Complex Inverse Trigonometric And Hyperbolic Functions
Let $a$, $b$ be real numbers and $z=a+bi=(a,b)$ a complex number, this calculator evaluates the following six functions.
- the inverse sine of the complex number $z$: \[\sin^{-1}(z)=-i\log\left(iz+\sqrt{1-z^2}\right)\]
- the inverse cosine of the complex number $z$: \[\cos^{-1}(z)=\frac{\pi}{2}+i\log\left(iz+\sqrt{1-z^2}\right)\]
- the inverse tangent of the complex number $z$: \[\tan^{-1}(z)=\frac{i}{2}(\log(1-iz)-\log(iz+1))\]
- the inverse hyperbolic sine of the complex number $z$: \[\sinh^{-1}(z)=\log\left(z+\sqrt{z^2+1}\right)\]
- the inverse hyperbolic cosine of the complex number $z$: \[\cosh^{-1}(z)=\log\left(z+\sqrt{z-1}\sqrt{z+1}\right)\]
- the inverse hyperbolic tangent of the complex number $z$: \[\tanh^{-1}(z)=\frac{1}{2}(\log(1+z)-\log(1-z))\]