An alternative form for the Legendre polynomial expansion coefficients (Q1086719)

Let \(a_ j\) be the coefficient of the Legendre expansions of f(x), that is, \[ f(x)=(1/2)\sum^{\infty}_{j=0}(2j+1)a_ jP_ j(x),a_ j=\int^{1}_{-1}f(x)P_ j(x)dx. \] Then the new form for \(a_ j\) considered in the paper is \[ a_ j=\frac{2^{1/2}}{\pi}\int^{1}_{- 1}\cos [(j+1/2)\cos^{-1}y](1-y)^{-1/2}L(y)dy, \] with \[ L(y)=\int^{y}_{-1}f(x)(y-x)^{-1/2}dx. \] The author discusses a number of properties of this form and an application from the theory of electron scattering.