I^i

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i^i = (e^(i*pi/2))^i = e^(-pi/2) = (e^pi)^(-1/2) = (Gelfond's constant)^(-1/2) = 1/sqrt(Gelfond's constant). Since Gelfond's constant is a transcendental number, this implies that i^i = 1/sqrt(Gelfond's constant) is also transcendental.

Decimal expansion for i^i

$i^{i}=e^{(-{\tfrac {\pi }{2}})}=0.20787957635076190854695561983497877003387\ldots \,$

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, ...}

Continued fraction expansion for i^i

The simple continued fraction expansion for i^i = exp(-pi/2) is

$i^{i}=e^{(-{\tfrac {\pi }{2}})}=0~+~{\cfrac {1}{4+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{3+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}}}\,$

A049007 Continued fraction for i^i = exp(-Pi/2).

{0, 4, 1, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 20, 1, 3, 6, 10, 3, 2, 1, 1, 7, 2, 2, 1, 1, 1, 2, 7, 1, 23, 28, 2, 1, 2, 3, 138, 1, 4, 2, 3, 1, 1, 50, 1, 2, 1, 1, 6, 1, 24, 1, 2, 2, 1, 1, 1, 1, 1, 4, 6, 11, 1, 16, 3, 3, 1, 1, 1, 2, ...}

Reciprocal of i^i

1/(i^i) = i^(-i) = (e^(i*pi/2))^(-i) = e^(pi/2) = (e^pi)^(1/2) = (Gelfond's constant)^(1/2) = sqrt(Gelfond's constant). Since Gelfond's constant is a transcendental number, this implies that 1/(i^i) = sqrt(Gelfond's constant) is also transcendental. Note that

${\sqrt[{i}]{i}}=i^{\frac {1}{i}}=i^{-i}=(i^{i})^{-1}={\frac {1}{i^{i}}}.\,$

Decimal expansion for 1/(i^i)

${\frac {1}{i^{i}}}=e^{({\tfrac {\pi }{2}})}=4.810477380965351655473035666703\ldots \,$

A042972 Decimal expansion of i^(-i), i=sqrt(-1).

{4, 8, 1, 0, 4, 7, 7, 3, 8, 0, 9, 6, 5, 3, 5, 1, 6, 5, 5, 4, 7, 3, 0, 3, 5, 6, 6, 6, 7, 0, 3, 8, 3, 3, 1, 2, 6, 3, 9, 0, 1, 7, 0, 8, 7, 4, 6, 6, 4, 5, 3, 4, 9, 4, 0, 0, 2, 0, 8, 1, 5, 4, 8, 9, 2, 4, 2, 5, 5, 1, 9, 0, 4, 8, 9, 1, 5, 8, ...}

Continued fraction expansion for 1/(i^i)

The continued fraction expansion for 1/(i^i) is trivially derived from the continued fraction expansion for i^i (just omit the initial term, i.e. 0, of A049007).

See also

• Gelfond's constant = 1/(i^i)^2