Addendum to: "A new numerical method for obtaining gluon distribution functions G(x,Q²)=xg(x,Q²), from the proton structure function F₂γᵖ(x,Q²)."

الأصل · EN

In a recent Letter entitled "A new numerical method for obtaining gluon distribution functions G(x,Q²)=xg(x,Q²), from the proton structure function F₂γᵖ(x,Q²)" [arXiv:0907.4790], we derived an accurate and fast algorithm for numerically inverting Laplace transforms, which we used in obtaining gluon distributions from the proton structure function F₂γᵖ(x,Q²). We inverted the function g(s), where s is the variable in Laplace space, to G(v), where v is the variable in ordinary space. Since publication, we have discovered that the algorithm does not work if g(s)→ 0 less rapidly than 1/s, as s→∞. Although we require that g(s)→ 0 as s→∞, it can approach 0 as 1 sβ, with 0<β<1, and still be a proper Laplace transform. In this note, we derive a new numerical algorithm for just such cases, and test it for g(s)=√ π √ s, the Laplace transform of 1 √ v.