The first Debye function is defined by
 |  |  |
(1)
|
 |  | ![x^n[1/n-x/(2(n+1))+sum_(k=1)^(infty)(B_(2k)x^(2k))/((2k+n)(2k!))],](/images/equations/DebyeFunctions/Inline6.svg) |
(2)
|
for 
,
, and 
are Bernoulli numbers.
Particular values are given by
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
where 
is a polylogarithm and 
is the Riemann zeta
function. Abramowitz and Stegun (1972, p. 998) tabulate numerical values
of 
for 
to 4 and 
to 10.
The second Debye function is defined by
 |  |  |
(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))],](/images/equations/DebyeFunctions/Inline29.svg) |
(7)
|
for 
and 
.
The sum of these two integrals is
 |  |  |
(8)
|
 |  |  |
(9)
|
where 
is the Riemann zeta function.
