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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  ,  .
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