Solve -(x)=1/2(S)^x

$$- ( x ) = \frac { 1 } { 2 } ( S ) ^ { x }$$

Solve for S (complex solution)

$$S=e^{\frac{arg(-x)Im(x)+iarg(-x)Re(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}-\frac{2\pi n_{1}iRe(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}-\frac{2\pi n_{1}Im(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}}\times \left(2|x|\right)^{\frac{Re(x)-iIm(x)}{\left(Re(x)\right)^{2}+\left(Im(x)\right)^{2}}}$$

$$n_{1}\in \mathrm{Z}$$

Solve for S

$$\left\{\begin{matrix}S=\left(-2x\right)^{\frac{1}{x}}\text{, }&\left(Numerator(x)\text{bmod}2=1\text{ and }x>0\text{ and }Denominator(x)\text{bmod}2=1\right)\text{ or }\left(\left(-2x\right)^{\frac{1}{x}}<0\text{ and }x<0\text{ and }Denominator(x)\text{bmod}2=1\right)\text{ or }\left(\left(-2x\right)^{\frac{1}{x}}>0\text{ and }x<0\right)\\S=-\left(-2x\right)^{\frac{1}{x}}\text{, }&\left(Numerator(x)\text{bmod}2=1\text{ and }x>0\text{ and }Numerator(x)\text{bmod}2=0\text{ and }Denominator(x)\text{bmod}2=1\right)\text{ or }\left(x<0\text{ and }\left(-2x\right)^{\frac{1}{x}}>0\text{ and }Numerator(x)\text{bmod}2=0\text{ and }Denominator(x)\text{bmod}2=1\right)\text{ or }\left(x<0\text{ and }\left(-2x\right)^{\frac{1}{x}}<0\text{ and }Numerator(x)\text{bmod}2=0\right)\end{matrix}\right.$$