naginterfaces.library.specfun.bessel_​j_​complex

naginterfaces.library.specfun.bessel_j_complex(fnu, z, n, scal)

bessel_j_complex returns a sequence of values for the Bessel functions $J_{\nu +n}\left(z\right)$ for complex $z$, non-negative $\nu$ and $n=0,1,\dots ,N-1$, with an option for exponential scaling.

For full information please refer to the NAG Library document for s17de

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s17def.html

Parameters

fnufloat

$\nu$, the order of the first member of the sequence of functions.

zcomplex

The argument $z$ of the functions.

nint

$N$, the number of members required in the sequence $J_{\nu }\left(z\right),J_{\nu +1}\left(z\right),\dots ,J_{\nu +N-1}\left(z\right)$.

scalstr, length 1

The scaling option.

$scal=\text{'U'}$

The results are returned unscaled.

$scal=\text{'S'}$

The results are returned scaled by the factor $e^{-\left|Im\left(z\right)\right|}$.

Returns

cycomplex, ndarray, shape $\left(n\right)$

The $N$ required function values: $cy[i-1]$ contains $J_{\nu +i-1}\left(z\right)$, for $\text{i}=1,2,\dots ,N$.

nzint

The number of components of $cy$ that are set to zero due to underflow. If $nz>0$, then elements $cy[n-nz]$, $cy[n-nz+1],\dots ,cy[n-1]$ are set to zero.

Raises

NagValueError

(errno $1$)

On entry, $fnu=⟨value⟩$.

Constraint: $fnu\ge 0.0$.

(errno $1$)

On entry, $scal$ has an illegal value: $scal=⟨value⟩$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $2$)

No computation because $Im\left(z\right)=⟨value⟩>⟨value⟩$, $scal=\text{'U'}$.

(errno $4$)

No computation because $\left|z\right|=⟨value⟩>⟨value⟩$.

(errno $4$)

No computation because $fnu+n-1=⟨value⟩>⟨value⟩$.

(errno $5$)

No computation – algorithm termination condition not met.

Warns

NagAlgorithmicWarning

(errno $3$)

Results lack precision because $\left|z\right|=⟨value⟩>⟨value⟩$.

(errno $3$)

Results lack precision because $fnu+n-1=⟨value⟩>⟨value⟩$.

Notes

bessel_j_complex evaluates a sequence of values for the Bessel function $J_{\nu }\left(z\right)$, where $z$ is complex, $-\pi <argz\le \pi$, and $\nu$ is the real, non-negative order. The $N$-member sequence is generated for orders $\nu$, $\nu +1,\dots ,\nu +N-1$. Optionally, the sequence is scaled by the factor $e^{-\left|Im\left(z\right)\right|}$.

Note: although the function may not be called with $\nu$ less than zero, for negative orders the formula $J_{-\nu }\left(z\right)=J_{\nu }\left(z\right)\cos \left(\pi \nu \right)-Y_{\nu }\left(z\right)\sin \left(\pi \nu \right)$ may be used (for the Bessel function $Y_{\nu }\left(z\right)$, see bessel_y_complex()).

The function is derived from the function CBESJ in Amos (1986). It is based on the relations $J_{\nu }\left(z\right)=e^{\nu \pi i/2}{I}_{\nu }(-iz)$, $Im\left(z\right)\ge 0.0$, and $J_{\nu }\left(z\right)=e^{-\nu \pi i/2}{I}_{\nu }\left(iz\right)$, $Im\left(z\right)<0.0$.

The Bessel function $I_{\nu }\left(z\right)$ is computed using a variety of techniques depending on the region under consideration.

When $N$ is greater than $1$, extra values of $J_{\nu }\left(z\right)$ are computed using recurrence relations.

For very large $\left|z\right|$ or $(\nu +N-1)$, argument reduction will cause total loss of accuracy, and so no computation is performed. For slightly smaller $\left|z\right|$ or $(\nu +N-1)$, the computation is performed but results are accurate to less than half of machine precision. If $Im\left(z\right)$ is large, there is a risk of overflow and so no computation is performed. In all the above cases, a warning is given by the function.

References

NIST Digital Library of Mathematical Functions

Amos, D E, 1986, Algorithm 644: A portable package for Bessel functions of a complex argument and non-negative order, ACM Trans. Math. Software (12), 265–273