A059526 -id:A059526 - cp's OEIS frontend

A030178 Decimal expansion of LambertW(1): the solution to x*exp(x) = 1.

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

Offset: 0

N. J. A. Sloane

Sometimes called the Omega constant.
Infinite power tower for c = 1/E, i.e., c^c^c^..., where c = 1/A068985. - Stanislav Sykora, Nov 03 2013
Notice the narrow interval exp(-gamma) < w(1) < gamma, with gamma = A001620. - Jean-François Alcover, Dec 18 2013
Also the solution to x = -log(x). - Robert G. Wilson v, Feb 22 2014

			0.5671432904097838729999686622103555497538157871865125081351310792230457930866...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.

G. C. Greubel and Stanislav Sykora, Table of n, a(n) for n = 0..10000 (terms 0..1999 from Stanislav Sykora)
Mateus Araújo, Andrew J. P. Garner, and Miguel Navascues, Non-commutative optimization problems with differential constraints, arXiv:2408.02572 [quant-ph], 2024. See p. 13.
blackpenredpen, Finding Omega, featuring Newton's method, video (2018).
Daniel Cummerow, Sound of Mathematics, Constants.
István Mező, An integral representation for the Lambert W function, arXiv:2012.02480 [math.CA], 2020.
Simon Plouffe, Lambert W(1, 0).
Simon Plouffe, The omega constant or W(1).
Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, Stan's Library, Vol. VI, 2016.
Eric Weisstein's World of Mathematics, Omega Constant.
Eric Weisstein's World of Mathematics, Lambert W-Function.
Wikipedia, Omega constant.
Wadim Zudilin, Diophantine problems related to the Omega constant, arXiv:2004.11029 [math.NT], 2020.
Index entries for transcendental numbers.

Cf. A019474, A059526, A059527, A238274.

Cf. A276759 (another fixed point of -exp(z)).

Maple
```
evalf(LambertW(1));
```

RealDigits[ ProductLog[1], 10, 111][[1]] (* Robert G. Wilson v, May 19 2004 *)

solve(x=0,1,x*exp(x)-1) \\ Charles R Greathouse IV, Mar 20 2012

lambertw(1) \\ Stanislav Sykora, Nov 03 2013

Equals 1/A030797.

Equals (1/Pi) * Integral_{x=0..Pi} log(1 + sin(x)*exp(x*cot(x))/x) dx (Mező, 2020). - Amiram Eldar, Jul 04 2021

A059527 Decimal expansion of imaginary part of solution to z = log z.

Original entry on oeis.org

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

Offset: 1

Fabian Rothelius, Jan 21 2001

Repeatedly take logs, starting from any number not equal to 0, 1, e, e^e, e^(e^e), etc. and you will converge to 0.31813150... + 1.33723570...*I.

			z = 0.31813150520476413531265425158766451720351761387139986692237... + 1.33723570143068940890116214319371061253950213846051241887631... *i.

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, 2016.
Wolfram Research, FixedPoint

Real part is A059526.

Cf. A030178.

RealDigits[ Im[ N[ FixedPoint[ Log, 1 + I, 910], 105]]] [[1]]
RealDigits[ N[ Im[ ProductLog[-1]], 105]][[1]] (* Jean-François Alcover, Feb 01 2012 *)

z=I;for(k=1,16000,z=log(z));imag(z)  \\ Using realprecision \p 2010. - Stanislav Sykora, Jun 07 2015

z=I; for(k=1, 10, z-=(z-log(z))/(1-1/z)); imag(z) \\ Jeremy Tan, Sep 23 2017

More terms from Vladeta Jovovic, Feb 26 2001

Edited and extended by Robert G. Wilson v, Aug 22 2002

A276759 Decimal expansion of the real part of the fixed point of -exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Nov 12 2016

The negated exponential mapping -exp(z) has in C a denumerable set of fixed points z_k with even k, which are the solutions of exp(z)+z = 0. The solutions with positive and negative indices k form mutually conjugate pairs, such as this z_2 and z_-2. A similar situation arises also for the fixed points of the mapping +exp(z). My link explains why is it convenient to use even indices for the fixed points of -exp(z) and odd ones for those of +exp(z). Setting K = sign(k)*floor(|k|/2), an even-indexed z_k is also a solution of z = log(-z)+2*Pi*K*i. Moreover, an even-indexed z_k equals -W_L(1), where W_L is the L-th branch of the Lambert W function, with L=-floor((k+1)/2). For any nonzero K, the mapping M_K(z) = log(-z)+2*Pi*K*i has the even-indexed z_k as its unique attractor, convergent from any nonzero point in C (the case K=0 is an exception, discussed in my linked document).

