Debye,s Asymptotic Representation
المؤلف:
Itô, K. (Ed.)
المصدر:
Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 4. Cambridge, MA: MIT Press
الجزء والصفحة:
p. 1805
24-3-2019
2061
Debye's Asymptotic Representation
Debye's asymptotic representation is an asymptotic expansion for a Hankel function of the first kind with 
. For
, 
, 
, and 
,
![H_nu^((1))(x)∼1/(sqrt(pi))exp<span style=]() {ix[cosalpha+(alpha-1/2pi)sinalpha]}[(e^(ipi/4))/X+(1/8+5/(24)tan^2alpha)(3e^(3pii/4))/(2X^3)+(3/(128)+(77)/(576)tan^alpha+(385)/(3456)tan^4alpha)(3·5e^(5pii/4))/(2^2X^5)+...]. " src="http://mathworld.wolfram.com/images/equations/DebyesAsymptoticRepresentation/NumberedEquation1.gif" style="height:89px; width:495px" /> |
(1)
|
For 
, 
, 
, 
, and 
,
![H_nu^((1))(x)∼1/(sqrt(pi))exp[x(sigmacoshsigma-sinhsigma)][1/X+(1/8-5/(24)coth^2sigma)3/(2X^3)+(3/(128)-(77)/(576)coth^2sigma+(385)/(3456)coth^4sigma)(3·5)/(2^2X^5)+...].](http://mathworld.wolfram.com/images/equations/DebyesAsymptoticRepresentation/NumberedEquation2.gif) |
(2)
|
Finally, for 
, 
, and 
,
![H_nu^((1))∼(6^(1/3)e^(ipi/3))/(pisqrt(3))[Gamma(1/3)x^(-1/3)-6^(1/3)e^(ipi/3)deltaGamma(2/3)x^(-2/3)+(2/5delta-delta^3)Gamma(4/3)x^(-4/3)+(3/(140)-1/4delta^2+1/4delta^4)6^(1/3)e^(ipi/3)Gamma(5/3)x^(-5/3)+...].](http://mathworld.wolfram.com/images/equations/DebyesAsymptoticRepresentation/NumberedEquation3.gif) |
(3)
|
REFERENCES:
Itô, K. (Ed.). Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 4. Cambridge, MA: MIT Press, p. 1805, 1986.
0
0
الاكثر قراءة في التفاضل و التكامل
اخر الاخبار
اخبار العتبة العباسية المقدسة
