Solución detallada
Tenemos la ecuación
$$\frac{\log{\left(y \right)}}{2} = - \log{\left(x + 1 \right)}$$
Transpongamos la parte derecha de la ecuación miembro izquierdo de la ecuación con el signo negativo
$$\log{\left(x + 1 \right)} = - \frac{\log{\left(y \right)}}{2}$$
Es la ecuación de la forma:
log(v)=p
Por definición log
v=e^p
entonces
$$x + 1 = e^{\frac{\left(-1\right) \frac{1}{2} \log{\left(y \right)}}{1}}$$
simplificamos
$$x + 1 = \frac{1}{\sqrt{y}}$$
$$x = -1 + \frac{1}{\sqrt{y}}$$
/atan2(im(y), re(y))\ /atan2(im(y), re(y))\
cos|-------------------| I*sin|-------------------|
\ 2 / \ 2 /
x1 = -1 + ------------------------ - --------------------------
_________________ _________________
4 / 2 2 4 / 2 2
\/ im (y) + re (y) \/ im (y) + re (y)
$$x_{1} = -1 - \frac{i \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{2} \right)}}{\sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}} + \frac{\cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{2} \right)}}{\sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}}$$
x1 = -1 - i*sin(atan2(im(y, re(y))/2)/(re(y)^2 + im(y)^2)^(1/4) + cos(atan2(im(y), re(y))/2)/(re(y)^2 + im(y)^2)^(1/4))
Suma y producto de raíces
[src]
/atan2(im(y), re(y))\ /atan2(im(y), re(y))\
cos|-------------------| I*sin|-------------------|
\ 2 / \ 2 /
-1 + ------------------------ - --------------------------
_________________ _________________
4 / 2 2 4 / 2 2
\/ im (y) + re (y) \/ im (y) + re (y)
$$-1 - \frac{i \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{2} \right)}}{\sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}} + \frac{\cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{2} \right)}}{\sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}}$$
/atan2(im(y), re(y))\ /atan2(im(y), re(y))\
cos|-------------------| I*sin|-------------------|
\ 2 / \ 2 /
-1 + ------------------------ - --------------------------
_________________ _________________
4 / 2 2 4 / 2 2
\/ im (y) + re (y) \/ im (y) + re (y)
$$-1 - \frac{i \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{2} \right)}}{\sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}} + \frac{\cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{2} \right)}}{\sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}}$$
/atan2(im(y), re(y))\ /atan2(im(y), re(y))\
cos|-------------------| I*sin|-------------------|
\ 2 / \ 2 /
-1 + ------------------------ - --------------------------
_________________ _________________
4 / 2 2 4 / 2 2
\/ im (y) + re (y) \/ im (y) + re (y)
$$-1 - \frac{i \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{2} \right)}}{\sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}} + \frac{\cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{2} \right)}}{\sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}}$$
-I*atan2(im(y), re(y))
_________________ -----------------------
4 / 2 2 2
- \/ im (y) + re (y) + e
-------------------------------------------------
_________________
4 / 2 2
\/ im (y) + re (y)
$$\frac{- \sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + e^{- \frac{i \operatorname{atan_{2}}{\left(\operatorname{im}{\left(y\right)},\operatorname{re}{\left(y\right)} \right)}}{2}}}{\sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}}$$
(-(im(y)^2 + re(y)^2)^(1/4) + exp(-i*atan2(im(y), re(y))/2))/(im(y)^2 + re(y)^2)^(1/4)