Complex Inverse Sine/Examples/2

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Examples of Complex Inverse Sine Function

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

for $k \in \Z$.


Proof

By definition of complex inverse sine:

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

Thus:

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

$\blacksquare$


Sources

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