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
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $7$