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Decimal expansion of the imaginary part of the infinite nested power (1+(1+(1+...)^i)^i)^i, with i being the imaginary unit.
+10
6
4, 0, 7, 5, 6, 3, 9, 3, 0, 5, 4, 5, 6, 2, 1, 8, 4, 4, 7, 3, 9, 6, 6, 3, 2, 4, 3, 3, 9, 4, 1, 5, 2, 0, 8, 8, 6, 4, 0, 6, 2, 7, 9, 9, 2, 8, 6, 6, 7, 7, 5, 1, 0, 3, 0, 4, 8, 7, 4, 8, 3, 5, 6, 7, 7, 0, 4, 0, 2, 1, 5, 5, 3, 9, 4, 8, 2, 2, 1, 5, 4, 2, 1, 4, 9, 1, 3, 9, 2, 7, 4, 8, 9, 9, 2, 3, 5, 0, 4, 0, 4, 8, 5, 8, 0
COMMENTS
The real part and the modulus of this complex constant are in A272875 and A272877, respectively. For more information, see A272875.
EXAMPLE
0.40756393054562184473966324339415208864062799286677510304874835...
MATHEMATICA
RealDigits[Im[z /. FindRoot[(1 + z)^I == z, {z, 0}, WorkingPrecision -> 120]]][[1]] (* Amiram Eldar, May 26 2023 *)
PROG
(PARI) \\ f(x) computes (x+(x+...)^i)^i, provided that it converges:
f(x)={my(z=1.0, zlast=0.0, eps=10.0^(1-default(realprecision))); while(abs(z-zlast)>eps, zlast=z; z=(x+z)^I); return(z)}
\\ To compute this constant, use:
z0 = f(1); imag(z0)
Decimal expansion of the modulus of the infinite nested power (1+(1+(1+...)^i)^i)^i, with i being the imaginary unit.
+10
6
7, 8, 7, 5, 4, 3, 2, 7, 2, 3, 9, 6, 8, 3, 7, 0, 1, 0, 9, 6, 7, 6, 6, 0, 2, 4, 0, 5, 3, 9, 4, 3, 6, 4, 2, 4, 5, 8, 9, 4, 5, 9, 2, 7, 7, 2, 8, 1, 3, 8, 8, 4, 0, 8, 2, 7, 6, 0, 9, 3, 8, 9, 9, 6, 0, 7, 4, 4, 4, 6, 5, 6, 3, 3, 1, 5, 6, 6, 9, 4, 3, 1, 9, 2, 0, 3, 0, 4, 6, 6, 0, 9, 1, 7, 3, 6, 4, 6, 9, 5, 6, 9, 9, 1, 7
COMMENTS
The real and imaginary parts giving rise to this constant are in A272875 and A272876, respectively. For more information, see A272875.
EXAMPLE
0.78754327239683701096766024053943642458945927728138840827609389960...
PROG
(PARI) \\ f(x) computes (x+(x+...)^i)^i, provided that it converges:
f(x)={my(z=1.0, zlast=0.0, eps=10.0^(1-default(realprecision))); while(abs(z-zlast)>eps, zlast=z; z=(x+z)^I); return(z)}
\\ To compute this constant, use:
z0 = f(1); abs(z0)
Decimal expansion of the real part of the solution of z = (i+z)^(-i) in C (i is the imaginary unit).
+10
3
1, 3, 3, 9, 2, 0, 9, 1, 6, 8, 5, 2, 9, 1, 1, 1, 9, 6, 8, 3, 5, 9, 2, 6, 9, 9, 8, 5, 7, 6, 2, 7, 6, 4, 1, 7, 0, 8, 8, 5, 9, 8, 8, 2, 6, 3, 2, 6, 9, 0, 4, 3, 3, 8, 4, 7, 7, 3, 9, 6, 7, 5, 8, 0, 8, 7, 2, 1, 1, 2, 9, 5, 3, 8, 1, 3, 9, 8, 0, 1, 2, 4, 4, 8, 7, 3, 7, 7, 1, 1, 3, 7, 7, 2, 4, 7, 7, 4, 1, 6, 6, 5, 5, 2, 5
COMMENTS
In C, the unique invariant point of the mapping M(z) = (i+z)^(-i) is not the attractor of the mapping (unstable behavior), but it is an attractor of the modified mapping M'(z) = (z+M(z))/2. For M', it takes 5000 iterations to reduce the value of |z - M'(z)| below 10^(-3400). Interestingly, the imaginary part of z seems to be equal to -1/2 (verified to 5000 digits). If this conjecture holds, and considering the definition, one can symbolically write (i+(i+(i+...)^(-i))^(-i))^(-i) = a - i/2.
