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.

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