A211268 Integral of a Gaussian peak with unit height and unit half-height width.

1, 0, 6, 4, 4, 6, 7, 0, 1, 9, 4, 3, 1, 2, 2, 6, 1, 7, 9, 3, 1, 5, 2, 6, 7, 5, 9, 6, 2, 3, 4, 6, 2, 0, 1, 0, 6, 9, 4, 2, 9, 4, 3, 0, 9, 4, 2, 4, 1, 7, 2, 0, 2, 3, 2, 5, 3, 8, 5, 2, 4, 7, 7, 9, 2, 7, 3, 2, 6, 7, 4, 6, 0, 9, 6, 9, 4, 2, 3, 1, 3, 9, 1, 9, 2, 6, 1, 7, 5, 5, 4, 0, 2, 4, 2, 0, 7, 4, 4, 8, 1, 2, 6, 2, 8

Offset

1,3

Comments

In spectroscopy, when comparing absorbtion peak shapes, the functions are first scaled vertically and horizontally to a canonical form with unit height and unit half-height width. The 4 most common canonical shapes are: rectangular R(x)=1 for |x|<=1/2 (0 otherwise), Lorentzian L(x)=1/(1+(2x)^2), Gaussian G(x)=exp(-log(2)(2x)^2), and sinc-type S(x) (see A211269). The areas A under such canonical peaks (integral from -inf to +inf) are 1.0 for R(x), (Pi/2)=A019669 for L(x), this constant for G(x), and A211269 for S(x). For a generic peak with height H and half-height width W belonging to the same canonical family, the area is A*H*W. Hence the practical importance of the constant A.

References

M. Quack and F. Merckt, Editors, Handbook of High Resolution Spectroscopy, Wiley, 2011.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Formula

Equals sqrt(Pi/(4*log(2))).

Example

1.064467019431226179315267...

Mathematica

RealDigits[Sqrt[Pi/(4*Log[2])], 10, 50][[1]] (* G. C. Greubel, Mar 30 2017 *)

Prog

(PARI) sqrt(Pi/(4*log(2))) \\ G. C. Greubel, Mar 30 2017

Crossrefs

Cf. A211269 (for sinc-shaped peaks).

Cf. A019669 (for Lorentzian-shaped peaks).

Sequence in context: A316162 A332431 A198840 * A021612 A329281 A201587

Adjacent sequences: A211265 A211266 A211267 * A211269 A211270 A211271

Keyword

nonn,cons,easy

Author

Stanislav Sykora, Apr 07 2012

Status

approved