Another way of generalizing the derivative to noninteger 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 noninteger 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 Cauchytype fractional derivative
 (10.2)  
where supposing that the branch line starts at t=z and passes through z_{0}, the contour C starts at t=z_{0}, encircles t=z once in the positive sense and returns to t=z_{0} where now the integrand has a different value.
It can be proved that this generalization of the derivative is equivalent to the RiemannLiouville derivative with a lower limit of z_{0} for the appropriate values of a in which both derivatives are defined.
