scipy.special.ive#

scipy.special.ive(v,z,out=None)=<ufunc'ive'>#

Exponentially scaled modified Bessel function of the first kind.

Defined as:

ive(v,z)=iv(v,z)*exp(-abs(z.real))

For imaginary numbers without a real part, returns the unscaledBessel function of the first kindiv.

Parameters:

varray_like of float

Order.

zarray_like of float or complex

Argument.

outndarray, optional

Optional output array for the function values

Returns:

scalar or ndarray

Values of the exponentially scaled modified Bessel function.

See also

iv

Modified Bessel function of the first kind

i0e

Faster implementation of this function for order 0

i1e

Faster implementation of this function for order 1

Notes

For positivev, the AMOS[1]zbesi routine is called. It uses apower series for smallz, the asymptotic expansion for largeabs(z), the Miller algorithm normalized by the Wronskian and aNeumann series for intermediate magnitudes, and the uniform asymptoticexpansions for\(I_v(z)\) and\(J_v(z)\) for large orders.Backward recurrence is used to generate sequences or reduce orders whennecessary.

The calculations above are done in the right half plane and continuedinto the left half plane by the formula,

\[I_v(z \exp(\pm\imath\pi)) = \exp(\pm\pi v) I_v(z)\]

(valid when the real part ofz is positive). For negativev, theformula

\[I_{-v}(z) = I_v(z) + \frac{2}{\pi} \sin(\pi v) K_v(z)\]

is used, where\(K_v(z)\) is the modified Bessel function of thesecond kind, evaluated using the AMOS routinezbesk.

ive is useful for large argumentsz: for these,iv easily overflows,whileive does not due to the exponential scaling.

References

[1]

Donald E. Amos, “AMOS, A Portable Package for Bessel Functionsof a Complex Argument and Nonnegative Order”,http://netlib.org/amos/

Examples

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In the following exampleiv returns infinity whereasive still returnsa finite number.

>>>fromscipy.specialimportiv,ive>>>importnumpyasnp>>>importmatplotlib.pyplotasplt>>>iv(3,1000.),ive(3,1000.)(inf, 0.01256056218254712)

Evaluate the function at one point for different orders byproviding a list or NumPy array as argument for thev parameter:

>>>ive([0,1,1.5],1.)array([0.46575961, 0.20791042, 0.10798193])

Evaluate the function at several points for order 0 by providing anarray forz.

>>>points=np.array([-2.,0.,3.])>>>ive(0,points)array([0.30850832, 1.        , 0.24300035])

Evaluate the function at several points for different orders byproviding arrays for bothv forz. Both arrays have to bebroadcastable to the correct shape. To calculate the orders 0, 1and 2 for a 1D array of points:

>>>ive([[0],[1],[2]],points)array([[ 0.30850832,  1.        ,  0.24300035],       [-0.21526929,  0.        ,  0.19682671],       [ 0.09323903,  0.        ,  0.11178255]])

Plot the functions of order 0 to 3 from -5 to 5.

>>>fig,ax=plt.subplots()>>>x=np.linspace(-5.,5.,1000)>>>foriinrange(4):...ax.plot(x,ive(i,x),label=fr'$I_{i!r}(z)\cdot e^{{-|z|}}$')>>>ax.legend()>>>ax.set_xlabel(r"$z$")>>>plt.show()

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