(Inverse sine) Developing uV I - Z2 and integrating, show that arcsin z = z + () ~ +

Chapter 15, Problem 15.1.88

(choose chapter or problem)

(Inverse sine) Developing \(1 / \sqrt{1-z^{2}}\) and integrating, show that

\(\arcsin z= z+\left(\frac{1}{2}\right) \frac{z^{3}}{3}+\left(\frac{1 \cdot 3}{2 \cdot 4}\right) \frac{z^{5}}{5}\)

\(+\left(\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\right) \frac{z^{7}}{7}+\cdot \quad(|z|<1)\)

Show that this series represents the principal value of arcsin z (defined in Team Project 30. Sec. 13.7).

Text Transcription:

1/sqrt{1 - z^2}

arcsin z =  z + (1 / 2) z^3 / 3 + ({1 cdot 3 / 2 cdot 4) z^5 / 5

+ (1 cdot 3 cdot 5 / 2 cdot 4 cdot 6) z^7 / 7 + cdot (|z| < 1)