The series expansion for \(\displaystyle \sqrt{1-x^{2}}=1-\frac{x^{2}}{2}-\frac{x^{4}}{8}-\frac{x^{6}}{16}+O(x^{8})\)
Thus, if we write it as
\(\displaystyle (-(1-x^{2}))^{\frac{1}{2}}=(-1)^{\frac{1}{2}}(1-x^{2})^{\frac{1}{2}}\)
\(\displaystyle =i-\frac{ix^{2}}{2}-\frac{ix^{4}}{8}-\frac{ix^{6}}{16}+O(x^{8})\)
Also, you could write it in terms of e: \(\displaystyle e^{\frac{i\pi}{2}}=i\)
