Is sin-1(2i) Equal to 0.5 + 1.31696i?

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The discussion revolves around finding the value of sin-1(2i) using the formula sin(x) = (exp(ix) - exp(-ix)) / 2i. The initial approach involves substituting y = exp(ix) and solving the equation (y + 1/y) / 2i = 2i, which leads to a quadratic function with potentially two solutions for y. The subsequent step of taking the logarithm of y introduces additional complexities due to the multi-valued nature of the logarithm, resulting in multiple values for x. There is also confusion regarding the correct formulation of the equation, with a suggestion that it should be (y - 1/y) / 2i = 2i instead. The thread highlights the complexities of solving for sin-1 in the complex plane.
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I could not found any answer to this question: What is sin-1(2i) equal?
 
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sin(x) = (exp(ix)-exp(-ix))/2i

Firstly let y=exp(ix) and subs in

( y + 1/y)/2i=2i

solve for y, then for x.
 


Since you titled this thread "Multi-Valuedness", note that solving (y+ 1/y)/2i= 2i will involve solving a quadratic function so you may have two values for y. Then solving y= exp(ix) with both values of involves taking the logarithm which adds multiples of 2\pi i.
 


For the two last posts, isn't it supposed to be (y - 1/y)/2i = 2i ?
 


Yes - I can't even seem to follow my own suggestion.
 

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