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Expresiones semejantes
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Expresiones con funciones
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sqrt(1-y^2)+sqrt(1-x^2)y=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
________ ________ / 2 / 2 \/ 1 - y + \/ 1 - x *y = 0
$$y \sqrt{1 - x^{2}} + \sqrt{1 - y^{2}} = 0$$
Solución detallada
Tenemos la ecuación
$$y \sqrt{1 - x^{2}} + \sqrt{1 - y^{2}} = 0$$
$$\sqrt{1 - y^{2}} = - y \sqrt{1 - x^{2}}$$
Elevemos las dos partes de la ecuación a la potencia 2
$$1 - y^{2} = y^{2} \left(1 - x^{2}\right)$$
$$1 - y^{2} = - x^{2} y^{2} + y^{2}$$
Transpongamos la parte derecha de la ecuación miembro izquierdo de la ecuación con el signo negativo
$$x^{2} y^{2} - 2 y^{2} + 1 = 0$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = x^{2} - 2$$
$$b = 0$$
$$c = 1$$
, entonces
La ecuación tiene dos raíces.
o
$$y_{1} = \frac{\sqrt{8 - 4 x^{2}}}{2 x^{2} - 4}$$
$$y_{2} = - \frac{\sqrt{8 - 4 x^{2}}}{2 x^{2} - 4}$$
$$y \sqrt{1 - x^{2}} + \sqrt{1 - y^{2}} = 0$$
$$\sqrt{1 - y^{2}} = - y \sqrt{1 - x^{2}}$$
Elevemos las dos partes de la ecuación a la potencia 2
$$1 - y^{2} = y^{2} \left(1 - x^{2}\right)$$
$$1 - y^{2} = - x^{2} y^{2} + y^{2}$$
Transpongamos la parte derecha de la ecuación miembro izquierdo de la ecuación con el signo negativo
$$x^{2} y^{2} - 2 y^{2} + 1 = 0$$
Es la ecuación de la forma
a*y^2 + b*y + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = x^{2} - 2$$
$$b = 0$$
$$c = 1$$
, entonces
D = b^2 - 4 * a * c =
(0)^2 - 4 * (-2 + x^2) * (1) = 8 - 4*x^2
La ecuación tiene dos raíces.
y1 = (-b + sqrt(D)) / (2*a)
y2 = (-b - sqrt(D)) / (2*a)
o
$$y_{1} = \frac{\sqrt{8 - 4 x^{2}}}{2 x^{2} - 4}$$
$$y_{2} = - \frac{\sqrt{8 - 4 x^{2}}}{2 x^{2} - 4}$$
Gráfica
Suma y producto de raíces
[src]
suma
/ / / 2 2 \ \\ / / / 2 2 \ \\ / / / 2 2 \ \\ / / / 2 2 \ \\
| | 2*im(x)*re(x) -\-2 + re (x) - im (x)/ || | | 2*im(x)*re(x) -\-2 + re (x) - im (x)/ || | | 2*im(x)*re(x) -\-2 + re (x) - im (x)/ || | | 2*im(x)*re(x) -\-2 + re (x) - im (x)/ ||
_____________________________________________________________________________________________ |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 + re (x) - im (x)/ 4*im (x)*re (x) | \\-2 + re (x) - im (x)/ + 4*im (x)*re (x) \-2 + re (x) - im (x)/ + 4*im (x)*re (x)/| / \-2 + re (x) - im (x)/ 4*im (x)*re (x) | \\-2 + re (x) - im (x)/ + 4*im (x)*re (x) \-2 + re (x) - im (x)/ + 4*im (x)*re (x)/| / \-2 + re (x) - im (x)/ 4*im (x)*re (x) | \\-2 + re (x) - im (x)/ + 4*im (x)*re (x) \-2 + re (x) - im (x)/ + 4*im (x)*re (x)/| / \-2 + re (x) - im (x)/ 4*im (x)*re (x) | \\-2 + re (x) - im (x)/ + 4*im (x)*re (x) \-2 + re (x) - im (x)/ + 4*im (x)*re (x)/|
- / -------------------------------------------- + -------------------------------------------- *cos|-------------------------------------------------------------------------------------------| - I* / -------------------------------------------- + -------------------------------------------- *sin|-------------------------------------------------------------------------------------------| + / -------------------------------------------- + -------------------------------------------- *cos|-------------------------------------------------------------------------------------------| + I* / -------------------------------------------- + -------------------------------------------- *sin|-------------------------------------------------------------------------------------------|
/ 2 2 \ 2 / / 2 2 \ 2 / / 2 2 \ 2 / / 2 2 \ 2 /
/ / 2 \ / 2 \ / / 2 \ / 2 \ / / 2 \ / 2 \ / / 2 \ / 2 \
4 / |/ 2 2 \ 2 2 | |/ 2 2 \ 2 2 | 4 / |/ 2 2 \ 2 2 | |/ 2 2 \ 2 2 | 4 / |/ 2 2 \ 2 2 | |/ 2 2 \ 2 2 | 4 / |/ 2 2 \ 2 2 | |/ 2 2 \ 2 2 |
