Gauss's continued fraction

Some manipulations of exponential

For

   as in 0327

when a=1 and b=1, it is known that

    in special form

The other way is start from the resulting formula

as in 0401. That is,

     in special form

We can express error function and Gaussian cdf in terms of 1F1(a,b;x).

Error function is defined as

That can also be expressed as

or

   in special form

from the formula

 as in 0401

and after shifting the negative signs from up to front by the rule of signs.

We can also express Gaussian cdf function as

because