Get solution: Expand the given function in a Laurent series that converges for 0 < I:: -

Chapter 16, Problem 16.1.8

(choose chapter or problem)

Laurent Series.

(a) Uniqueness. Prove that the Laurent expansion of a given analytic function in a given annulus is unique.

(b) Accumulation of singularities. Does tan (1/z) have a Laurent series that converges in a region 0 < |z| < R? (Give a reason.)

(c) Integrals. Expand the following functions in a Laurent series that converges for |z| > 0:

\(\frac{1}{z^{2}} \int_{0}^{z} \frac{e^{t}-1}{t} d t, \quad \frac{1}{z^{3}} \int_{0}^{z} \frac{\sin t}{t} d t\).

Text Transcription:

1 / z^2  int_{0}^{z} e^t - 1 / t dt,     1 / z^3 int_0^z sin t / t dt

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