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Taylor and Laurent expansions

This article explains and demonstrates Taylor, Laurent and second Laurent expansions of the particular sample function, 1/(z-1)(z-2), on the complex plane.

The function being analysed is

$f(z) = 1 \over {(z-1)(z-2)}$

The function can be expanded for (around) the selected point, to get the expansion that would provide approximate value at or near this point. If you are not sure that the Taylor and Laurent expansions are, see the definitions.

At each point , our previously mentioned function has the following expansions around this point:

a Taylor expansion that converges in the red disk;
a Laurent expansion that converges in the green annulus;
a second Laurent expansion that converges in the cyan disk exterior.

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Taylor (red), Laurent (green) and second Laurent (cyan) expansions around the point

. Click on the grid to move the

.

1 Expansions
- 1.1 Taylor
- 1.2 First Laurent expansion
2 Second Laurent expansion
3 See also

[Edit]Expansions

[Edit]Taylor

The Taylor expansion of the function

at a point is given by

$\sum_{n=0}^\infty ( (1-z_0)^{-n-1} - (2-z_0)^{-n-1} ) (z - z_0)^n$

and converges in the region

$|z-z_0| \lt |1-z_0|, \ |2-z_0|$

that is marked in the diagram as the red disk.

[Edit]First Laurent expansion

In the region

$|1-z_0| \lt |z-z_0| \lt |2-z_0|$

the Laurent series is instead given by

$- \sum_{n=0}^\infty- (2-z_0)^{-n-1} (z - z_0)^n -$ $\sum_{n=-\infty}^{-1} (1-z_0)^{-n-1} (z-z_0)^n$

In the region

$|2-z_0| \lt |z-z_0| \lt |1-z_0|$

the Laurent series is given instead by

$\sum_{n=0}^\infty- (1-z_0)^{-n-1} (z - z_0)^n \plus$ $\sum_{n=-\infty}^{-1} (2-z_0)^{-n-1} (z-z_0)^n$

These two regions are marked on the diagram as the green anulus.

[Edit]Second Laurent expansion

In the region

$|z-z_0| \gt |1-z_0|, \ |2-z_0|$

the Laurent series is given by

$\sum_{n=-\infty}^{-1} ((2-z_0)^{-n-1} - (1-z_0)^{-n-1}) (z-z_0)^n$

This region is marked on the diagram in cyan.

There are some white circles for which none of the three series converge.

[Edit]See also

W8 reading about series.

Categories: Mathematics | Articles | Complex plane