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], 

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

  • $\begingroup$ I believe the problem is with Re and Conjugate, which are not differentiable. Simpler example of failure: Series[Re@Conjugate[E^x], {x, 0, 2}] $\endgroup$ – Michael E2 May 20 at 18:43
  • $\begingroup$ I applied this solution so that Mathematica knows not to differentiate Conjugate (mathematica.stackexchange.com/a/137443/8459). Can I do the same thing for Re? $\endgroup$ – user85503 May 20 at 18:58

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