Integral of a complex variable/Integral of a complex…

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Problem: Calculate the integral and characterize all singular points of the functions under the integral sign (including point \( \infty \) ). \[ \int_{\partial \Omega} \frac{z^{3}}{z+1} e^{\frac{1}{z}} d z \text {, where } \Omega=\{|z|<2\} \text {. } \]

10.1.16 Integral of a complex variable

3.06 $

Problem: Calculate the integral and characterize all singular points of the functions under the integral sign (including point \( \infty \) ). \[ \int_{\partial \Omega} z \sin \frac{z+1}{z-1} d z \text {, where } \Omega=\{|z|<2\} . \]

10.1.17 Integral of a complex variable

3.06 $

Problem: Calculate the integral and characterize all singular points of the functions under the integral sign (including point \( \infty \) ). \[ \int_{\partial \Omega} \frac{\sin z}{\left(z^{3}-z\right)(z-i)} d z \text {, where } \Omega=\{|z-1|<1\} \text {. } \]

10.1.18 Integral of a complex variable

3.06 $

Problem: Make sure that the multi-valued analytic functions, under the integral sign, allow the selection of a given domain \( \Omega \) of single-valued branches that satisfy the given conditions, and calculate the integral of this branch. \[ \int_{\partial \Omega} \frac{d z}{(2+\sqrt{z-1}) \sin z}, \] where \( \Omega=\left\{|z|<\frac{1}{2}\right\} \) and \( \left.\sqrt{z-1}\right|_{z=0}=i \).

10.1.19 Integral of a complex variable

3.06 $

Problem: Make sure that the multi-valued analytical functions, standing under the sign of the integral, allow selection of the given domain \( \Omega \) of singlevalued branches that satisfy the given conditions, and calculate the integral of this branch. \[ \int_{\partial \Omega} \frac{d z}{\ln z-3 \pi i}, \] where \( \Omega=\left\{|z+2|<\frac{3}{2}\right\} \) and \( \left.\ln z\right|_{z=-e}=1-\pi i \).

10.1.20 Integral of a complex variable

1.53 $

Problem: Make sure that the multi-valued analytical functions standing under the sign of the integral, allow selection of the given domain \( \Omega \) of single-valued branches that satisfy the given conditions, and calculate the integral of this branch. \[ \int_{\partial \Omega} \frac{z+2}{2 \pi i-\ln (1+z)}, \] where \( \Omega=\left\{|z-2|<\frac{5}{2}\right\} \) and \( \left.\ln (1+z)\right|_{z=e-1}=1-2 \pi i \).

10.1.21 Integral of a complex variable

2.55 $

Problem: Make sure that the multi-valued analytical functions standing under the sign of the integral, allow selection of the given domain \( \Omega \) of singevalued branches, that satisfy the given conditions, and calculate the integral of this branch. \[ \int_{\partial \Omega} \frac{z+2}{z+\ln (1-z)}, \] where \( \Omega=\left\{|z+2|<\frac{5}{2}\right\} \) and \( \left.\ln (1+z)\right|_{z=e-1}=1 \).

10.1.22 Integral of a complex variable

4.34 $

Problem: Appropriately applying the Cauchy residue theorem, calculate the following integrals along the boundaries of infinite domains: \[ \int_{\partial \Omega} \frac{z e^{-z}}{z^{2}-1} d z, \quad \text { where } \Omega=\{\operatorname{Re} z>0\} . \]

10.1.23 Integral of a complex variable

4.34 $

Problem: Appropriately applying the Cauchy residue theorem, calculate the following integrals along the boundaries of infinite domains: \[ \int_{\partial \Omega} \frac{e^{z}}{\sinh 2 z} d z \text {, where } \Omega=\left\{-\frac{\pi}{4}<\operatorname{Im} z<\frac{\pi}{4}\right\} \text {. } \]

10.1.24 Integral of a complex variable

4.34 $

Problem: Appropriately applying the Cauchy residue theorem, calculate the following integrals along the boundaries of infinite domains: \[ \int_{\partial \Omega} \frac{z^{3}}{(z-1)^{2}} e^{-z^{2}} d z \text {, where } \Omega=\left\{-\frac{\pi}{6}<\arg z<\frac{\pi}{6}\right\} \text {. } \]

10.1.25 Integral of a complex variable

5.1 $

Problem: Appropriately applying the Cauchy residue theorem, calculate the following integrals along the boundaries of infinite domains: \[ \int_{\partial \Omega} \frac{d z}{\left(z^{4}+1\right) \sqrt{z^{2}+1}} d z \] where \( \Omega=\left\{\operatorname{Re} z>(\operatorname{Im} z)^{2}\right\} \), and \( \left.\sqrt{z^{2}+1}\right|_{z=0}=1 \).

10.1.26 Integral of a complex variable

7.65 $

Problem: Calculate the integral of the complex variable: \[ \oint_{|z-3|=1} \frac{1-\cos z}{z^{2}-\pi^{2}} d z . \]

10.1.27 Integral of a complex variable

0 $

Problem: Calculate the integral of the complex variable: \[ \oint_{|z|=2} \frac{4 z^{4} \alpha z}{(z+1)^{4}} d z . \]

10.1.28 Integral of a complex variable

1.28 $

Problem: Calculate the integral of the complex variable: \[ \oint_{|z|=\frac{1}{2}} \frac{\operatorname{ch} 3 z-\cos 3 z}{z^{2} \operatorname{sh} 4 z} d z . \]

10.1.29 Integral of a complex variable

3.06 $

Problem: Calculate the integral along the curve \( \mathcal{L} \). \[ \begin{array}{l} \int_{\mathcal{L}} z \operatorname{Re} z^{2} d z, \\ \mathcal{L}=\widetilde{A B}=\left\{z:|z|=2, \quad \operatorname{Im} z \geq 0, \quad z_{A}=2, \quad z_{B}=-2 .\right. \end{array} \]

10.1.30 Integral of a complex variable

1.28 $

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