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- La ecuación:
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Ecuación x^2+11=0
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Ecuación x^3-x^2-x-1=0
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Ecuación 5x+12=8x+30
-
Ecuación 1/x^2-3/x-4=0
- Expresar {x} en función de y en la ecuación:
- -7*x-13*y=5
- 15*x+1*y=19
- 7*x-1*y=-7
- 20*x+5*y=-3
-
Expresiones idénticas
- log(a^ dos *b^ tres)= cero
- logaritmo de (a al cuadrado multiplicar por b al cubo ) es igual a 0
- logaritmo de (a en el grado dos multiplicar por b en el grado tres) es igual a cero
- log(a2*b3)=0
- loga2*b3=0
- log(a²*b³)=0
- log(a en el grado 2*b en el grado 3)=0
- log(a^2b^3)=0
- log(a2b3)=0
- loga2b3=0
- loga^2b^3=0
- log(a^2*b^3)=O
-
Expresiones con funciones
- Logaritmo log
- log(u)+1/u=c-log(x)
- log((x^2)-x)/log(2)=(1/2)*(log(4)/log(2))
- log(x-7)25=2
- log(2)*|x-3|-log(2)*|x^3-8|=0
- log(x)=log(3*x-2)
log(a^2*b^3)=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Gráfica
Respuesta rápida
[src]
/ / / 2 2 \\ / / 2 2 \\\ / / 2 2 \\ / / 2 2 \\
| | | -2*im(a)*re(a) re (a) im (a) || | | -2*im(a)*re(a) re (a) im (a) ||| | | -2*im(a)*re(a) re (a) im (a) || | | -2*im(a)*re(a) re (a) im (a) ||
| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------||| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------||
| / 2 | | 2 2 2|| / 2 | | 2 2 2||| / 2 | | 2 2 2|| / 2 | | 2 2 2||
| / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ ||| / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ ||
| / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| ___ / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /|| / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| ___ / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /|
| / |------------------ - ------------------| + ------------------ *sin|------------------------------------------------------------------| \/ 3 * / |------------------ - ------------------| + ------------------ *cos|------------------------------------------------------------------|| / |------------------ - ------------------| + ------------------ *cos|------------------------------------------------------------------| \/ 3 * / |------------------ - ------------------| + ------------------ *sin|------------------------------------------------------------------|
| / | 2 2| 4 \ 3 / / | 2 2| 4 \ 3 /| / | 2 2| 4 \ 3 / / | 2 2| 4 \ 3 /
| 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ | 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \
| \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ | \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/
b1 = I*|- ------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------
\ 2 2 / 2 2
$$b_{1} = i \left(- \frac{\sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2}\right) - \frac{\sqrt{3} \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2}$$
/ / / 2 2 \\ / / 2 2 \\\ / / 2 2 \\ / / 2 2 \\
| | | -2*im(a)*re(a) re (a) im (a) || | | -2*im(a)*re(a) re (a) im (a) ||| | | -2*im(a)*re(a) re (a) im (a) || | | -2*im(a)*re(a) re (a) im (a) ||
| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------||| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------||
| / 2 | | 2 2 2|| / 2 | | 2 2 2||| / 2 | | 2 2 2|| / 2 | | 2 2 2||
| / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ ||| / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ ||
| / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| ___ / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /|| / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| ___ / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /|
| / |------------------ - ------------------| + ------------------ *sin|------------------------------------------------------------------| \/ 3 * / |------------------ - ------------------| + ------------------ *cos|------------------------------------------------------------------|| / |------------------ - ------------------| + ------------------ *cos|------------------------------------------------------------------| \/ 3 * / |------------------ - ------------------| + ------------------ *sin|------------------------------------------------------------------|
| / | 2 2| 4 \ 3 / / | 2 2| 4 \ 3 /| / | 2 2| 4 \ 3 / / | 2 2| 4 \ 3 /
| 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ | 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \
| \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ | \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/
b2 = I*|- ------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------
\ 2 2 / 2 2
$$b_{2} = i \left(- \frac{\sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2}\right) + \frac{\sqrt{3} \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2}$$
