naginterfaces.library.specfun.bessel_​j_​seq_​complex

naginterfaces.library.specfun.bessel_j_seq_complex(z, a, nl)

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

For full information please refer to the NAG Library document for s18gk

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s18gkf.html

Parameters

zcomplex

The argument $z$ of the function.

afloat

The order $\alpha$ of the first member in the required sequence of function values.

nlint

The value of $N$.

Returns

bcomplex, ndarray, shape $(\left|nl\right|+1)$

With the function exits successfully or $errno$ = 3, the required sequence of function values: $b[\text{n}-1]$ contains $J_{\alpha +\text{n}-1}\left(z\right)$ if $nl\ge 0$ and $J_{\alpha -\text{n}+1}\left(z\right)$ otherwise, for $\text{n}=1,2,\dots ,abs\left(nl\right)+1$.

Raises

NagValueError

(errno $1$)

On entry, $\left|nl\right|=⟨value⟩$.

Constraint: $\left|nl\right|\le 101$.

(errno $1$)

On entry, $a=⟨value⟩$.

Constraint: $a<1.0$.

(errno $1$)

On entry, $a=⟨value⟩$.

Constraint: $a\ge 0.0$.

(errno $1$)

On entry, $nl=⟨value⟩$.

Constraint: when $nl<0$, $z\ne (0.0,0.0)$.

(errno $2$)

Computation abandoned due to the likelihood of overflow.

(errno $4$)

Computation abandoned due to total loss of precision.

(errno $5$)

Computation abandoned due to failure to satisfy the termination condition.

Warns

NagAlgorithmicWarning

(errno $3$)

Computation completed but some precision has been lost.

Notes

bessel_j_seq_complex 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 $(\left|N\right|+1)$-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

$$J_{-q}\left(z\right)=\cos \left(\pi q\right)J_q\left(z\right)-\sin \left(\pi q\right)Y_q\left(z\right)$$

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 bessel_j_complex() and bessel_y_complex().

References

NIST Digital Library of Mathematical Functions