ComplexExpand in Mathematica 10 doesn't work with series data:
Taylor = Series[Exp[I x], {x, 0, 4}];
ComplexExpand[Taylor\[Conjugate]]
gives
1+I x-x^2/2-(I x^3)/6+x^4/24+O[x]^5
(I is not changed to -I) whereas converting it to a normal expression first results in the correct answer:
ComplexExpand[Normal[Taylor]\[Conjugate]]
1-x^2/2+x^4/24+I (-x+x^3/6)
What's the easiest way to take the complex conjugate of a series expansion? Simply replacing I with -I (and vice versa) doesn't work either (coefficient of x^3 below doesn't change sign):
Taylor/.{I->-I, -I->I}
1-I x-x^2/2 -I x^3/6 + x^4/24 + O[x]^5
Edited after further exploration
Using Refine with assumptions, as suggested in question 77456, rather than ComplexExpand, seems to imply the root source of the problem is MMA not being able to conjugate series data:
Refine[Conjugate@Taylor, Assumptions->x\[Element]Reals]
gives
Conjugate[1+I X-x^2/2+I (x^3/6)+x^4/24+O[x]^5]
while
Refine[Conjugate@Normal@Taylor, Assumptions->x\[Element]Reals]
actually applies the conjugation correctly, giving
1-I X-x^2/2+I (x^3/6)+x^4/24
I tried to think why this might be a 'feature', but couldn't come up with any subtlety that would prevent you from naively conjugating order by order (assuming that the expansion parameter is small and real, of course). Anyone have any ideas?
ComplexExpand
is not documented to work withSeriesData
, like in your example. Also inComplexExpand@Conjugate@funky[x, 0, {I}, 0, 5, 1]
,Conjugate
is simply removed, which is the same behaviour as in your example, only I wrotefunky
instead ofSeriesData
. $\endgroup$