Functions and Analysis

1404 Submissions

[2] viXra:1404.0072 [pdf] submitted on 2014-04-10 02:19:40

On The Leibniz Rule And Fractional Derivative For Differentiable And Non-Differentiable Functions

Authors: Xiong Wang
Comments: 7 Pages.

In the recent paper {\it Communications in Nonlinear Science and Numerical Simulation. Vol.18. No.11. (2013) 2945-2948}, it was demonstrated that a violation of the Leibniz rule is a characteristic property of derivatives of non-integer orders. It was proved that all fractional derivatives ${\cal D}^{\alpha}$, which satisfy the Leibniz rule ${\cal D}^{\alpha}(fg)=({\cal D}^{\alpha}f) \, g + f \, ({\cal D}^{\alpha}g)$, should have the integer order $\alpha=1$, i.e. fractional derivatives of non-integer orders cannot satisfy the Leibniz rule. However, it should be noted that this result is only for differentiable functions. We argue that the very reason for introducing fractional derivative is to study non-differentiable functions. In this note, we try to clarify and summarize the Leibniz rule for both differentiable and non-differentiable functions. The Leibniz rule holds for differentiable functions with classical integer order derivative. Similarly the Leibniz rule still holds for non-differentiable functions with a concise and essentially local definition of fractional derivative. This could give a more unified picture and understanding for Leibniz rule and the geometrical interpretation for both integer order and fractional derivative.
Category: Functions and Analysis

[1] viXra:1404.0026 [pdf] submitted on 2014-04-03 22:36:37

An Elementary Primer on Gaussian Integrals

Authors: William O. Straub
Comments: 9 Pages.

Gaussian integrals appear frequently in mathematics and physics, especially probability, statistics and quantum mechanics. One of the truly odd things about these integrals is that they cannot be evaluated in closed form over finite limits but are generally exactly integrable over +/- infinity. Yet their evaluation is still often difficult, particularly multi-dimensional integrals and those involving quadratics, vectors and matrices in the exponential. An added complication is that Gaussian integrals can involve ordinary real or complex variables as well as the less familiar Grassmann variables, which are important in the description of fermions. In this elementary primer we present some of the more common Gaussian integrals of both types, along with methods for their evaluation.
Category: Functions and Analysis