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equation question


Hello.

Consider the following simple examples of FindRoot application.

FindRoot[Sin[x] == 2, {x, I}]
{x -> 1.5707963267948966 + 1.3169578969248168*I}

FindRoot[Sin[x^2] == 2, {x, I + 1}]
{x -> 1.3454777060580754 + 0.4894016047219337*I}

FindRoot[Sin[x^2] == 2, {x, 3*I + 2}]
{x -> 0.3004695589886017 + 2.1914997002654357*I}

Is it possible for FindRoot (or in general in another way) to search
for solutions
in the complex plane in an particular domain e.g. searching in the
domain that
is made by the lines Re[x]=a1, Re[x]=a2 and Im[x]=b1, Im[b]=b2 ?

I really appreciate any assistance.

Regards
Dimitris


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