\/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/
$$\left(- i \sqrt[4]{\frac{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)} - \sqrt[4]{\frac{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)}\right) + \left(i \sqrt[4]{\frac{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)} + \sqrt[4]{\frac{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)}\right)$$
=
0
$$0$$
producto
/ / / / 2 2 \ \\ / / / 2 2 \ \\\ / / / / 2 2 \ \\ / / / 2 2 \ \\\ | | | 2*im(x)*re(x) -\-2 + re (x) - im (x)/ || | | 2*im(x)*re(x) -\-2 + re (x) - im (x)/ ||| | | | 2*im(x)*re(x) -\-2 + re (x) - im (x)/ || | | 2*im(x)*re(x) -\-2 + re (x) - im (x)/ ||| | _____________________________________________________________________________________________ |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 + re (x) - im (x)/ 4*im (x)*re (x) | \\-2 + re (x) - im (x)/ + 4*im (x)*re (x) \-2 + re (x) - im (x)/ + 4*im (x)*re (x)/| / \-2 + re (x) - im (x)/ 4*im (x)*re (x) | \\-2 + re (x) - im (x)/ + 4*im (x)*re (x) \-2 + re (x) - im (x)/ + 4*im (x)*re (x)/|| | / \-2 + re (x) - im (x)/ 4*im (x)*re (x) | \\-2 + re (x) - im (x)/ + 4*im (x)*re (x) \-2 + re (x) - im (x)/ + 4*im (x)*re (x)/| / \-2 + re (x) - im (x)/ 4*im (x)*re (x) | \\-2 + re (x) - im (x)/ + 4*im (x)*re (x) \-2 + re (x) - im (x)/ + 4*im (x)*re (x)/|| |- / -------------------------------------------- + -------------------------------------------- *cos|-------------------------------------------------------------------------------------------| - I* / -------------------------------------------- + -------------------------------------------- *sin|-------------------------------------------------------------------------------------------||*| / -------------------------------------------- + -------------------------------------------- *cos|-------------------------------------------------------------------------------------------| + I* / -------------------------------------------- + -------------------------------------------- *sin|-------------------------------------------------------------------------------------------|| | / 2 2 \ 2 / / 2 2 \ 2 /| | / 2 2 \ 2 / / 2 2 \ 2 /| | / / 2 \ / 2 \ / / 2 \ / 2 \ | | / / 2 \ / 2 \ / / 2 \ / 2 \ | | 4 / |/ 2 2 \ 2 2 | |/ 2 2 \ 2 2 | 4 / |/ 2 2 \ 2 2 | |/ 2 2 \ 2 2 | | |4 / |/ 2 2 \ 2 2 | |/ 2 2 \ 2 2 | 4 / |/ 2 2 \ 2 2 | |/ 2 2 \ 2 2 | | \ \/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ / \\/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ /
$$\left(- i \sqrt[4]{\frac{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)} - \sqrt[4]{\frac{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)}\right) \left(i \sqrt[4]{\frac{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)} + \sqrt[4]{\frac{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)}\right)$$
=
/ 2 2 \
| 2*im(x)*re(x) 2 + im (x) - re (x) |
I*atan2|----------------------------------------, ----------------------------------------|
| 2 2 |
|/ 2 2 \ 2 2 / 2 2 \ 2 2 |
\\2 + im (x) - re (x)/ + 4*im (x)*re (x) \2 + im (x) - re (x)/ + 4*im (x)*re (x)/
-e
----------------------------------------------------------------------------------------------
__________________________________________
/ 2
/ / 2 2 \ 2 2
\/ \2 + im (x) - re (x)/ + 4*im (x)*re (x)
$$- \frac{e^{i \operatorname{atan_{2}}{\left(\frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},\frac{- \left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2}{\left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}}{\sqrt{\left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}}$$
-exp(i*atan2(2*im(x)*re(x)/((2 + im(x)^2 - re(x)^2)^2 + 4*im(x)^2*re(x)^2), (2 + im(x)^2 - re(x)^2)/((2 + im(x)^2 - re(x)^2)^2 + 4*im(x)^2*re(x)^2)))/sqrt((2 + im(x)^2 - re(x)^2)^2 + 4*im(x)^2*re(x)^2)
