Construction

If one defines k₀ = 1/√2 and

$${\displaystyle k_{n+1}={\frac {1-{\sqrt {1-k_{n}^{2}}}}{1+{\sqrt {1-k_{n}^{2}}}}}}$$

for n > 0, then the sequence^[3]

$${\displaystyle (4/k_{n+1})^{2^{-n}}}$$

converges rapidly to e^π.

Ramanujan's constant

$${\displaystyle e^{\pi {\sqrt {163}}}=({\text{Gelfond's constant}})^{\sqrt {163}}}$$

This is known as Ramanujan's constant. It is an application of Heegner numbers, where 163 is the Heegner number in question.

Similar to e^π - π, e^π√163 is very close to an integer:

${\displaystyle e^{\pi {\sqrt {163}}}=}$ 262537412640768743.9999999999992500725971981856888793538563373369908627075374103782106479101186073129... ${\displaystyle \approx 640\,320^{3}+744}$

This number was discovered in 1859 by the mathematician Charles Hermite.^[5] In a 1975 April Fool article in Scientific American magazine,^[6] "Mathematical Games" columnist Martin Gardner made the hoax claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name.

The coincidental closeness, to within 0.000 000 000 000 75 of the number 640320³ + 744 is explained by complex multiplication and the q-expansion of the j-invariant, specifically:

$${\displaystyle j((1+{\sqrt {-163}})/2)=(-640\,320)^{3}}$$

and,

$${\displaystyle (-640\,320)^{3}=-e^{\pi {\sqrt {163}}}+744+O\left(e^{-\pi {\sqrt {163}}}\right)}$$

where O(e^-π√163) is the error term,

$${\displaystyle {\displaystyle O\left(e^{-\pi {\sqrt {163}}}\right)=-196\,884/e^{\pi {\sqrt {163}}}\approx -196\,884/(640\,320^{3}+744)\approx -0.000\,000\,000\,000\,75}}$$

which explains why e^π√163 is 0.000 000 000 000 75 below 640320³ + 744.

(For more detail on this proof, consult the article on Heegner numbers.)

The number e^π − π

The decimal expansion of e^π − π is given by A018938:

${\displaystyle e^{\pi }-\pi =}$ 19.9990999791894757672664429846690444960689368432251061724701018172165259444042437848889371717254321516...

Despite this being nearly the integer 20, no explanation has been given for this fact and it is believed to be a mathematical coincidence.

The number π^e

The decimal expansion of π^e is given by A059850:

${\displaystyle \pi ^{e}=}$ 22.4591577183610454734271522045437350275893151339966922492030025540669260403991179123185197527271430315...

It is not known whether or not this number is transcendental. Note that, by Gelfond-Schneider theorem, we can only infer definitively that a^b is transcendental if a is algebraic and b is not rational (a and b are both considered complex numbers, also a ≠ 0, a ≠ 1).

In the case of e^π, we are only able to prove this number transcendental due to properties of complex exponential forms, where π is considered the modulus of the complex number e^π, and the above equivalency given to transform it into (-1)^-i, allowing the application of Gelfond-Schneider theorem.

π^e has no such equivalence, and hence, as both π and e are transcendental, we can make no conclusion about the transcendence of π^e.

The number e^π − π^e

As with π^e, it is not known whether e^π − π^e is transcendental. Further, no proof exists to show whether or not it is irrational.

The decimal expansion for e^π − π^e is given by A063504:

${\displaystyle e^{\pi }-\pi ^{e}=}$ 0.6815349144182235323019341634048123526767911086035197442420438554574163102913348711984522443404061881...

The number iⁱ

Using the principal value of the complex logarithm,

$${\displaystyle i^{i}=(e^{i\pi /2})^{i}=e^{-\pi /2}=(e^{\pi })^{-1/2}}$$

The decimal expansion of is given by A049006:

${\displaystyle i^{i}=}$ 0.2078795763507619085469556198349787700338778416317696080751358830554198772854821397886002778654260353...

Because of the equivalence, we can use the Gelfond-Schneider theorem to prove that the reciprocal square root of Gelfond's constant is also transcendental:

i is both algebraic (a solution to the polynomial x² + 1 = 0), and not rational, hence iⁱ is transcendental.