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arcsin() return value with tiny imaginary part ?

asked 2020-11-08 13:41:23 +0100

ortollj gravatar image

updated 2020-11-08 13:43:47 +0100

Hi

where does this imaginary small quantity come from ?

SageMath notebook 9.1 W10

show(arcsin(sqrt(2)*sqrt(1/2)).n()) 

show(arcsin(sqrt(2)*sqrt(2^(-1))).n())

show(arcsin(sqrt(2)/sqrt(2)).n(),"\t ok no small imaginary part")
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answered 2020-11-08 18:56:27 +0100

Florentin Jaffredo gravatar image

updated 2020-11-08 18:56:56 +0100

It comes from machine precision. sqrt(2)*sqrt(1/2) is slightly bigger than one. In fact sqrt(2)*sqrt(1/2)-1 returns 2.22044604925031e-16 on my computer, instead of zero. To avoid the issue, you can simply replace .n() by .simplify().n().

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Hi Florentin

but arcsin((2.22044604925031e-16)).n() does not have tiny imaginary part .

ortollj gravatar imageortollj ( 2020-11-08 20:18:40 +0100 )edit

You forgot the +1. Arcsin is a complex (multivalued) function outside $[-1, 1]$. Try to compute arcsin(1+1/1000). (Also It's not accepting floats, since reals values outside the interval define a branch cut, you can find the plots on wikipedia).

Florentin Jaffredo gravatar imageFlorentin Jaffredo ( 2020-11-09 08:56:48 +0100 )edit

Thank you Florentin

ortollj gravatar imageortollj ( 2020-11-10 08:26:32 +0100 )edit

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Asked: 2020-11-08 13:41:23 +0100

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Last updated: Nov 08 '20