Definition:First Lemniscate Constant
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Definition
The first lemniscate constant is the value of the expression:
| \(\ds L_1\) | \(=\) | \(\ds \int_0^1 \dfrac {\d x} {\sqrt {1 - x^4} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 2 \int_0^\pi \dfrac {\d \theta} {\sqrt {1 + \sin^2 \theta} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {4 \sqrt {2 \pi} } \paren {\map \Gamma {\dfrac 1 4} }^2\) | ||||||||||||
| \(\ds \) | \(\approx\) | \(\ds 1 \cdotp 31102 \, 87771 \, 46059 \, 90523 \ldots\) |
This sequence is A085565 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Also see
Sources
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1,31102 87771 46059 90523 \ldots$
- Weisstein, Eric W. "Lemniscate Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LemniscateConstant.html