A202501 -id:A202501 - cp's OEIS frontend

A049006 Decimal expansion of i^i = exp(-Pi/2).

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

Offset: 0

Deepak R. N (deepak_rama(AT)bigfoot.com)

Equals 1/A042972. - Lekraj Beedassy, Sep 02 2005
Euler knew this number to be purely real, and called the fact "remarkable" in a letter to Goldbach dated June 14, 1746. - Alonso del Arte, Nov 30 2012
The value follows immediately from Euler's formula i = exp(i Pi/2) and the rule (a^b)^c = a^(b*c). - The value given by Uhler has the final digits ...14 instead ...08, which is compatible with the claimed accuracy of 52 digits. - M. F. Hasler, May 17 2018

			0.20787957635076190854695561983497877003387...

Florian Cajori, History of Mathematics. New York: Chelsea Publishing Company for the American Mathematical Society (1991): 236.
Ian Connell, Modern Algebra: A Constructive Introduction. New York: Elsevier (1981) p. 363.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.
Roger Penrose, "The Road to Reality, A complete guide to the Laws of the Universe", Jonathan Cape, London, 2004, page 97.
Reinhold Remmert, Theory of Complex Functions: Readings in Mathematics. New York: Springer-Verlag (1991): 162.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 26.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Leonhard Euler, Letter to Christian Goldbach, Berlin, June 14 1746.
Simon Plouffe, exp(-Pi/2) also i**i to 10000 digits.
H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116.
Eric Weisstein's World of Mathematics, i.
Index entries for transcendental numbers.

Cf. A042972, A049007, A097665, A202501 (tetration).

Cf. A077589 and A077590 for i^i^i^...

Mathematica
```
RealDigits[Re[N[I^I, 100]]][[1]]
```

{ default(realprecision, 20080); x=10*exp(-Pi/2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b049006.txt", n, " ", d)); } \\ Harry J. Smith, Apr 28 2009, corrected May 19 2009

digits(exp(-Pi/2)\.1^default(realprecision))[^-1] \\ M. F. Hasler, May 17 2018

Equals 1/A042972 = 2*A097665. - Hugo Pfoertner, Aug 21 2024

A202348 Decimal expansion of x satisfying x = exp(x-2).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 20 2011

For many choices of u and v, there is just one value of x satisfying x = exp(u*x+v). Guide to related sequences, with graphs included in Mathematica programs:
    u       v        x
  -----    --     -------
    1      -2     A202348
    1      -3     A202494
   -1      -1     A202357
   -1      -2     A202496
   -2      -2     A202497
   -2       0     A202498
   -3       0     A202499
   -Pi      0     A202500
  -Pi/2     0     A202501
  -2*Pi    -1     A202495
Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v) = 0. We call the graph of z = g(u,v) an implicit surface of f.
For an example related to this sequence, take f(x,u,v) = x - exp(u*x+v) and g(u,v) = a nonzero solution x of f(x,u,v) = 0. If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous. A portion of an implicit surface is plotted by Program 2 in the Mathematica section.
Actually there are two solutions to x = exp(x-2). This sequence gives the lesser one, x = -LambertW(-exp(-2)), and A226572 gives the greater one, x = -LambertW(-1,-exp(-2)) = 3.14619322062... - Jianing Song, Dec 30 2018

			x = 0.158594339563039362153395341987513893949...

Index entries for transcendental numbers

Cf. A202320, A226572.

(* Program 1: A202348 *)
u = 1; v = -2;
f[x_] := x; g[x_] := E^(u*x + v)
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .15, .16}, WorkingPrecision -> 110]
RealDigits[r]  (* A202348 *)
(* Program 2: implicit surface of x=e^(ux+v) *)
f[{x_, u_, v_}] := x - E^(u*x + v);
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .1, .3}]}, {v, 1, 5}, {u, -5, -.1}];
ListPlot3D[Flatten[t, 1]] (* for A202348 *)
RealDigits[-ProductLog[-1/E^2], 10, 99] // First (* Jean-François Alcover, Feb 26 2013 *)

solve(x=0,1,exp(x-2)-x) \\ Charles R Greathouse IV, Feb 26 2013

Equals -LambertW(-exp(-2)) = 2 - A202320. - Jianing Song, Dec 30 2018

Digits from a(93) on corrected by Jean-François Alcover, Feb 26 2013

A248472 Decimal expansion of C_1 = gamma + log(log(2)) - 2*Ei(-log(2)), one of the Tauberian constants, where Ei is the exponential integral function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Oct 27 2014

			0.96804483044204448704848730112285492269036397005924632964...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Steven R. Finch, Tauberian Constants, August 30, 2004 [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 68.
Eric Weisstein's MathWorld, Exponential Integral

Cf. A001620, A074785, A249385.

evalf(gamma + log(log(2)) - 2*Ei(-log(2)), 120); # Vaclav Kotesovec, Oct 27 2014

C1 = EulerGamma + Log[Log[2]] - 2*ExpIntegralEi[-Log[2]]; RealDigits[C1, 10, 103] // First

Euler + log(log(2)) + 2*eint1(log(2)) \\ Altug Alkan, Sep 05 2018

C_1 also equals gamma + log(log(2)) + 2*Gamma(0, log(2)), where Gamma is the incomplete gamma function.

A249385 Decimal expansion of gamma - 2*Ei(-1), one of the Tauberian constants, where Ei is the exponential integral function.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 27 2014

			1.01598353369257340796083964100264572910425392275374...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Tauberian Constants, August 30, 2004 [Cached copy, with permission of the author]
Eric Weisstein's MathWorld, Exponential Integral

Cf. A001620, A073003, A099285, A248472.

evalf(gamma - 2*Ei(-1), 120); # Vaclav Kotesovec, Oct 27 2014

RealDigits[ EulerGamma - 2*ExpIntegralEi[-1], 10, 103] // First

default(realprecision, 100); Euler + 2*eint1(1) \\ G. C. Greubel, Sep 04 2018

Also equals gamma + 2*G/e, where G is the Euler-Gompertz constant 0.596347...

Equals A001620 + 2*A073003/e. - G. C. Greubel, Sep 04 2018

cp's OEIS Frontend

A049006 Decimal expansion of i^i = exp(-Pi/2).

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A202348 Decimal expansion of x satisfying x = exp(x-2).

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A248472 Decimal expansion of C_1 = gamma + log(log(2)) - 2*Ei(-log(2)), one of the Tauberian constants, where Ei is the exponential integral function.

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Examples

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Programs

Maple

Mathematica

PARI

Formula

A249385 Decimal expansion of gamma - 2*Ei(-1), one of the Tauberian constants, where Ei is the exponential integral function.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula