Definition:Gelfond's Constant

Definition

Gelfond's constant is the number obtained by raising Euler's number $e$ to the power of $\pi$ (pi):

$e^\pi$

Its decimal expansion gives its value to be approximately:

$e^\pi \approx 23 \cdotp 14069 \, 26327 \, 79269 \, 00572 \, 90863 \, 67948 \, 54738 \ldots$

This sequence is A039661 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Also see

• Results about Gelfond's constant can be found here.

Source of Name

This entry was named for Alexander Osipovich Gelfond.

Historical Note

Gelfond's constant arises in the context of the $7$th problem of the Hilbert $23$.

It was proved to be transcendental in $1929$ by Alexander Osipovich Gelfond.

It was since determined to be a special case of the Gelfond-Schneider Theorem, established $\text {1934}$ – $\text {1935}$.

Sources

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): Table $1.1$. Mathematical Constants
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 1$: Special Constants: $1.10$
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $23 \cdotp 140 \, 692 \, 632 \, 7792 \, 69 \, 005 \, 729 \, 086 \ldots$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $23 \cdotp 14069 \, 26327 \, 79269 \, 00572 \, 9086 \ldots$