/ / 2 2 \\ / / 2 2 \\
| | -2*im(a)*re(a) re (a) im (a) || | | -2*im(a)*re(a) re (a) im (a) ||
_________________________________________________________________ |atan2|------------------, ------------------ - ------------------|| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------||
/ 2 | | 2 2 2|| / 2 | | 2 2 2||
/ / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ ||
/ | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /|
b3 = / |------------------ - ------------------| + ------------------ *cos|------------------------------------------------------------------| + I* / |------------------ - ------------------| + ------------------ *sin|------------------------------------------------------------------|
/ | 2 2| 4 \ 3 / / | 2 2| 4 \ 3 /
6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \
\/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/
$$b_{3} = i \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)} + \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}$$
b3 = i*((re(a)^2/(re(a)^2 + im(a)^2)^2 - im(a)^2/(re(a)^2 + im(a)^2)^2)^2 + 4*re(a)^2*im(a)^2/(re(a)^2 + im(a)^2)^4)^(1/6)*sin(atan2(-2*re(a)*im(a)/(re(a)^2 + im(a)^2^2, re(a)^2/(re(a)^2 + im(a)^2)^2 - im(a)^2/(re(a)^2 + im(a)^2)^2)/3) + ((re(a)^2/(re(a)^2 + im(a)^2)^2 - im(a)^2/(re(a)^2 + im(a)^2)^2)^2 + 4*re(a)^2*im(a)^2/(re(a)^2 + im(a)^2)^4)^(1/6)*cos(atan2(-2*re(a)*im(a)/(re(a)^2 + im(a)^2)^2, re(a)^2/(re(a)^2 + im(a)^2)^2 - im(a)^2/(re(a)^2 + im(a)^2)^2)/3))
Suma y producto de raíces
[src]
suma
/ / / 2 2 \\ / / 2 2 \\\ / / 2 2 \\ / / 2 2 \\ / / / 2 2 \\ / / 2 2 \\\ / / 2 2 \\ / / 2 2 \\
| | | -2*im(a)*re(a) re (a) im (a) || | | -2*im(a)*re(a) re (a) im (a) ||| | | -2*im(a)*re(a) re (a) im (a) || | | -2*im(a)*re(a) re (a) im (a) || | | | -2*im(a)*re(a) re (a) im (a) || | | -2*im(a)*re(a) re (a) im (a) ||| | | -2*im(a)*re(a) re (a) im (a) || | | -2*im(a)*re(a) re (a) im (a) ||
| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------||| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|| | _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------||| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------||
| / 2 | | 2 2 2|| / 2 | | 2 2 2||| / 2 | | 2 2 2|| / 2 | | 2 2 2|| | / 2 | | 2 2 2|| / 2 | | 2 2 2||| / 2 | | 2 2 2|| / 2 | | 2 2 2||
| / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ ||| / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || | / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ ||| / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || / / 2 2 \\ / / 2 2 \\
| / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| ___ / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /|| / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| ___ / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| | / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| ___ / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /|| / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| ___ / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| | | -2*im(a)*re(a) re (a) im (a) || | | -2*im(a)*re(a) re (a) im (a) ||
| / |------------------ - ------------------| + ------------------ *sin|------------------------------------------------------------------| \/ 3 * / |------------------ - ------------------| + ------------------ *cos|------------------------------------------------------------------|| / |------------------ - ------------------| + ------------------ *cos|------------------------------------------------------------------| \/ 3 * / |------------------ - ------------------| + ------------------ *sin|------------------------------------------------------------------| | / |------------------ - ------------------| + ------------------ *sin|------------------------------------------------------------------| \/ 3 * / |------------------ - ------------------| + ------------------ *cos|------------------------------------------------------------------|| / |------------------ - ------------------| + ------------------ *cos|------------------------------------------------------------------| \/ 3 * / |------------------ - ------------------| + ------------------ *sin|------------------------------------------------------------------| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------||
| / | 2 2| 4 \ 3 / / | 2 2| 4 \ 3 /| / | 2 2| 4 \ 3 / / | 2 2| 4 \ 3 / | / | 2 2| 4 \ 3 / / | 2 2| 4 \ 3 /| / | 2 2| 4 \ 3 / / | 2 2| 4 \ 3 / / 2 | | 2 2 2|| / 2 | | 2 2 2||
| 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ | 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ | 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ | 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ ||
