Complex Inverse Sine/Examples/2

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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 $k \in \Z$

$\blacksquare$


Sources