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{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)+...].](/images/equations/DebyesAsymptoticRepresentation/NumberedEquation1.svg) |
(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)+...].](/images/equations/DebyesAsymptoticRepresentation/NumberedEquation2.svg) |
(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)+...].](/images/equations/DebyesAsymptoticRepresentation/NumberedEquation3.svg) |
(3)
|
Debye's Asymptotic Representation
See also
Hankel Function of the First KindExplore with Wolfram|Alpha
References
Itô, K. (Ed.). Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 4. Cambridge, MA: MIT Press, p. 1805, 1986.Referenced on Wolfram|Alpha
Debye's Asymptotic RepresentationCite this as:
Weisstein, Eric W. "Debye's Asymptotic Representation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DebyesAsymptoticRepresentation.html