# NAG Library Function Document

## 1Purpose

nag_bessel_j_alpha (s18ekc) returns a sequence of values for the Bessel functions ${J}_{\alpha +n-1}\left(x\right)$ or ${J}_{\alpha -n+1}\left(x\right)$ for real $x$, non-negative $\alpha <1$ and $n=1,2,\dots ,\left|N\right|+1$.

## 2Specification

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

## 3Description

nag_bessel_j_alpha (s18ekc) evaluates a sequence of values for the Bessel function of the first kind ${J}_{\alpha }\left(x\right)$, where $x$ is real 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 $\alpha ,\alpha +1,\dots ,\alpha +N$when $N\ge 0$. Note that $+$ is replaced by $-$ when $N<0$. For positive orders the function may also be called with $x=0$, since ${J}_{q}\left(0\right)=0$ when $q>0$. For negative orders the formula
 $J -q x = cosπq J q x - sinπq Y q x$
is used to generate the required sequence.

## 4References

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

## 5Arguments

1:    $\mathbf{x}$doubleInput
On entry: the argument $x$ of the function.
Constraint: if ${\mathbf{nl}}<0$, ${\mathbf{x}}\ne 0.0$.
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:    $\mathbf{b}\left[×\right]$ComplexOutput
On exit: with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=\mathrm{NE_NOERROR}$ or ${\mathbf{fail}}\mathbf{.}\mathbf{code}={\mathbf{NW_SOME_PRECISION_LOSS}}$, the required sequence of function values: b$\left(\mathit{n}\right)$ contains ${J}_{\alpha +\mathit{n}-1}\left(x\right)$ if ${\mathbf{nl}}\ge 0$ and ${J}_{\alpha -\mathit{n}+1}\left(x\right)$ 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_INT
On entry, ${\mathbf{nl}}=〈\mathit{\text{value}}〉$.
Constraint: $\mathrm{abs}\left({\mathbf{nl}}\right)\le 101$.
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
The evaluation has been abandoned due to the likelihood of overflow.
NE_REAL
On entry, ${\mathbf{a}}=〈\mathit{\text{value}}〉$.
Constraint: $0.0\le {\mathbf{a}}<1.0$.
NE_REAL_INT
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$, ${\mathbf{nl}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\ne 0.0$ when ${\mathbf{nl}}<0$.
NE_TERMINATION_FAILURE
The evaluation has been abandoned due to failure to satisfy the termination condition.
NE_TOTAL_PRECISION_LOSS
The evaluation has been abandoned due to total loss of precision.
NW_SOME_PRECISION_LOSS
The evaluation has been completed but some precision has been lost.

## 7Accuracy

All constants in the underlying functions are 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 the underlying functions are, the actual number of correct digits is limited, in general, by $p-s$, where $s\approx \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\left|{\mathrm{log}}_{10}\left|x\right|\right|,\left|{\mathrm{log}}_{10}\left|\alpha \right|\right|\right)$ represents the number of digits lost due to the argument reduction. Thus the larger the values of $\left|x\right|$ and $\left|\alpha \right|$, the less the precision in the result.

## 8Parallelism and Performance

nag_bessel_j_alpha (s18ekc) is not threaded in any implementation.

None.

## 10Example

The example program evaluates ${J}_{0}\left(x\right),{J}_{1}\left(x\right),{J}_{2}\left(x\right)$ and ${J}_{3}\left(x\right)$ at $x=0.5$, and prints the results.

### 10.1Program Text

Program Text (s18ekce.c)

### 10.2Program Data

Program Data (s18ekce.d)

### 10.3Program Results

Program Results (s18ekce.r)

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