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Decimal expansion of the solution to Ei(x) = x.
0

%I #11 Feb 16 2025 08:34:06

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

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

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

%N Decimal expansion of the solution to Ei(x) = x.

%C Fixed point of exponential integral.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ExponentialIntegral.html">Exponential Integral</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Exponential_integral">Exponential integral</a>.

%e 0.5276123472017420...

%p Digits:=101: fsolve(Ei(1,x)-x, x);

%t RealDigits[FindRoot[ExpIntegralE[1, x] - x, {x, 0.5}, WorkingPrecision -> 101][[1, 2]], 10, 101][[1]]

%o (PARI) default(realprecision, 101); solve(x=0.5,0.6,eint1(x)-x)

%o (PARI) solve(x=0.5,0.6,-Euler()-log(x)-suminf(k=1,(-x)^k/(k*k!))-x)

%Y Cf. A091723, A091725, A099285.

%K nonn,cons,changed

%O 0,1

%A _Michal Paulovic_, Aug 15 2023