Spherical Bessel expansions of sine, cosine, and exponential integrals (Q1567373)

The following spherical Bessel expansions of exponential integrals are presented and verified: \[ Ci(2x)= \gamma+ \log(2x)+ \int^{2x}_0 {\cos t-1\over t} dt= \gamma+ \log(2x+ \sum^\infty_{n=1} a_n[J_n(x)]^2,\tag{1} \] \[ Ei(x)= \int^x_{-\infty} {e^t\over t} dt= \gamma+\log|x|+ \sum^\infty_{n= 0}(- 1)^n(x+ a_n)\Biggl[i_n \Biggl({x\over 2}\Biggr)\Biggr]^2.\tag{2} \] A new expansion of \(\cos(2x)\) is also given: \[ \cos(2x)= \sum^\infty_{n=0} (-1)^n(2n+ 1)(2n^2+ 2n+ 1)[j_n(x)]^2. \]