The study of derivatives (including antiderivatives) to fractional orders. For instance, rather than the second derivative of sin(x), you would take the halfth derivative of sin(x).

Denote the nth derivative D^{n} and the n-fold integral D^{-n}. Then

D^{-1}f(t) = ∫_{0}^{t}f(ξ)dξ

Now, if

D^{-n}f(t) = (n-1)!^{-1} ∫_{0}^{t}(t-ξ)^{n-1}f(ξ) dξ

is true for n, then

D^{-(n+1)}f(t) = n!^{-1}∫_{0}^{t}(t-ξ)^{n
}f(ξ)dξ

But the former equation is true for n = 1, so it is also true for all integral n by induction.

The fractional derivative can only be given in terms of elementary functions for a small number of functions. For example:

D^{-ν}t^{-λ} = Γ(λ+1) / Γ(λ+ν+1) t^{λ+ν}

Where Γ(x) is the Gamma function. (CRC Concise Encyclopedia of Mathematics, Eric W. Weisstein)

In practice, the fractional calculus is not a very useful tool (except perhaps on a theoretical level), and is a pure figment of recreational mathematics.