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I have a real-valued function of four variables, namely $L[\phi, m0, m1, m2]$. I am interested in taking the partial derivative $\partial_{\phi}L$ thereof and setting the partial derivative to zero, that is $\partial_{\phi}L = 0$. Then solving for $\phi \in [0,2\pi]$ in terms of $m0, m1, m2 \in \mathbb{N}_{0}$. I have reached the point in the attached code where I have generated the partial derivative $\partial_{\phi}L$, but I am having difficulty setting the partial derivative to zero and getting an analytic expression for $\phi$ in terms of the other variables $m0,~m1$ and $m2$. Please advise if there is a clear way of doing this analytically and numerically.

    P1[ϕ_] := 
    2^-4 ((E^(I*(ϕ/2)) + E^(-I*(ϕ/2))) (E^(I*(-(ϕ/2))) + E^(I*(ϕ/2))))^2; 
    P2[ϕ_] :=  
     2^-3 (E^(I*(ϕ)) - E^(-I*(ϕ))) (E^(I*(-ϕ)) - E^(I*(ϕ))); 
    P3[ϕ_] := 
     2^-4 ((E^(I*(ϕ/2)) - E^(-I*(ϕ/2))) (E^(I*(-(ϕ/2))) - E^(I*(ϕ/2))))^2;
    
    L[ϕ_, m0_, m1_, m2_] := ((P1[ϕ])^m0)*((P2[ϕ])^m1)*((P3[ϕ])^m2) 
    
    Refine[Simplify[P1[ϕ] + P2[ϕ] + P3[ϕ]], ϕ ∈ Reals] 
    
    Refine[Simplify[L[ϕ, m0, m1, m2]], ϕ ∈ Reals] 
    
    DL[ϕ_, m0_, m1_, m2_] := D[Refine[Simplify[L[ϕ, m0, m1, m2]], ϕ ∈ Reals] , ϕ]
    
    Simplify[DL[ϕ, m0, m1, m2]]

Just a note that the function $L[\phi, m0, m1, m2]$ is defined in terms of probabilities $P1, P2$ and $P3$, hence it has a real-valued output for any $\phi \in \mathbb{R}$.

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  • 3
    $\begingroup$ (1) Use cosines and sines instead of complex exponentials. (2) Optimize the log of L. $\endgroup$ Mar 9 at 15:36
  • $\begingroup$ (3) And link to/from the Wolfram Community cross-post. $\endgroup$ Mar 10 at 22:29

1 Answer 1

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  1. The functions used to define L are themselves undefined here.
  2. Don't begin user-defined names with capital letters.
  3. Use D to partially differentiate and expression with respect to its symbols.
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