A276761 -id:A276761 - cp's OEIS frontend

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.

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).

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)

cp's OEIS Frontend

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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Examples

Links

Crossrefs

Programs

Mathematica

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Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A277683 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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.

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.