By Smirnov V.I.
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Additional info for A course of higher mathematics, vol. 2
This can also be accomplished by decreasing the angle until θ = −π , which also gets us to the ‘branch-cut’. This ‘branch-cut’ lies at z = r exp(± jπ ) for all positive r . Further increasing the angle eventually gets to θ = ± j2π . Further increasing the angle θ > π makes us go ‘underneath’, the primary Riemann sheet, inside the negative real axis of z − complex plane. 5 + 1) −1 is described by thus two Riemann sheets. Return to the first Riemann sheet on the z − complex plane, the branch cut begins at z = 0 , the origin and extends out to the negative real axis to infinity.
Also the applications and discussions are limited to fixed fractional order differintegrals and the variable order of differintegration is kept as future research subject. Perhaps the Fractional Calculus will be the calculus of XXI century. In this book attempt is made to make this topic application oriented for regular science and engineering applications. Therefore rigorous mathematics is kept minimal. In this introductory chapter list in tabular form is provided to readers to have feel of fractional derivatives of some commonly occurring functions.
A very recent approach to solve General Dynamic Systems with action reaction process is described in the Chapter-11, which gives approximate analytic solution close to physics; the mathematics of Adomian Decomposition Method is applied to describe the physical reaction chain. A simple examples in this section elaborates the concept of Fractional Differential equation. 1 Abel’s Fractional Integral Equation of Tautochrone The Abel’s problem is to find a curve where the time of decent is same irrespective of the position of release of ball in frictionless system.
A course of higher mathematics, vol. 2 by Smirnov V.I.