In my calculations, I found this equation:
Re[-Conjugate[MathieuCPrime[-4, -2 \[Epsilon], 0.7853981633974483`]] Conjugate[MathieuS[-4, -2 \[Epsilon], 0.7853981633974483`]] + MathieuC[-4, -2 \[Epsilon], 0.7853981633974483`] MathieuSPrime[-4, -2 \[Epsilon], 0.7853981633974483`] As $\epsilon$ is small, I expanded it in a series and tried to obtain the coefficient of the $\epsilon^2$ term, but Mathematica gave me this:
In[1083]:= Coefficient[Series[ Re[-Conjugate[ MathieuCPrime[-4, -2 \[Epsilon], 0.7853981633974483`]] Conjugate[ MathieuS[-4, -2 \[Epsilon], 0.7853981633974483`]] + MathieuC[-4, -2 \[Epsilon], 0.7853981633974483`] MathieuSPrime[-4, -2 \[Epsilon], 0.7853981633974483`]], {\[Epsilon], 0, 2}], \[Epsilon], 1] Out[1083]= 0 As getting an output of $0$ doesn't make sense to me, I tried expanding the complex conjugate of a Mathieu function as a series. However, this is where I am having trouble:
In[1078]:= $Assumptions = \[Epsilon] > 0 Out[1078]= \[Epsilon] > 0 In[1079]:= Normal[Series[Refine[Conjugate[MathieuC[1, \[Epsilon], 3.]]], {\[Epsilon], 0, 2}]] Out[1079]= Conjugate[MathieuC[1, \[Epsilon], 3.]] However, if I instead do the series expansion first, then apply the complex conjugate, I find:
In[1080]:= Refine[Conjugate[Normal[Series[MathieuC[1, \[Epsilon], 3.], {\[Epsilon], 0, 2}]]]] Out[1080]= -0.989992 - (1.11614 + 1.31405 I) \[Epsilon] + (0.416372 - 0.0728342 I) \[Epsilon]^2 How do I get Out[1079] to appear the same as Out[1080]? Or more generally, how do I get Mathematica to expand the complex conjugate of a Mathieu function (and their derivatives) as a series?
If it is important, I am using Mathematica 11.2.0.0
ReandConjugate, which are not differentiable. Simpler example of failure:Series[Re@Conjugate[E^x], {x, 0, 2}]$\endgroup$