Complex Inverse Sine/Examples/2

Examples of Complex Inverse Sine Function

$\sin^{-1} \paren 2 = \dfrac {4 k + 1} 2 \pi - i \ln \paren {2 \pm \sqrt 3}$

for $k \in \Z$.

Proof

By definition of complex inverse sine:

$\sin^{-1} \paren 2 := \set {z \in \C: \sin z = 2}$

Thus:

 $\displaystyle \sin z$ $=$ $\displaystyle 2$ $\displaystyle \leadsto \ \$ $\displaystyle \dfrac {\exp \paren {i z} - \exp \paren {-i z} } {2 i}$ $=$ $\displaystyle 2$ Sine Exponential Formulation $\displaystyle \leadsto \ \$ $\displaystyle \exp \paren {2 i z} - 4 i \exp \paren {i z} - 1$ $=$ $\displaystyle 0$ rearranging $\displaystyle \leadsto \ \$ $\displaystyle \exp \paren {i z}$ $=$ $\displaystyle \dfrac {4 i \pm \sqrt {\paren {-4 i}^2 - 4 \times 1 \times \paren {-1} } } 2$ Quadratic Formula $\displaystyle$ $=$ $\displaystyle \paren {2 \pm \sqrt 3} i$ simplifying $\displaystyle$ $=$ $\displaystyle \paren {2 \pm \sqrt 3} \exp \paren {i \dfrac \pi 2}$ Euler's Formula: $e^{i \pi / 2}$ $\displaystyle \leadsto \ \$ $\displaystyle i z$ $=$ $\displaystyle \ln \paren {2 \pm \sqrt 3} + i \paren {\dfrac \pi 2 + 4 k \pi}$ Definition of Complex Logarithm $\displaystyle \leadsto \ \$ $\displaystyle z$ $=$ $\displaystyle \dfrac {4 k + 1} 2 \pi - i \ln \paren {2 \pm \sqrt 3}$ for $k \in \Z$

$\blacksquare$