login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A194567 Decimal expansion of the positive solution to x = 3*(1-exp(-x)). 6

%I #25 Aug 09 2023 16:59:28

%S 2,8,2,1,4,3,9,3,7,2,1,2,2,0,7,8,8,9,3,4,0,3,1,9,1,3,3,0,2,9,4,4,8,5,

%T 1,9,5,3,4,5,8,8,1,7,4,4,0,7,3,1,1,4,0,9,2,2,7,9,8,5,7,6,9,3,9,4,1,2,

%U 1,4,3,0,4,5,0,5,5,1,7,3,9,1,2,4,5,6,8,6,5,6,5,3,4,7,8,3,9,6,4,4,3,8,9,5,9

%N Decimal expansion of the positive solution to x = 3*(1-exp(-x)).

%C The positive solution to x=3*(1-exp(-x)) is the dimensionless coefficient corresponding to the maximum brightness in Planck's law of radiation.

%C It can be symbolically expressed as 3+W(-3/e^3), where W stands for Lambert (a.k.a. "ProductLog") function.

%H Stanislav Sykora, <a href="/A194567/b194567.txt">Table of n, a(n) for n = 1..2000</a>

%H SpectralCalc, <a href="http://www.spectralcalc.com/blackbody/blackbody.html">Calculation of Blackbody Radiance</a>, Appendix C.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Planck%27s_law">Planck's law</a>

%e 2.821439372...

%t RealDigits[ N[x /. ToRules[ Reduce[x > 0 && x == 3*(1 - E^-x), x, Reals]], 100]][[1]]

%t RealDigits[3 + ProductLog[-3/E^3], 10, 111][[1]] (* _Robert G. Wilson v_, Oct 16 2013 *)

%t RealDigits[x/.FindRoot[x==3(1-Exp[-x]),{x,2},WorkingPrecision->120]][[1]] (* _Harvey P. Dale_, Aug 09 2023 *)

%o (PARI) a3=solve(x=0.1,10,x-3*(1-exp(-x))) \\ Use real precision in excess

%o (PARI) 3+lambertw(-3/exp(3)) \\ _Charles R Greathouse IV_, Sep 13 2022

%Y Cf. A094090, A256500, A256501.

%K nonn,cons

%O 1,1

%A _Jean-François Alcover_, Aug 29 2011

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 13:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)