Taylor and Laurent expansions
This applet displays the various Taylor and Laurent expansions of the function
f(z) = 1/(z-1)(z-2). At each point z_0, the function
f has a Taylor expansion around z_0, which converges
in the red disk; a Laurent expansion around z_0, which converges
in the green annulus; and a second Laurent expansion around z_0,
which converges in the cyan disk exterior. Note that there are some
white circles for which none of the three series converge; these include
the singularities of f at 1 and 2. Click on the grid to
move the point z_0.
To read off all the displayed terms of a Taylor or Laurent series, one
may have to use the cursor keys to navigate the text windows.
Note that the co-efficients of the Taylor series become much larger
as one approaches a singularity.
Thanks to Brandt Kronholm for pointing out an error in an earlier version of this applet.
Previous applet: The complex integral
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