Integer multiples The fractional parts of the expressions for various exponents . The fractional parts of the expressions for various exponents . The fractional parts of the expressions for various exponents . Density plots of the fractional parts of the expressions as a function of and exponents . Cumulative sums Cumulative sums of the fractional parts minus one‐half of the expressions for various exponents . Cumulative sums of the fractional parts minus one‐half of the expressions for various exponents . Cumulative sums of the fractional parts minus one‐half of the expressions for various exponents . Density plots of the cumulative sums of the fractional parts minus one‐half of the expressions as a function of and exponents . Cumulative sums of the fractional parts minus one half for various . Cumulative sums of the e fractional parts for various exponents and . Eigenvalue plots Eigenvalues in the complex plane of the square matrices formed from the fractional parts of . The eigenvalues are colored from red () to blue (). Eigenvalues in the complex plane of the square matrices formed from the fractional parts of . The matrix formed is . The eigenvalues are colored from red () to blue (). Root plots Zeros of the polynomials over the complex plane. The zeros are colored from red () to blue (). Nested differences, sums, products, and quotients Iterated pairwise differences modulo 2 and 3 starting with the first 200 . Fourier series Fourier series over the ,‐plane. Fejer averages of the exponential Fourier series , over the complex plane. The coloring of the curves varies from red () to violet (). Fourier sums . Power series plots 3D plots of the power series . The left graphic shows the real part and the coloring is according to the imaginary part. The right graphic shows a contour plot of the argument over the complex plane and the coloring is cyclic. Density plots of coefficients of of the power series . The left graphic shows the coefficients modulo 2 and the right graphic shows the coefficients modulo . Polar plots Plots of the points . Iterated fractional parts The numbers , .
