Gauss's continued fraction

Recall that

It is known that

When we regard

we can use the following formula

   (regular expression)

   in special form of 0218.

If we regard

we can use the following formulas

as in 0215. When a=1/2, b=1, c=3/2, and x=-x^2/2,

Applying gamma-shaped multipliers repeatedly, we finally get

We also know that

we can say that

That’s one of the ways to get pi.