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7. Riemann-Liouville Derivative
 
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Riemann-Liouville derivative is the most used generalization of the derivative. It is based on Cauchy's formula for calculating iterated integrals. If the first integral of a function, which must equal to deriving it to -1, is as follows

(7.1)

the calculation of the second can be simplified by interchanging the integration order

(7.2)

This method can be applied repeatedly, resulting in the following formula for calculating iterated integrals,

(7.3)

Now this can be easily generalized to non-integer values, in what is the Riemann-Liouville derivative,

(7.4)

Note however that in the above formulas the election of 0 as the lower limit of integration has been arbitrary, and any other number could be chosen. Generally, the election of the integration limits in this and other generalizations of the derivative is indicated with subscripts. The Riemann-Liouville derivative with the lower integration limit a would be

(7.5)

The problem with this generalization is that if the real part of a is positive or zero the integral diverges. So it only can be used to calculate generalized integrals. However, this can be solved easily by deriving first by ordinary derivative more than the amount necessary, thus making the remaining necessary differentiation negative and then applying the generalized derivative for completing the rest in which will be a negative differentiation,

(7.6)

Once seen how integration limits must be specified in fractional derivatives, we can return to the derivative of the exponential (2.1) and the derivative of powers (3.2) to see how their disagreement (3.7) was caused by having different limits. With the exponential, supposing b a positive number,

(7.7)

while with the powers

(7.8)

So both results were different in the case of the exponential function because the limits of the derivatives were different. And the case of this discrepancy is not casual. Differences in the results of derivatives with different differentiation limits are indeed important.

Finally, it can be proved that the Grunwald-Letnikov derivative with any given integration limits is equal to the Riemann-Liouville derivative with the same limits for any complex number a with a negative real part. This important result means that our derivative generalized by the binomial formula in (4.4) and (4.6) is equivalent to the Riemann-Liouville derivative with a lower limit of negative infinity provided that the real part of a is negative,

(7.9)

This kind of Riemann-Liouville derivative with a lower limit of negative infinity is known as the Weyl derivative.