The value listed here is the real part of z_2 = a + i*A276760.

			1.533913319793574507919741082072733779785298610650766671733076...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
S. Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, Stan's Library, Vol.VI, Oct 2016.
Eric Weisstein's World of Mathematics, Exponential Function.
Wikipedia, Exponential function.

Fixed points of -exp(z): z_0: A030178 (real-valued), and z_2: A276760 (imaginary part), A276761 (modulus).

Fixed points of +exp(z): z_1: A059526, A059527, A238274, and z_3: A277681, A277682, A277683.

RealDigits[Re[-ProductLog[-1, 1]], 10, 105][[1]] (* Jean-François Alcover, Nov 12 2016 *)

default(realprecision,2050);eps=5.0*10^(default(realprecision))
M(z,K)=log(-z)+2*Pi*K*I; \\ the convergent mapping (any K!=0)
K=1;z=1+I;zlast=z;
while(1,z=M(z,K);if(abs(z-zlast)

Let z_2 = A276759+i*A276760. Then z_2 = -exp(z_2) = log(-z_2)+2*Pi*i = -W_-1(1).

A277681 Decimal expansion of the real part of the fixed point of exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Nov 12 2016

The exponential mapping exp(z) has in C a denumerable set of fixed points z_k with odd k, which are the solutions of exp(z) = z. The solutions with positive and negative indices k form mutually conjugate pairs, such as z_3 and z_-3. A similar situation arises also for the related fixed points of the mapping -exp(z). My link explains why is it convenient to use odd indices for the fixed points of +exp(z) and even indices for those of -exp(z). Setting K = sign(k)*floor(|k|/2), an odd-indexed z_k is also a fixed point of the logarithmic function in its K-th branch, i.e., a solution of z = log(z)+2*Pi*K*i. Moreover, an odd-indexed z_k equals -W_L(-1), where W_L is the L-th branch of the Lambert W function, with L = -floor((k+1)/2). For any K, the mapping M_K(z) = log(z)+2*Pi*K*i has z_k as its unique attractor, convergent from any nonzero point in C (an exception occurs for K=0, for which M_0(z) has two attractors, z_1 and z_-1, as described in my linked document).

The value listed here is the real part of z_3 = a + i*A277682.

			2.062277729598283884978486720008045951283592306704591613100984...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
S. Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, Stan's Library, Vol.VI, Oct 2016.
Eric Weisstein's World of Mathematics, Exponential Function.
Wikipedia, Exponential function.

Fixed points of +exp(z): z_1: A059526, A059527, A238274, and z_3: A277682 (imaginary part), A277683 (modulus).

Fixed points of -exp(z): z_0: A030178, and z_2: A276759, A276760, A276761.

RealDigits[Re[-ProductLog[-2, -1]], 10, 105][[1]] (* Jean-François Alcover, Nov 12 2016 *)

default(realprecision,2050);eps=5.0*10^(default(realprecision))
M(z,K)=log(z)+2*Pi*K*I; \\ the convergent mapping (any K)
K=1;z=1+I;zlast=z;
while(1,z=M(z,K);if(abs(z-zlast)

Let z_3 = A277681+i*A277682. Then z_3 = exp(z_3) = log(z_3)+2*Pi*i = -W_-2(-1).

A276760 Decimal expansion of the imaginary part of the fixed point of -exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Nov 12 2016

Imaginary part of the complex constant z_2 whose real part is in A276759 (see the latter entry for more information).

			4.375185153061898385470906564852584291623823114677011864961044...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, 2016.

Fixed points of -exp(z): z_0: A030178, and z_2: A276759 (real part), A276761 (modulus).

Fixed points of +exp(z): z_1: A059526, A059527, A238274, and z_3: A277681, A277682, A277683.

RealDigits[Im[-ProductLog[-1, 1]], 10, 105][[1]] (* Jean-François Alcover, Nov 12 2016 *)

default(realprecision,2050);eps=5.0*10^(default(realprecision))
M(z,K)=log(-z)+2*Pi*K*I; \\ the convergent mapping (any K)
K=1;z=1+I;zlast=z;
while(1,z=M(z,K);if(abs(z-zlast)

Let z_2 = A276759+i*A276760. Then z_2 = -exp(z_2) = log(-z_2)+2*Pi*i = -W_-1(1).

A276761 Decimal expansion of the modulus of the fixed point of -exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Nov 12 2016

Modulus of z_2 = A276759+i*A276760. See A276759 for more information.

			4.636284632786625189544952318034205387044699355677575252963935...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, 2016.

Fixed points of -exp(z): z_0: A030178, and z_2: A276759 (real part), A276760 (imaginary part).

Fixed points of +exp(z): z_1: A059526, A059527, A238274, and z_3: A277681, A277682, A277683.

