Fractional differentiation and integration

The n-th order derivative or integral of a function is calculated through Fourier series expansion.

% The n-th order derivative or integral of a function defined in a given
% range [a,b] is calculated through Fourier series expansion, where n is
% any real number and not necessarily integer. The necessary integrations
% are performed with the Gauss-Legendre quadrature rule. Selection for the
% number of desired Fourier coefficient pairs is made as well as for the
% number of the Gauss-Legendre integration points.
% Unlike many publicly available functions, the Gauss integration points k
% can be calculated for k>=46. The algorithm does not rely on the build-in
% Matlab routine 'roots' to determine the roots of the Legendre polynomial,
% but finds the roots by looking for the eigenvalues of an alternative
% version of the companion matrix of the k'th degree Legendre polynomial.
% The companion matrix is constructed as a symmetrical matrix, guaranteeing
% that all the eigenvalues (roots) will be real. On the contrary, the
% 'roots' function uses a general form for the companion matrix, which
% becomes unstable at higher values of k, leading to complex roots.
%
%__________________________________________________________________________
% Copyright (c) 13-Mar-2014
% George Papazafeiropoulos
% First Lieutenant, Infrastructure Engineer, Hellenic Air Force
% Civil Engineer, M.Sc., Ph.D. candidate, NTUA
% Email: [e-mail]
% Website: http://users.ntua.gr/gpapazaf/
%
%