Respuesta rápida
[src]
/ / / 2 2 \ \\ / / / 2 2 \ \\
| | 2*im(x)*re(x) -\-2 + re (x) - im (x)/ || | | 2*im(x)*re(x) -\-2 + re (x) - im (x)/ ||
_____________________________________________________________________________________________ |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 + re (x) - im (x)/ 4*im (x)*re (x) | \\-2 + re (x) - im (x)/ + 4*im (x)*re (x) \-2 + re (x) - im (x)/ + 4*im (x)*re (x)/| / \-2 + re (x) - im (x)/ 4*im (x)*re (x) | \\-2 + re (x) - im (x)/ + 4*im (x)*re (x) \-2 + re (x) - im (x)/ + 4*im (x)*re (x)/|
y1 = - / -------------------------------------------- + -------------------------------------------- *cos|-------------------------------------------------------------------------------------------| - I* / -------------------------------------------- + -------------------------------------------- *sin|-------------------------------------------------------------------------------------------|
/ 2 2 \ 2 / / 2 2 \ 2 /
/ / 2 \ / 2 \ / / 2 \ / 2 \
4 / |/ 2 2 \ 2 2 | |/ 2 2 \ 2 2 | 4 / |/ 2 2 \ 2 2 | |/ 2 2 \ 2 2 |
\/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/
$$y_{1} = - i \sqrt[4]{\frac{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)} - \sqrt[4]{\frac{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)}$$
/ / / 2 2 \ \\ / / / 2 2 \ \\
| | 2*im(x)*re(x) -\-2 + re (x) - im (x)/ || | | 2*im(x)*re(x) -\-2 + re (x) - im (x)/ ||
_____________________________________________________________________________________________ |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 + re (x) - im (x)/ 4*im (x)*re (x) | \\-2 + re (x) - im (x)/ + 4*im (x)*re (x) \-2 + re (x) - im (x)/ + 4*im (x)*re (x)/| / \-2 + re (x) - im (x)/ 4*im (x)*re (x) | \\-2 + re (x) - im (x)/ + 4*im (x)*re (x) \-2 + re (x) - im (x)/ + 4*im (x)*re (x)/|
y2 = / -------------------------------------------- + -------------------------------------------- *cos|-------------------------------------------------------------------------------------------| + I* / -------------------------------------------- + -------------------------------------------- *sin|-------------------------------------------------------------------------------------------|
/ 2 2 \ 2 / / 2 2 \ 2 /
/ / 2 \ / 2 \ / / 2 \ / 2 \
4 / |/ 2 2 \ 2 2 | |/ 2 2 \ 2 2 | 4 / |/ 2 2 \ 2 2 | |/ 2 2 \ 2 2 |
\/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/ \\-2 + re (x) - im (x)/ + 4*im (x)*re (x)/
$$y_{2} = i \sqrt[4]{\frac{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)} + \sqrt[4]{\frac{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} + \frac{4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}}{\left(\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\frac{2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)}}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}},- \frac{\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2}{\left(\left(\operatorname{re}{\left(x\right)}\right)^{2} - \left(\operatorname{im}{\left(x\right)}\right)^{2} - 2\right)^{2} + 4 \left(\operatorname{re}{\left(x\right)}\right)^{2} \left(\operatorname{im}{\left(x\right)}\right)^{2}} \right)}}{2} \right)}$$
y2 = i*((re(x)^2 - im(x)^2 - 2)^2/((re(x)^2 - im(x)^2 - 2)^2 + 4*re(x)^2*im(x)^2)^2 + 4*re(x)^2*im(x)^2/((re(x)^2 - im(x)^2 - 2)^2 + 4*re(x)^2*im(x)^2)^2)^(1/4)*sin(atan2(2*re(x)*im(x)/((re(x)^2 - im(x)^2 - 2)^2 + 4*re(x)^2*im(x)^2, -(re(x)^2 - im(x)^2 - 2)/((re(x)^2 - im(x)^2 - 2)^2 + 4*re(x)^2*im(x)^2))/2) + ((re(x)^2 - im(x)^2 - 2)^2/((re(x)^2 - im(x)^2 - 2)^2 + 4*re(x)^2*im(x)^2)^2 + 4*re(x)^2*im(x)^2/((re(x)^2 - im(x)^2 - 2)^2 + 4*re(x)^2*im(x)^2)^2)^(1/4)*cos(atan2(2*re(x)*im(x)/((re(x)^2 - im(x)^2 - 2)^2 + 4*re(x)^2*im(x)^2), -(re(x)^2 - im(x)^2 - 2)/((re(x)^2 - im(x)^2 - 2)^2 + 4*re(x)^2*im(x)^2))/2))