The solution to x^i = x + i with the real part greater than 1. - Michal Paulovic, Jul 06 2023
EXAMPLE
1.3392091685291119683592699857627641708859882632690433847739675808721...
MATHEMATICA
RealDigits[Re[z /. FindRoot[(I + z)^(-I) == z, {z, 0}, WorkingPrecision -> 120]]][[1]] (* Amiram Eldar, May 30 2023 *)
PROG
(PARI) \p 4000 \\ Set precision
Mp(z)=0.5*(z+I)^(-I); \\ Mapping M'
z=1.0; for(k=1, 5000, z=Mp(z)); \\ Initialize and iterate
d = -floor(log(abs(z-Mp(z)))/log(10)) \\ Crude convergence test (3438)
real(z) \\ The result; keep << d digits, and test for stability.
(PARI) \p 120
x=1; for(a=1, 1000, x=x-(x^I-x-I)/(I*x^(I-1)-1)); x \\ Michal Paulovic, Jul 06 2023
Decimal expansion of the real part of the solution of z = (i+z)^i in C (i is the imaginary unit).
+10
3
2, 6, 9, 2, 9, 3, 4, 3, 7, 1, 6, 9, 3, 1, 1, 2, 2, 7, 1, 9, 0, 8, 6, 8, 0, 2, 1, 2, 6, 8, 8, 6, 2, 0, 1, 0, 5, 3, 2, 9, 1, 1, 0, 0, 6, 0, 3, 7, 6, 8, 4, 6, 7, 1, 7, 1, 2, 7, 1, 6, 0, 1, 5, 1, 5, 2, 8, 3, 9, 2, 3, 1, 5, 2, 6, 4, 9, 8, 1, 7, 6, 1, 9, 8, 3, 1, 3, 6, 8, 0, 1, 9, 9, 1, 0, 9, 8, 9, 9, 9, 4, 8, 8, 4, 1
COMMENTS
In C, the unique invariant point of the mapping M(z) = (i+z)^i is also its attractor. The convergence is linear and takes about 1650 iterations to reduce the value of |z - M(z)| by 1000 decimal digits. The imaginary part of the invariant point is in A290410. Considering the definition, one can symbolically write A290409 + i* A290410 = (i+(i+(i+...)^i)^i)^i.
EXAMPLE
0.269293437169311227190868021268862010532911006037684671712716015...
MATHEMATICA
RealDigits[Re[z /. FindRoot[(I + z)^I == z, {z, 0}, WorkingPrecision -> 120]]][[1]] (* Amiram Eldar, May 30 2023 *)
PROG
(PARI) \p 3000 \\ Set precision
M(z)=(z+I)^I; \\ Mapping M
z=1.0; for(k=1, 2000, z=M(z)); \\ Initialize and iterate
d = -floor(log(abs(z-M(z)))/log(10)) \\ A crude convergence test
real(z) \\ The result; keep << d digits, and test for stability.
Decimal expansion of the imaginary part of the solution of z = (i+z)^i in C (i is the imaginary unit).
+10
3
1, 2, 5, 7, 6, 4, 5, 4, 5, 7, 3, 8, 6, 3, 8, 3, 2, 3, 8, 1, 5, 6, 1, 8, 5, 0, 8, 4, 1, 1, 8, 3, 6, 7, 7, 0, 1, 4, 4, 2, 3, 2, 4, 3, 4, 7, 3, 4, 0, 7, 6, 3, 5, 2, 0, 3, 4, 1, 7, 6, 7, 4, 9, 7, 6, 5, 0, 5, 0, 8, 9, 7, 0, 3, 3, 6, 7, 9, 1, 0, 8, 7, 1, 9, 7, 7, 9, 4, 7, 5, 7, 7, 3, 8, 8, 0, 9, 9, 3, 3, 3, 0, 9, 5, 7
COMMENTS
For more comments, see the real part in A290409.
EXAMPLE
0.012576454573863832381561850841183677014423243473407635203417674...
MATHEMATICA
RealDigits[Im[z /. FindRoot[(I + z)^I == z, {z, 0}, WorkingPrecision -> 120]]][[1]] (* Amiram Eldar, May 30 2023 *)
PROG
(PARI) \p 3000 \\ Set precision
M(z)=(z+I)^I; \\ Mapping M
z=1.0; for(k=1, 2000, z=M(z)); \\ Initialize and iterate
d = -floor(log(abs(z-M(z)))/log(10)) \\ A crude convergence test
imag(z) \\ The result; keep << d digits, and test for stability.
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