Mathematica Floor Function Sum

A number of geometric like sequences with a floor function in the numerator can be done analytically.

Mathematica floor function sum.

Sum f i imin imax j jmin jmax evaluates the multiple sum sum i imin imax sum j jmin jmax. The floor function really throws the wrench in the works. For simplicity let s assume x 2 without the floor function this is more or less trivial it s just n 2 ln x discretization corrections which can be computed. Changed t to tt feven to feven tt a n 4 integrate feven tt cos n tt tt 0 2.

Sum f i i1 i2 uses successive values i1 i2. X 1 π k 11 ksin 2πkx x 1 2. Sum k n xn lfloor frac n 2 k rfloor where n is an integer and x is a small number x ll n. One can use the fourier series representation of the floor function.

And. For the rest we can verify that it gives zero contribution. Browse other questions tagged real analysis number theory ceiling and floor functions or ask your own question. For a unit fraction 3 sums of this form lead to devil s staircase like behavior.

A 0 2 integrate feven tt tt 0 1. Floor or the greatest integer function gives the greatest integer less than or equal to a given value. Integrate n x 1 2 x 0 1 n 2 1 2. Round or the nearest integer function gives the nearest integer.

Sum f i imax evaluates the sum sum i 1 imax f. Sum f i imin imax starts with i imin. Changed t to tt feven to feven tt fourier approximation function fapp t n a 0 sum a 2 x 1 cos 2 x 1 t x 1 n. Integrating the x 1 2 we obtain your expected answer.

Inverseztransform z x x x x if the function whose ztranform you are trying to find can t be evaluated mathematica does not give a ztransform for. 2 can be done analytically for rational. Since all the functions evaluated to same value which is x. Ztransform f x z all the functions f produced the same ztransform.

Sum f i imin imax di uses steps d i. Floor x gives the greatest integer less than or equal to x. For instance sums of the form. Floor x a gives the greatest multiple of a less than or equal to x.