Series with complex terms

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Problem: Determine the circle of convergence of the given series. Find out if the series converges at the given point \( z_{1}, z_{2}, z_{3} \) (if yes, then how: absolutely or conditionally). Make the drawing. \begin{tabular}{|c|c|c|c|} \hline Series & \( z_{1} \) & \( z_{2} \) & \( z_{3} \) \\ \hline\( \sum_{n=1}^{\infty} \frac{(z+1-2 i)^{n}}{2^{n}(n+1) \ln ^{2}(n+1)} \) & 0 & \( 1+2 i \) & -1 \\ \hline \end{tabular}

10.8.1 Series with complex terms

5.1 $

Problem: Find all expansions of the given function \( f(z) \) in powers of \( z-a \) and indicate the domains of these expansions. Remark 1. For a multivalued function \( \sqrt[3]{z} \mathrm{p} \) we consider the branch, which that takes real values on the positive part of the real axis. Remark 2. For the multivalued function \( \arctan z \) we consider the branch that takes real values on the positive part of the real axis. In this case, there is a representation: \[ \begin{array}{l} \arctan z=\int_{0}^{z} \frac{d z}{1+z^{2}}=\frac{\pi}{2}+\int_{\infty}^{z} \frac{d z}{1+z^{2}} \\ f(z)=\frac{z-1}{\sqrt[3]{z^{3}-3 z^{2}+3 z}} \text { in powers of }(z-1) . \end{array} \]

10.8.2 Series with complex terms

3.82 $

Problem: Find all Laurent expansions in powers of \( z \). \[ \frac{3 z-4}{z^{3}-z^{2}-6 z} \text {. } \]

10.8.3 Series with complex terms

3.82 $

Problem: Let's find all Laurent expansions in powers of \( z-z_{0} \). \[ \frac{z+6}{z^{2}-4}, \quad z_{0}=1+2 i . \]

10.8.4 Series with complex terms

3.82 $

Problem: Expand \( f(z) \) into a Laurent series in a neighborhood of the point \( z_{0} \). \[ z^{2} \cos \frac{z}{z+2}, \quad z_{0}=-2 . \]

10.8.5 Series with complex terms

3.82 $

Problem: Expand \( f(z) \) into Laurant series (Taylor) in a given ring or neighborhood of a given point (in the latter case, indicate the area of convergence of the resulting series). \[ f(z)=\frac{z-4}{z^{2}(z+4)}, \quad 1<|z-1|<5 . \]

10.8.6 Series with complex terms

2.55 $

Problem: Expand \( f(z) \) into the Laurant series (Taylor) in the given ring or the neighbourhood of the given point (in this case indicate the domain of convergence of the obtained series). \[ f(z)=\frac{z}{z+1}+e \frac{z}{z-1}, \quad z_{0}=1 . \]

10.8.7 Series with complex terms

2.55 $

Problem: Expand \( f(z) \) into Laurent series (Taylor) into the given ring or neighborhood of the given point (in this case indicate the domain of convergence of the obtained series). \[ f(z)=\frac{1}{(z+1)(z-3)^{2}}, \quad 1<|z|<3 . \]

10.8.8 Series with complex terms

2.55 $

Problem: Expand \( f(z) \) into Laurent series (Taylor) into the given ring or neighborhood of the given point (in this case indicate the domain of convergence of the obtained series). \[ f(z)=\frac{\cos z}{z^{2}}+\sin \frac{1}{z}, \quad z_{0}=0 . \]

10.8.9 Series with complex terms

1.27 $