A Few Beautiful Applications
In the formulas below G is Catalan's constant, γ is the Euler's constant, β2n is the Bernoulli number of order 2n, H2n+1 is the harmonic number of order 2n+1 and Ei is the Exponential integral function.
The notation indicating the 2 periodic function coinciding with on (−1, 1).
If the measure ρ is reducible and let φ be the associated reducer, one has the equality
If the measure ρ is reducible with μ the associated reducer, then if f is square integrable for μ, and if g is square integrable for ρ and is orthogonal with P0 = 1 one has equivalence:
c1 indicates the moment of order 1 of ρ and Tρ the operator
Read more about this topic: Secondary Measure
Famous quotes containing the word beautiful:
“The point of the dragonfly’s terrible lip, the giant water bug, birdsong, or the beautiful dazzle and flash of sunlighted minnows, is not that it all fits together like clockwork--for it doesn’t ... but that it all flows so freely wild, like the creek, that it all surges in such a free, finged tangle. Freedom is the world’s water and weather, the world’s nourishment freely given, its soil and sap: and the creator loves pizzazz.”
—Annie Dillard (b. 1945)