We determine the Laurent series for a given function.

Consider the function . This function is analytic in the punctured plane . Using the Maclaurin series for we can create a series representation for as follows: The ratio test shows that this series converges for all except . This type of series is called a Laurent series.

(problem 1a) Find the Laurent series for centered at the origin.
(problem 1b) Find the Laurent series for centered at the origin.
(problem 1c) Find the Laurent series for centered at the origin.
(problem 1d) Find the Laurent series for centered at the origin.

Here is a video solution to problem 1 (all parts):

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Isolated singularities of a function can be classified according to the largest negative power appearing in its Laurent series.

(problem 2a) Classify the isolated singularity at the origin for the function
discontinuity simple pole pole of order 2 pole of order 5 essential singularity
(problem 2b) Classify the isolated singularity at the origin for the function
discontinuity simple pole pole of order 3 pole of order 6 essential singularity
(problem 2c) Classify the isolated singularity at the origin for the function
discontinuity simple pole pole of order 2 pole of order 5 essential singularity
(problem 2d) Classify the isolated singularity at the origin for the function
discontinuity simple pole pole of order 4 pole of order 8 essential singularity
(problem 2e) Classify the isolated singularity at the origin for the function
discontinuity simple pole pole of order 7 pole of order 9 essential singularity

Here is a video solution to problem 2 (all parts):

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The term in a Laurent series is of particular importance in the theory of integration.

(problem 3a) Find Res for the function
(problem 3b) Find Res for the function
(problem 3c) Find Res for the function
(problem 3d) Find Res for the function
(problem 3e) Find Res for the function

Here is a video solution to problem 3 (all parts):

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Consider the function . This function is analytic on . We may recognize this function as the sum of the geometric series This series represents the function inside of the unit disk, but what about the rest of the domain of analyticity of the function? On the set we can write a Laurent series centered at the origin for the function as follows: \begin{align*} \frac{1}{1-z} &= \frac{1}{z\left (\frac{1}{z} - 1\right )} = -\frac{1}{z}\cdot \frac{1}{1-\frac{1}{z}} \\ &= -\frac{1}{z} \sum _{k=0}^\infty \left (\frac{1}{z}\right )^k = -\sum _{k=0}^\infty \frac{1}{z^{k+1}}\\ &= -\frac{1}{z} - \frac{1}{z^2}- \frac{1}{z^3} - \cdots , \, |z|>1 \end{align*}

Hence we have multiple representations of with each representation being valid in a different region. We can generalize this idea.

(problem 4) Find the Laurent series for on the annuli and

Here is a video solution to problem 4:

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2024-09-27 14:05:43