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In mathematics, the combined differentiation/integration operator used in fractional calculus is called the differintegral, and it has a few different forms which are all equivalent, provided that they are initialized (used) properly.
It is noted:
- <math>{}_a \mathbb{D}^q_t<math>
and is most generally defined as:
<math>{}_a\mathbb{D}^q_t= \left\{\begin{matrix} \frac{d^q}{dx^q}, & \Re(q)>0 \\ 1, & \Re(q)=0 \\ \int^t_a(dx)^{-q}, & \Re(q)<0 \end{matrix}\right.<math>
The Riemann-Liouville differintegral (RL) is the simplest and easiest to use, and consequently it is the most often used.
Constructing the Riemann-Liouville differintegral
We first introduce the Riemann-Liouville fractional integral, which is a straightforward generalization of the Cauchy integral formula:
- <math>{}_a\mathbb{D}^{-q}_tf(x)=\frac{1}{\Gamma(q)} \int_{a}^{t}(t-\tau)^{q-1}f(\tau)d\tau<math>
This gives us integration to an arbitrary order. To get differentiation to an arbitrary order, we simply integrate to arbitrary order n − q, and differentiate the result to integer order n. (We choose n and q so that n is the smallest positive integer greater than or equal to q (that is, the ceiling of q)):
- <math>{}_a\mathbb{D}^q_tf(x)=\frac{d^n}{dx^n}{}_a\mathbb{D}^{-(n-q)}_tf(x)<math>
Thus, we have differentiated n − (n − q) = q times. The RL differintegral is thus defined as (the constant is brought to the front):
- <math>{}_a\mathbb{D}^q_tf(x)=\frac{1}{\Gamma(n-q)}\frac{d^n}{dx^n}\int_{a}^{t}(t-\tau)^{n-q-1}f(\tau)d\tau<math> definition
When we are taking the differintegral at the upper bound (t), it is usually written:
- <math>{}_a\mathbb{D}^q_tf(t)=\frac{d^qf(t)}{d(t-a)^q}=\frac{1}{\Gamma(n-q)} \frac{d^n}{dt^n} \int_{a}^{t}(t-\tau)^{n-q-1}f(\tau)d\tau<math> definition
And when we are assuming that the lower bound is zero, it is usually written:
<math>\mathbb{D}^q_tf(t)=\frac{d^qf(t)}{d(t)^q}=\frac{1}{\Gamma(n-q)} \frac{d^n}{dt^n} \int_{0}^{t}(t-\tau)^{n-q-1}f(\tau)d\tau<math> definition
That is, we are taking the differintegral of f(t) with respect to t.
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