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Einstein Functions

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EinsteinFunctions

Einstein functions are the functions

E_1(x)=(x^2e^x)/((e^x-1)^2)
(1)
E_2(x)=x/(e^x-1)
(2)
E_3(x)=ln(1-e^(-x))
(3)
E_4(x)=x/(e^x-1)-ln(1-e^(-x)).
(4)

For real x, E_1 and E_2 are defined for x!=0 and extend continuously to E_1(0)=E_2(0)=1. The real-valued functions E_3 and E_4 are defined for x>0.

The inflection points of E_1(x) satisfy, for x!=0,

E_1^('')(x)=1/8csch^4(1/2x)[(x^2+2)coshx+2(x^2-2xsinhx-1)]=0,
(5)

which can be solved numerically to give x=+/-2.34694130... (OEIS A118080).

See also

Bose-Einstein Distribution, Fermi-Dirac Distribution, Sigmoid Function

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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, pp. 999-1000, 1972.Sloane, N. J. A. Sequence A118080 in "The On-Line Encyclopedia of Integer Sequences."

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Einstein Functions

Cite this as:

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

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