In mathematics, the family of Debye functions is defined by $${\displaystyle D_{n}(x)={\frac {n}{x^{n}}}\int _{0}^{x}{\frac {t^{n}}{e^{t}-1}}\,dt.}$$

The functions are named in honor of Peter Debye, who came across this function (with n = 3) in 1912 when he analytically computed the heat capacity of what is now called the Debye model.

The Debye functions are closely related to the polylogarithm.

They have the series expansion^[1] $${\displaystyle D_{n}(x)=1-{\frac {n}{2(n+1)}}x+n\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k+n)(2k)!}}x^{2k},\quad |x|<2\pi ,\ n\geq 1,}$$ where ${\displaystyle B_{n}}$ is the n-th Bernoulli number.

$${\displaystyle \lim _{x\to 0}D_{n}(x)=1.}$$ If ${\displaystyle \Gamma }$ is the gamma function and ${\displaystyle \zeta }$ is the Riemann zeta function, then, for ${\displaystyle x\gg 0}$,^[2] $${\displaystyle D_{n}(x)={\frac {n}{x^{n}}}\int _{0}^{x}{\frac {t^{n}\,dt}{e^{t}-1}}\sim {\frac {n}{x^{n}}}\Gamma (n+1)\zeta (n+1),\qquad \operatorname {Re} n>0,}$$

The derivative obeys the relation $${\displaystyle xD_{n}^{\prime }(x)=n\left(B(x)-D_{n}(x)\right),}$$ where ${\displaystyle B(x)=x/(e^{x}-1)}$ is the Bernoulli function.

The Debye model has a density of vibrational states $${\displaystyle g_{\text{D}}(\omega )={\frac {9\omega ^{2}}{\omega _{\text{D}}^{3}}}\,,\qquad 0\leq \omega \leq \omega _{\text{D}}}$$ with the Debye frequency ω_D.

Inserting g into the internal energy $${\displaystyle U=\int _{0}^{\infty }d\omega \,g(\omega )\,\hbar \omega \,n(\omega )}$$ with the Bose–Einstein distribution $${\displaystyle n(\omega )={\frac {1}{\exp(\hbar \omega /k_{\text{B}}T)-1}}.}$$ one obtains $${\displaystyle U=3k_{\text{B}}T\,D_{3}(\hbar \omega _{\text{D}}/k_{\text{B}}T).}$$ The heat capacity is the derivative thereof.

The intensity of X-ray diffraction or neutron diffraction at wavenumber q is given by the Debye-Waller factor or the Lamb-Mössbauer factor. For isotropic systems it takes the form $${\displaystyle \exp(-2W(q))=\exp \left(-q^{2}\langle u_{x}^{2}\rangle \right).}$$ In this expression, the mean squared displacement refers to just once Cartesian component u_x of the vector u that describes the displacement of atoms from their equilibrium positions. Assuming harmonicity and developing into normal modes,^[3] one obtains $${\displaystyle 2W(q)={\frac {\hbar ^{2}q^{2}}{6Mk_{\text{B}}T}}\int _{0}^{\infty }d\omega {\frac {k_{\text{B}}T}{\hbar \omega }}g(\omega )\coth {\frac {\hbar \omega }{2k_{\text{B}}T}}={\frac {\hbar ^{2}q^{2}}{6Mk_{\text{B}}T}}\int _{0}^{\infty }d\omega {\frac {k_{\text{B}}T}{\hbar \omega }}g(\omega )\left[{\frac {2}{\exp(\hbar \omega /k_{\text{B}}T)-1}}+1\right].}$$ Inserting the density of states from the Debye model, one obtains $${\displaystyle 2W(q)={\frac {3}{2}}{\frac {\hbar ^{2}q^{2}}{M\hbar \omega _{\text{D}}}}\left[2\left({\frac {k_{\text{B}}T}{\hbar \omega _{\text{D}}}}\right)D_{1}{\left({\frac {\hbar \omega _{\text{D}}}{k_{\text{B}}T}}\right)}+{\frac {1}{2}}\right].}$$ From the above power series expansion of ${\displaystyle D_{1}}$ follows that the mean square displacement at high temperatures is linear in temperature $${\displaystyle 2W(q)={\frac {3k_{\text{B}}Tq^{2}}{M\omega _{\text{D}}^{2}}}.}$$ The absence of ${\displaystyle \hbar }$ indicates that this is a classical result. Because ${\displaystyle D_{1}(x)}$ goes to zero for ${\displaystyle x\to \infty }$ it follows that for ${\displaystyle T=0}$ $${\displaystyle 2W(q)={\frac {3}{4}}{\frac {\hbar ^{2}q^{2}}{M\hbar \omega _{\text{D}}}}}$$ (zero-point motion).

• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 27". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 998. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
• "Debye function" entry in MathWorld, defines the Debye functions without prefactor n/xⁿ

• Ng, E. W.; Devine, C. J. (1970). "On the computation of Debye functions of integer orders". Math. Comp. 24 (110): 405–407. doi:10.1090/S0025-5718-1970-0272160-6. MR 0272160.
• Engeln, I.; Wobig, D. (1983). "Computation of the generalized Debye functions delta(x,y) and D(x,y)". Colloid & Polymer Science. 261: 736–743. doi:10.1007/BF01410947. S2CID 98476561.
• MacLeod, Allan J. (1996). "Algorithm 757: MISCFUN, a software package to compute uncommon special functions". ACM Trans. Math. Software. 22 (3): 288–301. doi:10.1145/232826.232846. S2CID 37814348. Fortran 77 code
• Fortran 90 version
• Maximon, Leonard C. (2003). "The dilogarithm function for complex argument". Proc. R. Soc. A. 459 (2039): 2807–2819. Bibcode:2003RSPSA.459.2807M. doi:10.1098/rspa.2003.1156. S2CID 122271244.
• Guseinov, I. I.; Mamedov, B. A. (2007). "Calculation of Integer and noninteger n-Dimensional Debye Functions using Binomial Coefficients and Incomplete Gamma Functions". Int. J. Thermophys. 28 (4): 1420–1426. Bibcode:2007IJT....28.1420G. doi:10.1007/s10765-007-0256-1. S2CID 120284032.
• C version of the GNU Scientific Library