다음을 풀어보세요: f^(n)(x)=arcsinx

$$f ^ { ( n ) } ( x ) = \arcsin x$$

f에 대한 해 (complex solution)

$$\left\{\begin{matrix}f=\left(\frac{|\arcsin(x)|}{|x|}\right)^{\frac{Re(n)-iIm(n)}{\left(Re(n)\right)^{2}+\left(Im(n)\right)^{2}}}e^{\frac{\left(Im(n)+iRe(n)\right)arg(\frac{\arcsin(x)}{x})-2\pi n_{1}iRe(n)-2\pi n_{1}Im(n)}{\left(Re(n)\right)^{2}+\left(Im(n)\right)^{2}}}\text{, }n_{1}\in \mathrm{Z}\text{, }&x\neq 0\\f\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

n에 대한 해 (complex solution)

$$\left\{\begin{matrix}n=\frac{\ln(\frac{\arcsin(x)}{x})+2\pi n_{1}i}{\ln(f)}\text{, }n_{1}\in \mathrm{Z}\text{, }&x\neq 0\text{ and }f\neq 1\text{ and }f\neq 0\\n\in \mathrm{C}\text{, }&x=0\text{ or }\left(f=1\text{ and }\frac{\arcsin(x)}{x}=1\text{ and }x\neq 0\right)\end{matrix}\right.$$

f에 대한 해

$$\left\{\begin{matrix}f=\left(\frac{\arcsin(x)}{x}\right)^{\frac{1}{n}}\text{, }&\left(\frac{\arcsin(x)}{x}\right)^{\frac{1}{n}}\neq 0\text{ and }x\leq 1\text{ and }n\neq 0\text{ and }\left(Denominator(n)\text{bmod}2=1\text{ or }\left(\frac{\arcsin(x)}{x}\right)^{\frac{1}{n}}>0\right)\text{ and }\left(x<0\text{ or }\left(\frac{\arcsin(x)}{x}\right)^{\frac{1}{n}}>0\text{ or }Denominator(n)\text{bmod}2=1\right)\text{ and }x\geq -1\text{ and }\left(Denominator(n)\text{bmod}2=1\text{ or }\left(\frac{\arcsin(x)}{x}\right)^{\frac{1}{n}}>0\text{ or }x>0\right)\text{ and }x\neq 0\\f=-\left(\frac{\arcsin(x)}{x}\right)^{\frac{1}{n}}\text{, }&\left(\frac{\arcsin(x)}{x}\right)^{\frac{1}{n}}\neq 0\text{ and }x\neq 0\text{ and }x\geq -1\text{ and }x\leq 1\text{ and }n\neq 0\text{ and }Numerator(n)\text{bmod}2=0\text{ and }\left(Denominator(n)\text{bmod}2=1\text{ or }\left(\frac{\arcsin(x)}{x}\right)^{\frac{1}{n}}<0\right)\text{ and }\left(Denominator(n)\text{bmod}2=1\text{ or }\left(\frac{\arcsin(x)}{x}\right)^{\frac{1}{n}}<0\text{ or }x>0\right)\\f\neq 0\text{, }&|x|\leq 1\text{ and }n=0\text{ and }\frac{\arcsin(x)}{x}=1\text{ and }x\neq 0\\f<0\text{, }&Denominator(n)\text{bmod}2=1\text{ and }x=0\\f=0\text{, }&n>0\text{ and }x=0\\f>0\text{, }&x=0\end{matrix}\right.$$

n에 대한 해

$$\left\{\begin{matrix}n=\log_{f}\left(\frac{\arcsin(x)}{x}\right)\text{, }&x\neq 0\text{ and }x\geq -1\text{ and }x\leq 1\text{ and }f\neq 1\text{ and }f>0\\n\in \mathrm{R}\text{, }&\left(x=0\text{ and }f<0\text{ and }Denominator(n)\text{bmod}2=1\right)\text{ or }\left(x=0\text{ and }f>0\right)\text{ or }\left(|x|\leq 1\text{ and }f=-1\text{ and }\frac{\arcsin(x)}{x}=-1\text{ and }Denominator(n)\text{bmod}2=1\text{ and }Numerator(n)\text{bmod}2=1\text{ and }x\neq 0\right)\text{ or }\left(|x|\leq 1\text{ and }f=1\text{ and }\frac{\arcsin(x)}{x}=1\text{ and }x\neq 0\right)\\n>0\text{, }&x=0\text{ and }f=0\end{matrix}\right.$$