Chebyshev Polynomials

The applet below depicts Chebyshev polynomials of the first kind T_n(x) = cos(n·arccos(x)), n = 0, ..., 50.

The applet displays the sequence of polynomials P_Min through P_Max, inclusive. When sufficiently many (30 above) of the polynomial graphs are displayed simultanesoiusly, one can see "ghost lines" -- empty spaces left untouched by the graphs. What are they?

Both Min and Max numbers in the applet can be clicked on off their vertical center line. To increase the number click to the right of its center line, to decrease it click to the left. Dragging the cursor will speed up the process. 50 is the maximum value for both.

When plenty of the polynomials are graphed there appear some ghost images. What are these?

Copyright © 1996-2008 Alexander Bogomolny