Conversion from some integral to continued fraction

Euler’s constant

It is know that

  in continued fraction

whose coefficient is [0;1,1,2,1,2,1,4,3,13,5,1,1,8,1,2,4,1,1,40,1,11,3,7,1,7,1,1,5,1,49,…].

One of the expansions is

    in infinite series

The exponential integral is

   for x>0

One of the expansions of this is

We can convert it into continued fraction using the following formula

  (regular expression)

   in special form

  in special form

where  is Euler’s constant.