Gauss's continued fraction

Trigonometric functions such as sin(x) and cos(x) are expressed in terms of ₀F₁(a;x).

That is

and

Recall that the following relation

in special form as in 0329. Since 3/2=1/2+1 when a=1/2, we can say that

after applying gamma-shaped multipliers of 1/2 repeatedly.

Thus

   in special form

since sin(x)/cos(x)=tan(x). That is

Similarly, we can do the same thing for tanh(x) from sinh(x) and cosh(x).

Hyperbolic functions such as sinh(x) and cosh(x) are expressed in terms of ₀F₁(a;x).

That is

and

We use the same following relation

in special form as in 0329. Since 3/2=1/2+1 when a=1/2, we can say that

after applying gamma-shaped multipliers of 1/2 repeatedly.

Thus

   in special form

since sinh(x)/cosh(x)=tanh(x). That is,