Math Problems and solutions
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10.1.31 Integral of a complex variable
1.28 $
10.1.32 Integral of a complex variable
3.07 $
10.1.33 Integral of a complex variable
10.1.34 Integral of a complex variable
2.05 $
10.1.35 Integral of a complex variable
![Problem: Calculate the integral of the complex variable: \[ \int_{L} f(z) d z \text {, where } f(z)=z \bar{z} \text {; } \] \( L \) : the contour of the circular segment with the \( \operatorname{arc} z=e^{i t}, 0 \leq t \leq \pi / 2 \).](/storage/uploads/task_mls/1816/en/10.1.31.png){kind=link}
![Problem: Using the Cauchy theorem for a multiply connected domain and the Cauchy integral formula, calculate the integral: \[ \int_{L} f(z) d z \text {, where } f(z)=\frac{e^{3 z}}{z^{2}+\frac{\pi^{2}}{9}}, \quad L:|z|=2 . \]](/storage/uploads/task_mls/1817/en/10.1.32.png){kind=link}
![Problem: Using the Cauchy theorem for the multiply connected domain and the Cauchy integral formula for the analytic function and its derivatives, calculate the integral: \[ \int_{L} f(z) d z \] where \( f(z)=\frac{2 z^{2}-z+1}{(z-i)^{2}(z+i)}, \quad L:|z+i|=3 \).](/storage/uploads/task_mls/1818/en/10.1.33.png){kind=link}
![Problem: Calculate complex integral using residues: \[ \oint_{C} \frac{1-e^{z}}{z\left(z^{2}-4 z+3\right)} d z, \quad \text { where } \quad C:|z|=4 . \]](/storage/uploads/task_mls/1819/en/10.1.34.png){kind=link}
![Problem: Calculate complex integral using residues: \[ \oint_{C} \frac{\operatorname{ch} z-1}{z^{2}(z-1)^{2}} d z, \quad C:|z|=2 . \]](/storage/uploads/task_mls/1820/en/10.1.35.png){kind=link}