| \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ | \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ | \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ | \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /|
I*|- ------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------- + I*|- ------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------- + / |------------------ - ------------------| + ------------------ *cos|------------------------------------------------------------------| + I* / |------------------ - ------------------| + ------------------ *sin|------------------------------------------------------------------|
\ 2 2 / 2 2 \ 2 2 / 2 2 / | 2 2| 4 \ 3 / / | 2 2| 4 \ 3 /
6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \
\/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/
$$\left(i \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)} + \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}\right) + \left(\left(i \left(- \frac{\sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2}\right) + \frac{\sqrt{3} \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2}\right) + \left(i \left(- \frac{\sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2}\right) - \frac{\sqrt{3} \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2}\right)\right)$$
=
/ / / 2 2 \\ / / 2 2 \\\ / / / 2 2 \\ / / 2 2 \\\
| | | -2*im(a)*re(a) re (a) im (a) || | | -2*im(a)*re(a) re (a) im (a) ||| | | | -2*im(a)*re(a) re (a) im (a) || | | -2*im(a)*re(a) re (a) im (a) |||
| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------||| | _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|||
| / 2 | | 2 2 2|| / 2 | | 2 2 2||| | / 2 | | 2 2 2|| / 2 | | 2 2 2|||
| / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ ||| | / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ ||| / / 2 2 \\
| / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| ___ / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /|| | / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| ___ / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /|| | | -2*im(a)*re(a) re (a) im (a) ||
| / |------------------ - ------------------| + ------------------ *sin|------------------------------------------------------------------| \/ 3 * / |------------------ - ------------------| + ------------------ *cos|------------------------------------------------------------------|| | / |------------------ - ------------------| + ------------------ *sin|------------------------------------------------------------------| \/ 3 * / |------------------ - ------------------| + ------------------ *cos|------------------------------------------------------------------|| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------||
| / | 2 2| 4 \ 3 / / | 2 2| 4 \ 3 /| | / | 2 2| 4 \ 3 / / | 2 2| 4 \ 3 /| / 2 | | 2 2 2||
| 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ | | 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ | / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ ||
| \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ | | \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ | / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /|
I*|- ------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------| + I*|- ------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------| + I* / |------------------ - ------------------| + ------------------ *sin|------------------------------------------------------------------|
\ 2 2 / \ 2 2 / / | 2 2| 4 \ 3 /
6 / |/ 2 2 \ / 2 2 \ | / 2 2 \
\/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/
$$i \left(- \frac{\sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2}\right) + i \left(- \frac{\sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2}\right) + i \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}$$
producto
/ / / / 2 2 \\ / / 2 2 \\\ / / 2 2 \\ / / 2 2 \\\ / / / / 2 2 \\ / / 2 2 \\\ / / 2 2 \\ / / 2 2 \\\
| | | | -2*im(a)*re(a) re (a) im (a) || | | -2*im(a)*re(a) re (a) im (a) ||| | | -2*im(a)*re(a) re (a) im (a) || | | -2*im(a)*re(a) re (a) im (a) ||| | | | | -2*im(a)*re(a) re (a) im (a) || | | -2*im(a)*re(a) re (a) im (a) ||| | | -2*im(a)*re(a) re (a) im (a) || | | -2*im(a)*re(a) re (a) im (a) |||
| | _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------||| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------||| | | _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------||| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|||
| | / 2 | | 2 2 2|| / 2 | | 2 2 2||| / 2 | | 2 2 2|| / 2 | | 2 2 2||| | | / 2 | | 2 2 2|| / 2 | | 2 2 2||| / 2 | | 2 2 2|| / 2 | | 2 2 2|||
| | / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ ||| / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ ||| | | / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ ||| / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ ||| / / / 2 2 \\ / / 2 2 \\\
| | / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| ___ / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /|| / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| ___ / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /|| | | / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| ___ / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /|| / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| ___ / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /|| | | | -2*im(a)*re(a) re (a) im (a) || | | -2*im(a)*re(a) re (a) im (a) |||
| | / |------------------ - ------------------| + ------------------ *sin|------------------------------------------------------------------| \/ 3 * / |------------------ - ------------------| + ------------------ *cos|------------------------------------------------------------------|| / |------------------ - ------------------| + ------------------ *cos|------------------------------------------------------------------| \/ 3 * / |------------------ - ------------------| + ------------------ *sin|------------------------------------------------------------------|| | | / |------------------ - ------------------| + ------------------ *sin|------------------------------------------------------------------| \/ 3 * / |------------------ - ------------------| + ------------------ *cos|------------------------------------------------------------------|| / |------------------ - ------------------| + ------------------ *cos|------------------------------------------------------------------| \/ 3 * / |------------------ - ------------------| + ------------------ *sin|------------------------------------------------------------------|| | _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|| _________________________________________________________________ |atan2|------------------, ------------------ - ------------------|||
| | / | 2 2| 4 \ 3 / / | 2 2| 4 \ 3 /| / | 2 2| 4 \ 3 / / | 2 2| 4 \ 3 /| | | / | 2 2| 4 \ 3 / / | 2 2| 4 \ 3 /| / | 2 2| 4 \ 3 / / | 2 2| 4 \ 3 /| | / 2 | | 2 2 2|| / 2 | | 2 2 2|||
| | 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ | 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ | | | 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ | 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ | | / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ || / / 2 2 \ 2 2 | |/ 2 2 \ / 2 2 \ / 2 2 \ |||
| | \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ | \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ | | | \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ | \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ | | / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /| / | re (a) im (a) | 4*im (a)*re (a) | \\im (a) + re (a)/ \im (a) + re (a)/ \im (a) + re (a)/ /||
|I*|- ------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------|*|I*|- ------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------|*| / |------------------ - ------------------| + ------------------ *cos|------------------------------------------------------------------| + I* / |------------------ - ------------------| + ------------------ *sin|------------------------------------------------------------------||
\ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 / | / | 2 2| 4 \ 3 / / | 2 2| 4 \ 3 /|
|6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ 6 / |/ 2 2 \ / 2 2 \ | / 2 2 \ |
\\/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ \/ \\im (a) + re (a)/ \im (a) + re (a)/ / \im (a) + re (a)/ /
$$\left(i \left(- \frac{\sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2}\right) + \frac{\sqrt{3} \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2}\right) \left(i \left(- \frac{\sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2}\right) - \frac{\sqrt{3} \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}}{2}\right) \left(i \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)} + \sqrt[6]{\left(\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}}\right)^{2} + \frac{4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{4}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}},\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)^{2}} \right)}}{3} \right)}\right)$$
=
2 2
re (a) - im (a) - 2*I*im(a)*re(a)
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/ 2 2 \ / 4 4 2 2
\im (a) + re (a)/*\/ im (a) + re (a) + 2*im (a)*re (a)
$$\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2} - 2 i \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right) \sqrt{\left(\operatorname{re}{\left(a\right)}\right)^{4} + 2 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{4}}}$$
(re(a)^2 - im(a)^2 - 2*i*im(a)*re(a))/((im(a)^2 + re(a)^2)*sqrt(im(a)^4 + re(a)^4 + 2*im(a)^2*re(a)^2))