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10. Cauchy Integral Formula

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Another way of generalizing the derivative to non-integer order is given by the Cauchy integral formula that plays a key role in complex analysis,

 (10.1)

Despite its generalization to any complex number a seems straightforward, it must be taken into account that while being n integer there is an isolated singularity at t=z, being it non-integer there is a branch point, what means that the integration contour has to be chosen carefully. Otherwise, the generalization only involves changing the factorial to the gamma function, so defining what is known as the Cauchy-type fractional derivative

 (10.2)

where supposing that the branch line starts at t=z and passes through z0, the contour C starts at t=z0, encircles t=z once in the positive sense and returns to t=z0 where now the integrand has a different value.

It can be proved that this generalization of the derivative is equivalent to the Riemann-Liouville derivative with a lower limit of z0 for the appropriate values of a in which both derivatives are defined.