Integral of a complex variable

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Problem: Calculate the integral of a function of a complex variable along the given curve. \[ \int_{L} z \operatorname{Re} z^{2} d z, L:\{|z|=1, \operatorname{Im} z \geq 0\} \text {. } \]

10.1.1 Integral of a complex variable

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Problem: Calculate the integral. \[ \oint_{|z+3 i|=2} \frac{4}{(z-1+3 i)^{2}(z-3+3 i)} d z \]

10.1.2 Integral of a complex variable

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Problem: Calculate the integral. \[ \oint_{C} \frac{\sin z d z}{\left(z^{2}-\frac{\pi^{2}}{4}\right)^{2}}, \] \( C:\left|z-\frac{\pi}{2}\right|=1 \).

10.1.3 Integral of a complex variable

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Problem: Calculate the integral of the function of the complex variable \( f(z)=u(x, y)+i v(x, y) \) along the given arc \( \widetilde{A B} \) : \[ \begin{array}{l} f(z)=(2 x y-x-y+1)+i(3 x y+2 x+2 y+1), \\ \widetilde{A B}: y=x^{2}-2 x-3, A(-4 ; 5), B(0 ;-3) . \end{array} \]

10.1.4 Integral of a complex variable

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Problem: Calculate the integral of the function of the complex variable \( f(z)=u(x, y)+i v(x, y) \) along the given arc \( \widetilde{A B} \) : \[ \begin{array}{l} f(z)=(2 x y-x-y+1)+i(3 x y+2 x+2 y+1), \\ \widetilde{A B}: y=x^{2}-2 x-3, A(-1 ; 0), B(0 ;-3) . \end{array} \]

10.1.5 Integral of a complex variable

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Problem: Calculate the integral of the analytic function: \[ \int_{4-3 i}^{1+4 i}\left(3 z^{2}-2 z+1\right) d z . \]

10.1.6 Integral of a complex variable

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Problem: Calculate the integral of the analytic function: \[ \int_{2+2 i}^{-5+2 i}\left(3 z^{2}-4 z-1\right) d z \text {. } \]

10.1.7 Integral of a complex variable

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Problem: Apply Cauchy's integral formula to calculate a closed loop integral. \[ \oint_{|z+1|=2} \frac{z-2}{(z-3)(z+1-i)} d z . \]

10.1.8 Integral of a complex variable

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Problem: Apply Cauchy's integral formula to calculate a closed loop integral. \[ \oint_{|z+2|=\frac{3}{2}} \frac{z-1}{(z+4)(z+2-i)} d z . \]

10.1.9 Integral of a complex variable

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Problem: Calculate the curvilinear integral along the segment \( L: z_{1} z_{2} \), where \[ \int_{L} z \operatorname{Re} z d z, \quad L: z_{1}=1 \text { to } z_{2}=-1 . \]

10.1.10 Integral of a complex variable

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Problem: Calculate the integral of a complex variable along the closed loop: \[ \int_{\left|z-\frac{1}{2}\right|=1} \frac{i z(z-i)}{\sin \pi z} d z . \]

10.1.11 Integral of a complex variable

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Problem: Calculate the integral of a complex variable along a closed loop: \[ \int_{|z|=\frac{1}{3}} \frac{3-2 z+4 z^{4}}{z^{3}} d z . \]

10.1.12 Integral of a complex variable

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Problem: Calculate the integral of a complex variable along a closed loop: \[ \int_{|z|=0.05} \frac{e^{4 z}-1-\sin 4 z}{z^{3} \operatorname{sh} 16 \pi z} d z . \]

10.1.13 Integral of a complex variable

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Problem: Calculate the integral of a complex variable along a closed loop: \[ \int_{|z+5|=2}\left(z \sin \frac{1}{z+5}+\frac{2 \operatorname{ch}\left(\frac{\pi i z}{4}\right)}{(z+4)^{2}(z+2)}\right) d z . \]

10.1.14 Integral of a complex variable

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

10.1.15 Integral of a complex variable

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