Which numerical approximation of $\exp(1000 i \sqrt3 )$ is correct?
-0.512025 - 0.85897i or -0.115162 + 0.993347i
Could you argument your answer?
Numerical value of $\exp(1000 i \sqrt3 )$
-1
$\begingroup$
complex-numbers
numerical-methods
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0This is a somewhat odd question, since a simple computer calculation will yield the answer. Are you asking for a way to know the answer without a computer? – 2017-02-04
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0My problem is that, Mathematica returns first answer for N[E^(1000 I Sqrt[3])] and the second for N[(E^(1000 I ))^Sqrt[3]]. At this moment I don't know which one is correct. – 2017-02-04
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0http://imgur.com/3BQwIq6 – 2017-02-04
1 Answers
3
When you have complex numbers, in general, the exponentials are multivalued and $\exp(ab) \ne \exp(a)^b$ when you consider the principal value.
For example, $\exp(2\pi i \sqrt 2) = \cos(2\pi\sqrt 2)+i\sin(2\pi\sqrt 2)$
But $\exp(2 \pi i )^\sqrt 2 = 1$
Hence you should not try to simplify $\exp(1000 i \sqrt 3)$ as $\exp(1000 i)^\sqrt 3$.