# NAG Library Function Document

## 1Purpose

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 $n=1,2,\dots ,\left|N\right|+1$.

## 2Specification

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

## 3Description

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).

## 4References

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

## 5Arguments

1:    $\mathbf{z}$ComplexInput
On entry: the argument $z$ of the function.
Constraint: ${\mathbf{z}}\ne \left(0.0,0.0\right)$ when ${\mathbf{nl}}<0$.
2:    $\mathbf{a}$doubleInput
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:    $\mathbf{nl}$IntegerInput
On entry: the value of $N$.
Constraint: $\mathrm{abs}\left({\mathbf{nl}}\right)\le 101$.
4:    $\mathbf{b}\left[\mathrm{abs}\left({\mathbf{nl}}\right)+1\right]$ComplexOutput
On exit: with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ 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:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, $\left|{\mathbf{nl}}\right|=〈\mathit{\text{value}}〉$.
Constraint: $\left|{\mathbf{nl}}\right|\le 101$.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
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.

## 7Accuracy

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.

## 8Parallelism and Performance

nag_complex_bessel_j_seq (s18gkc) is not threaded in any implementation.

None.

## 10Example

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.

### 10.1Program Text

Program Text (s18gkce.c)

### 10.2Program Data

Program Data (s18gkce.d)

### 10.3Program Results

Program Results (s18gkce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017