Bei
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
"Kelvin Functions." §9.9 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
24-3-2019
2042
Bei
The 
function is defined through the equation
 |
(1)
|
where 
is a Bessel function of the first kind, so
![bei_nu(z)=I[J_nu(ze^(3pii/4))],](http://mathworld.wolfram.com/images/equations/Bei/NumberedEquation2.gif) |
(2)
|
where ![I[z]](http://mathworld.wolfram.com/images/equations/Bei/Inline3.gif)
is the imaginary part.
It is implemented in the Wolfram Language as KelvinBei[nu, z].
has the series expansion
![bei_nu(x)=(1/2x)^nusum_(k=0)^infty(sin[(3/4nu+1/2k)pi])/(k!Gamma(nu+k+1))(1/4x^2)^k,](http://mathworld.wolfram.com/images/equations/Bei/NumberedEquation3.gif) |
(3)
|
where 
is the gamma function (Abramowitz and Stegun 1972, p. 379), which can be written in closed form as
![bei_nu(x)=-1/2ie^(-3piinu/4)x^nu[(-1)^(1/4)x]^(-nu)×[e^(3piinu/2)I_nu((-1)^(1/4)x)-J_nu((-1)^(1/4)x)],](http://mathworld.wolfram.com/images/equations/Bei/NumberedEquation4.gif) |
(4)
|
where 
is a modified Bessel function of the first kind.
The special case 
, commonly denoted 
, corresponds to
 |
(5)
|
where 
is the zeroth order Bessel function of the first kind. The function 
has the series expansion
![bei(z)=sum_(n=0)^infty((-1)^n(1/2z)^(2+4n))/([(2n+1)!]^2).](http://mathworld.wolfram.com/images/equations/Bei/NumberedEquation6.gif) |
(6)
|
Closed forms include
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). "Kelvin Functions." §9.9 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 379-381, 1972.
Prudnikov, A. P.; Marichev, O. I.; and Brychkov, Yu. A. "The Kelvin Functions 
, 
, 
and 
." §1.7 in Integrals and Series, Vol. 3: More Special Functions. Newark, NJ: Gordon and Breach, pp. 29-30, 1990.
Spanier, J. and Oldham, K. B. "The Kelvin Functions." Ch. 55 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 543-554, 1987.
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