Gelfond's constant

In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is e^π, that is, e raised to the power pi. Like both e and pi, this constant is a transcendental number. This was first established by Gelfond and may now be considered as an application of the Gelfond–Schneider theorem, noting that where i is the imaginary unit. Since −i is algebraic but not rational, e^π is transcendental. The constant was mentioned in Hilbert's seventh problem. A related constant is 2^, known as the Gelfond–Schneider constant. The related value pi + e^π is also irrational. The decimal expansion of Gelfond's constant begins 23.1406926327792690057290863679485473802661062426002119934450464095243423506904527835169719970675492196... If one defines k_0 = 1/ and for n > 0, then the sequence converges rapidly to e^π. This is based on the digits for the simple continued fraction: As given by the integer sequence A058287. The volume of the n-dimensional ball (or n-ball), is given by where R is its radius, and Γ is the gamma function. Any even-dimensional ball has volume and, summing up all the unit-ball (R = 1) volumes of even-dimension gives 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 is very close to an integer: 262537412640768743.9999999999992500725971981856888793538563373369908627075374103782106479101186073129... As it was the Indian mathematician Srinivasa Ramanujan who first predicted this almost-integer number, it has been named after him, though the number was first discovered by the French mathematician Charles Hermite in 1859. The coincidental closeness, to within 0.000 000 000 000 75 of the number 640320^3 + 744 is explained by complex multiplication and the q-expansion of the j-invariant, specifically: and, where O(e^-π) is the error term, which explains why e^π is 0.000 000 000 000 75 below 640320^3 + 744. (For more detail on this proof, consult the article on Heegner numbers.) The decimal expansion of e^π − π is given by A018938: 19.