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Debye Functions

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DebyeFunctions1

The first Debye function is defined by

D_n^((1))(x)=int_0^x(t^ndt)/(e^t-1)
(1)
=x^n[1/n-x/(2(n+1))+sum_(k=1)^(infty)(B_(2k)x^(2k))/((2k+n)(2k)!)],
(2)

for |x|<2pi, n>=1 where B_n are Bernoulli numbers (Abramowitz and Stegun 1972, eqn. 27.1.1). Particular values are given by

D_1^((1))(x)=-1/2x^2+ln(1-e^x)x+Li_2(e^x)-zeta(2)
(3)
D_2^((1))(x)=-1/3x^3+ln(1-e^x)x^2+2Li_2(e^x)-2Li_3(e^x)+2zeta(3)
(4)
D_3^((1))(x)=-1/4x^4+ln(1-e^x)x^3+3Li_2(e^x)-6Li_3(e^x)+6Li_4(e^x)-6zeta(4),
(5)

where Li_n(x) is a polylogarithm and zeta(n) is the Riemann zeta function. Abramowitz and Stegun (1972, p. 998) tabulate numerical values of nD_n^((1))(x)/x^n for n=1 to 4 and x=0 to 10.

The second Debye function is defined by

D_n^((2))(x)=int_x^infty(t^ndt)/(e^t-1)
(6)
=sum_(k=1)^(infty)e^(-kx)[(x^n)/k+(nx^(n-1))/(k^2)+(n(n-1)x^(n-2))/(k^3)+...+(n!)/(k^(n+1))],
(7)

for x>0 and n>=1 (Abramowitz and Stegun 1972, eqn. 27.1.2).

The sum of these two integrals is

D_n^((1))(x)+D_n^((2))(x)=int_0^infty(t^ndt)/(e^t-1)
(8)
=n!zeta(n+1),
(9)

where zeta(z) is the Riemann zeta function (Abramowitz and Stegun 1972, eqn. 27.1.3).

See also

Polylogarithm

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Debye Functions." §27.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 998, 1972.Beattie, J. A. "Six-Place Tables of the Debye Energy and Specific Heat Functions." J. Math. Phys. 6, 1-32, 1926.Grüneisen, E. "Die Abhängigkeit des elektrischen Widerstandes reiner Metalle von der Temperatur." Ann. Phys. 16, 530-540, 1933.

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Debye Functions

Cite this as:

Weisstein, Eric W. "Debye Functions." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DebyeFunctions.html

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