RealDigits[Norm[ProductLog[1, 1]], 10, 105][[1]] (* Jean-François Alcover, Nov 12 2016 *)

default(realprecision,2050);eps=5.0*10^(default(realprecision))
M(z,K)=log(-z)+2*Pi*K*I; \\ the convergent mapping (any K)
K=1;z=1+I;zlast=z;
while(1,z=M(z,K);if(abs(z-zlast)

A277682 Decimal expansion of the imaginary part of the fixed point of exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Nov 12 2016

Imaginary part of the complex constant z_3 whose real part is in A277681 (see the latter entry for more information).

			7.588631178472512622568923954107584383013473671992856360409437...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, 2016.

Fixed points of +exp(z): z_1: A059526, A059527, A238274, and z_3: A277681 (real part), A277683 (modulus).

Fixed points of -exp(z): z_0: A030178, and z_2: A276759, A276760, A276761.

RealDigits[Im[ProductLog[1, -1]], 10, 105][[1]] (* Jean-François Alcover, Nov 12 2016 *)

default(realprecision,2050);eps=5.0*10^(default(realprecision))
M(z,K)=log(z)+2*Pi*K*I; \\ the convergent mapping (any K)
K=1;z=1+I;zlast=z;
while(1,z=M(z,K);if(abs(z-zlast)

Let z_3 = A277681+i*A277682. Then z_3 = exp(z_3) = log(z_3)+2*Pi*i = -W_-2(-1).

A277683 Decimal expansion of the modulus of the fixed point of exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Nov 12 2016

Modulus of z_3 = A277681 + i*A277682. See A277681 for more information.

			7.863861176094232668842573623487382321468320207779893460294144...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, 2016.

Fixed points of +exp(z): z_1: A059526, A059527, A238274, and z_3: A277681 (real part), A277682 (imaginary part).

Fixed points of -exp(z): z_0: A030178, and z_2: A276759, A276760, A276761.

RealDigits[Norm[ProductLog[1, -1]], 10, 105][[1]] (* Jean-François Alcover, Nov 12 2016 *)

default(realprecision,2050);eps=5.0*10^(default(realprecision))
M(z,K)=log(z)+2*Pi*K*I; \\ the convergent mapping (any K)
K=1;z=1+I;zlast=z;
while(1,z=M(z,K);if(abs(z-zlast)

A387131 Decimal expansion of the real part of the complex solutions to log(z) = -1/z on the principal branch of log(z).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Aug 17 2025

			0.16837637908722291055702904019641979887737474829...

Index entries for transcendental numbers.

Cf. A030797, A059526, A059527.

Cf. A387132 (absolute value of the imaginary part).

RealDigits[Re[-1/ProductLog[-1]],10,100][[1]]

Equals Re(-1/LambertW(-1)).

A387132 Decimal expansion of the absolute value of the imaginary part of the complex solutions to log(z) = -1/z on the principal branch of log(z).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Aug 17 2025

			0.7077541887847276164728458930765162394938408142...

Index entries for transcendental numbers.

Cf. A030797, A059526, A059527.

Cf. A387131 (real part).

RealDigits[Im[-1/ProductLog[-1]],10,100][[1]]

Equals Im(-1/LambertW(-1)).

Equals -Im(-1/LambertW(-1, -1)).

cp's OEIS Frontend

A030178 Decimal expansion of LambertW(1): the solution to x*exp(x) = 1.

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A059527 Decimal expansion of imaginary part of solution to z = log z.

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A276759 Decimal expansion of the real part of the fixed point of -exp(z) in C congruent with the branch K=1 of log(z)+2*Pi*K*i.

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A277681 Decimal expansion of the real part of the fixed point of exp(z) in C congruent with the branch K=1 of log(z)+2*Pi*K*i.

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A276760 Decimal expansion of the imaginary part of the fixed point of -exp(z) in C congruent with the branch K=1 of log(z)+2*Pi*K*i.

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A276761 Decimal expansion of the modulus of the fixed point of -exp(z) in C congruent with the branch K=1 of log(z)+2*Pi*K*i.

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A277682 Decimal expansion of the imaginary part of the fixed point of exp(z) in C congruent with the branch K=1 of log(z)+2*Pi*K*i.

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A276759 Decimal expansion of the real part of the fixed point of -exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

A277681 Decimal expansion of the real part of the fixed point of exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

A276760 Decimal expansion of the imaginary part of the fixed point of -exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

A276761 Decimal expansion of the modulus of the fixed point of -exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

A277682 Decimal expansion of the imaginary part of the fixed point of exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

A277683 Decimal expansion of the modulus of the fixed point of exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.