Operations with complex numbers

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Problem: Complex numbers \( z_{1}, z_{2} \) are given. Find \[ z_{1}+z_{2}, z_{1}-z_{2}, z_{1} \cdot z_{2}, \frac{z_{1}}{z_{2}}, \] where \( z_{1}=2-3 i, z_{2}=5-4 i \).

10.3.1 Operations with complex numbers

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Problem: Find all the values of the root \[ \sqrt[4]{-\frac{1}{16}} . \]

10.3.2 Operations with complex numbers

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Problem: Draw a domain, given by the inequalities. \[ |z| \leq 1, \quad \arg (z+i)>\frac{\pi}{4} . \]

10.3.3 Operations with complex numbers

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Problem: Separate the real and imaginary parts of the function. \[ f(z)=3 z^{3}+3 z^{2}+2 z+1, \quad z=x+i y . \]

10.3.4 Operations with complex numbers

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Problem: Separate the real and imaginary parts of the function. \[ f(z)=2 z^{3}-3 z^{2}+2 z+1, \quad z=x+i y . \]

10.3.5 Operations with complex numbers

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Problem: Calculate the value of the derivative of the function at the given point \( z_{0}=x_{0}+i y_{0} \). \[ f(z)=2 z^{3}-3 z^{2}+2 z+1, \quad z_{0}=-1+3 i . \]

10.3.6 Operations with complex numbers

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Problem: Find all the values of the root \( \sqrt[5]{32} \).

10.3.8 Operations with complex numbers

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Problem: Solve the equation in complex numbers: \[ \sin z-i \cos z=1 \text {. } \]

10.3.9 Operations with complex numbers

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Problem: A polynomial and one of its roots are given: \[ P(z)=a z^{4}+b z^{3}+c z^{2}+d z+e, \quad z_{1}=\frac{3+3 i \sqrt{3}}{2} . \] 1) Find all the other roots of the polynomial \( P(z) \). 2) Expand \( P(z) \) into irreducible factors in \( \mathbb{C}, \mathbb{R} \). \[ a=1, \quad b=-2, \quad c=8, \quad d=3, \quad e=18 . \]

10.3.11 Operations with complex numbers

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Problem: Find the values of \( \tan \left(\frac{\pi}{6}+\frac{i}{2} \ln 3\right) \).

10.3.7 Operations with complex numbers

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Problem: Find all solutions of the equation: \[ \sin z=\frac{4}{3} i \text {. } \]

10.3.12 Operations with complex numbers

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Problem: A complex number is given: \( z=\frac{-4 \sqrt{3}-4 i}{1+i \sqrt{3}} \). 1) Write the complex number in algebraic, trigonometric and exponential forms. 2) Write the algebraic, trigonometric and exponential forms of the number \( u=z^{n} \), where \( n(-1)^{N}(N+4), N=25 \). 3) Write the exponential and trigonometric forms of the roots \( W_{k}=\sqrt[3]{z}, k=0,1,2 \). 4) Draw \( z \) and \( W_{0}, W_{1}, W_{2} \) on the same complex plane.

10.3.10 Operations with complex numbers

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