Analytic functions/Analytic functions

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Problem: Prove that function \( f(z)=u(x, y)+i v(x, y) \) is differentiable on the entire complex plane and find its derivative \( f^{\prime}(z) \). \[ \begin{array}{l} f(z)=\left(3 x^{3}-6 x y^{2}+2 x^{2}-2 y^{2}-3 x+1\right)+ \\ +i\left(6 x^{2} y-2 y^{3}+4 x y-3 y\right) . \end{array} \]

10.6.3 Analytic functions

2.06 $

Problem: Prove that function \( f(z)=u(x, y)+i v(x, y) \) is differentiable on the entire complex plane and find its derivative \( f^{\prime}(z) \). \[ \begin{array}{l} f(z)=\left(2 x^{3}-6 x y^{2}-3 x^{2}+3 y^{2}+x+2\right)+ \\ +i\left(6 x^{2} y-2 y^{3}-6 x y+y\right) . \end{array} \]

10.6.4 Analytic functions

2.06 $

Problem: Prove that there is an analytic function with a given real \( (u(x, y)) \) or imaginary \( (v(x, y)) \) part and restore it under the given condition \( f\left(z_{0}\right)=z_{1} \). \[ \begin{array}{l} v(x, y)=3 x^{2} y-y^{3}-6 x y-3 y+2, \\ f(i)=3-2 i . \end{array} \]

10.6.5 Analytic functions

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Problem: Prove that there is an analytic function with a given real \( (u(x, y)) \) or imaginary \( (v(x, y)) \) part and restore it under the given condition \( f\left(z_{0}\right)=z_{1} \). \[ \begin{array}{l} v(x, y)=3 x^{2} y-y^{3}+6 x y+3 y+2, \\ f(-i)=-3 . \end{array} \]

10.6.6 Analytic functions

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

10.6.7 Analytic functions

1.29 $

Problem: Find the derivative of the function: \[ f(z)=z^{2}+2 i z-1 . \]

10.6.1 Analytic functions

1.55 $

Problem: For what values of the parameter \( a \) is this function the real (imaginary) part of some analytic function. Find this function. \[ u=x^{3}+6 x^{2} y-3 x y^{2}-a y^{3} . \]

10.6.2 Analytic functions

2.58 $

Problem: Using the argument principle, find the number of roots of the given equation that lie in the right halfplane: \[ z^{4}+2 z^{3}+3 z^{2}+z+2=0 . \]

10.6.8 Analytic functions

4.64 $

Problem: Using the argument principle, find the number of roots of the given equation lying in the right halfplane: \( 2 z^{3}-z^{2}-7 z+5=0 \).

10.6.9 Analytic functions

3.87 $

Problem: Using the argument principle, find the number of roots of the given equation, lying in the right halfplane: \( z^{5}+5 z^{4}-5=0 \).

10.6.10 Analytic functions

3.87 $

Problem: Using the argument principle, find the number of roots of the given equation, lying in the right halfplane: \( z^{12}-z+1=0 \).

10.6.11 Analytic functions

3.35 $

Problem: Using Rouche's theorem, find the number of roots of the equation in disk \( D(0 ; R) \). a) \( z^{3}+z+1=0, \quad R=\frac{1}{2} \). b) \( z^{2}-\cos z=0, \quad R=2 \).

10.6.12 Analytic functions

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Problem: Using Rouche's theorem, find the number of roots of the equation in disk \( D(0 ; R) \). a) \( z^{5}+z^{2}+1=0, \quad R=2 \). b) \( z^{4}-\sin z, \quad R=\pi \).

10.6.13 Analytic functions

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Problem: Using Rouche's theorem, find the number of roots of the equation in disk \( D(0, R) \). a) \( z^{8}+6 z+10=0 \), \( R=1 \). b) \( \cosh z=z^{2}-4 z, \quad R=1 \).

10.6.14 Analytic functions

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Problem: Using Rouche's theorem, find the number of roots of the equation in disk \( D(0 ; R) \). a) \( z^{8}-6 z^{6}-z^{3}+2=0, \quad R=1 \). b) \( 2^{z}=4 z, \quad R=1 \).

10.6.15 Analytic functions

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