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cosh(3z)=i la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
cosh(3*z) = I
$$\cosh{\left(3 z \right)} = i$$
Simplificar
Gráfica
Suma y producto de raíces [src] 
suma
              /   ____________\               /   ____________\             /   ____________\               /   ___________\             /   ___________\               /   ___________\
  5*pi*I      |3 /        ___ |     pi*I      |3 /        ___ |   pi*I      |3 /        ___ |     pi*I      |3 /       ___ |   pi*I      |3 /       ___ |   5*pi*I      |3 /       ___ |
- ------ + log\\/  -1 + \/ 2  / + - ---- + log\\/  -1 + \/ 2  / + ---- + log\\/  -1 + \/ 2  / + - ---- + log\\/  1 + \/ 2  / + ---- + log\\/  1 + \/ 2  / + ------ + log\\/  1 + \/ 2  /
    6                                6                             2                               2                            6                             6
$$\left(\left(\left(\left(\left(\log{\left(\sqrt[3]{-1 + \sqrt{2}} \right)} - \frac{5 i \pi}{6}\right) + \left(\log{\left(\sqrt[3]{-1 + \sqrt{2}} \right)} - \frac{i \pi}{6}\right)\right) + \left(\log{\left(\sqrt[3]{-1 + \sqrt{2}} \right)} + \frac{i \pi}{2}\right)\right) + \left(\log{\left(\sqrt[3]{1 + \sqrt{2}} \right)} - \frac{i \pi}{2}\right)\right) + \left(\log{\left(\sqrt[3]{1 + \sqrt{2}} \right)} + \frac{i \pi}{6}\right)\right) + \left(\log{\left(\sqrt[3]{1 + \sqrt{2}} \right)} + \frac{5 i \pi}{6}\right)$$ 
=
     /   ___________\        /   ____________\
     |3 /       ___ |        |3 /        ___ |
3*log\\/  1 + \/ 2  / + 3*log\\/  -1 + \/ 2  /
$$3 \log{\left(\sqrt[3]{-1 + \sqrt{2}} \right)} + 3 \log{\left(\sqrt[3]{1 + \sqrt{2}} \right)}$$ 
producto
/              /   ____________\\ /            /   ____________\\ /          /   ____________\\ /            /   ___________\\ /          /   ___________\\ /            /   ___________\\
|  5*pi*I      |3 /        ___ || |  pi*I      |3 /        ___ || |pi*I      |3 /        ___ || |  pi*I      |3 /       ___ || |pi*I      |3 /       ___ || |5*pi*I      |3 /       ___ ||
|- ------ + log\\/  -1 + \/ 2  /|*|- ---- + log\\/  -1 + \/ 2  /|*|---- + log\\/  -1 + \/ 2  /|*|- ---- + log\\/  1 + \/ 2  /|*|---- + log\\/  1 + \/ 2  /|*|------ + log\\/  1 + \/ 2  /|
\    6                          / \   6                         / \ 2                         / \   2                        / \ 6                        / \  6                         /
$$\left(\log{\left(\sqrt[3]{-1 + \sqrt{2}} \right)} - \frac{5 i \pi}{6}\right) \left(\log{\left(\sqrt[3]{-1 + \sqrt{2}} \right)} - \frac{i \pi}{6}\right) \left(\log{\left(\sqrt[3]{-1 + \sqrt{2}} \right)} + \frac{i \pi}{2}\right) \left(\log{\left(\sqrt[3]{1 + \sqrt{2}} \right)} - \frac{i \pi}{2}\right) \left(\log{\left(\sqrt[3]{1 + \sqrt{2}} \right)} + \frac{i \pi}{6}\right) \left(\log{\left(\sqrt[3]{1 + \sqrt{2}} \right)} + \frac{5 i \pi}{6}\right)$$ 
=
                                                                                                              /     /      ___\       \ /     /       ___\       \
/     /      ___\       \ /     /      ___\         \ /     /       ___\       \ /     /       ___\         \ |2*log\1 + \/ 2 /       | |2*log\-1 + \/ 2 /       |
\2*log\1 + \/ 2 / + pi*I/*\2*log\1 + \/ 2 / + 5*pi*I/*\2*log\-1 + \/ 2 / - pi*I/*\2*log\-1 + \/ 2 / - 5*pi*I/*|---------------- - pi*I|*|----------------- + pi*I|
                                                                                                              \       3               / \        3               /
------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                               5184
$$\frac{\left(\frac{2 \log{\left(-1 + \sqrt{2} \right)}}{3} + i \pi\right) \left(2 \log{\left(-1 + \sqrt{2} \right)} - 5 i \pi\right) \left(2 \log{\left(-1 + \sqrt{2} \right)} - i \pi\right) \left(\frac{2 \log{\left(1 + \sqrt{2} \right)}}{3} - i \pi\right) \left(2 \log{\left(1 + \sqrt{2} \right)} + i \pi\right) \left(2 \log{\left(1 + \sqrt{2} \right)} + 5 i \pi\right)}{5184}$$ 
(2*log(1 + sqrt(2)) + pi*i)*(2*log(1 + sqrt(2)) + 5*pi*i)*(2*log(-1 + sqrt(2)) - pi*i)*(2*log(-1 + sqrt(2)) - 5*pi*i)*(2*log(1 + sqrt(2))/3 - pi*i)*(2*log(-1 + sqrt(2))/3 + pi*i)/5184
Respuesta rápida [src] 
                   /   ____________\
       5*pi*I      |3 /        ___ |
z1 = - ------ + log\\/  -1 + \/ 2  /
         6
$$z_{1} = \log{\left(\sqrt[3]{-1 + \sqrt{2}} \right)} - \frac{5 i \pi}{6}$$ 
                 /   ____________\
       pi*I      |3 /        ___ |
z2 = - ---- + log\\/  -1 + \/ 2  /
        6
$$z_{2} = \log{\left(\sqrt[3]{-1 + \sqrt{2}} \right)} - \frac{i \pi}{6}$$ 
               /   ____________\
     pi*I      |3 /        ___ |
z3 = ---- + log\\/  -1 + \/ 2  /
      2
$$z_{3} = \log{\left(\sqrt[3]{-1 + \sqrt{2}} \right)} + \frac{i \pi}{2}$$ 
                 /   ___________\
       pi*I      |3 /       ___ |
z4 = - ---- + log\\/  1 + \/ 2  /
        2
$$z_{4} = \log{\left(\sqrt[3]{1 + \sqrt{2}} \right)} - \frac{i \pi}{2}$$ 
               /   ___________\
     pi*I      |3 /       ___ |
z5 = ---- + log\\/  1 + \/ 2  /
      6
$$z_{5} = \log{\left(\sqrt[3]{1 + \sqrt{2}} \right)} + \frac{i \pi}{6}$$ 
                 /   ___________\
     5*pi*I      |3 /       ___ |
z6 = ------ + log\\/  1 + \/ 2  /
       6
$$z_{6} = \log{\left(\sqrt[3]{1 + \sqrt{2}} \right)} + \frac{5 i \pi}{6}$$ 
z6 = log((1 + sqrt(2))^(1/3)) + 5*i*pi/6
Respuesta numérica [src] 
z1 = -0.293791195673181 - 2.61799387799149*i
z2 = -0.293791195673181 - 0.523598775598299*i
z3 = -0.293791195673181 + 1.5707963267949*i
z4 = 0.293791195673181 - 1.5707963267949*i
z5 = 0.293791195673181 + 0.523598775598299*i
z6 = 0.293791195673181 + 2.61799387799149*i
z6 = 0.293791195673181 + 2.61799387799149*i
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