{\alpha +n-1}\left(z\right)$ or ${J}_{\alpha -n+1}\left(z\right)$ for complex $z$, non-negative $\alpha <1$ and $n=1,2,\dots ,\left|N\right|+1$.

2  Specification

 #include #include
 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 ${J}_{\alpha }\left(z\right)$, where $z$ is complex and nonzero and $\alpha$ is the order with $0\le \alpha <1$. The $\left(\left|N\right|+1\right)$-member sequence is generated for orders $\alpha ,\alpha +1,\dots ,\alpha +\left|N\right|$ when $N\ge 0$. Note that $+$ is replaced by $-$ when $N<0$. For positive orders the function may also be called with $z=0$, since ${J}_{q}\left(0\right)=0$ when $q>0$. For negative orders the formula
 $J-qz=cosπqJqz-sinπqYqz$
is used to generate the required sequence. The appropriate values of ${J}_{q}\left(z\right)$ and ${Y}_{q}\left(z\right)$ 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:     zComplexInput
On entry: the argument $z$ of the function.
Constraint: ${\mathbf{z}}\ne \left(0.0,0.0\right)$ when ${\mathbf{nl}}<0$.
On entry: the order $\alpha$ of the first member in the required sequence of function values.
Constraint: $0.0\le {\mathbf{a}}<1.0$.
3:     nlIntegerInput
On entry: the value of $N$.
Constraint: $\mathrm{abs}\left({\mathbf{nl}}\right)\le 101$.
4:     b[$\mathrm{abs}{\mathbf{nl}}+1$]ComplexOutput
On exit: with NE_NOERROR or NW_SOME_PRECISION_LOSS, the required sequence of function values: ${\mathbf{b}}\left[\mathit{n}-1\right]$ contains ${J}_{\alpha +\mathit{n}-1}\left(z\right)$ if ${\mathbf{nl}}\ge 0$ and ${J}_{\alpha -\mathit{n}+1}\left(z\right)$ otherwise, for $\mathit{n}=1,2,\dots ,\mathrm{abs}\left({\mathbf{nl}}\right)+1$.
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, $\mathrm{abs}\left({\mathbf{nl}}\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathrm{abs}\left({\mathbf{nl}}\right)\le 101$.
On entry, ${\mathbf{nl}}=〈\mathit{\text{value}}〉$.
Constraint: when ${\mathbf{nl}}<0$, ${\mathbf{z}}\ne \left(0.0,0.0\right)$.
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, ${\mathbf{a}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}<1.0$.
On entry, ${\mathbf{a}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{a}}\ge 0.0$.
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 $p=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(t,18\right)$. 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 $s\approx \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\left|{\mathrm{log}}_{10}\left|z\right|\right|,\left|{\mathrm{log}}_{10}\left|\alpha \right|\right|\right)$ represents the number of digits lost due to the argument reduction. Thus the larger the values of $\left|z\right|$ and $\left|\alpha \right|$, the less the precision in the result.

None.

9  Example

This example evaluates ${J}_{0}\left(z\right),{J}_{1}\left(z\right),{J}_{2}\left(z\right)$ and ${J}_{3}\left(z\right)$ at $z=0.6-0.8i$, and prints the results.

9.1  Program Text

Program Text (s18gkce.c)

9.2  Program Data

Program Data (s18gkce.d)

9.3  Program Results

Program Results (s18gkce.r)