(*MODIFIEDCODE*)
In[]:=
P1[ϕ_]:=
-4
2
2
*
ϕ
2

+
-*
ϕ
2

*-
ϕ
2

+
*
ϕ
2

;P2[ϕ_]:=
-3
2
(
*(ϕ)

-
-*(ϕ)

)(
*(-ϕ)

-
*(ϕ)

);P3[ϕ_]:=
-4
2
2
*
ϕ
2

-
-*
ϕ
2

*-
ϕ
2

-
*
ϕ
2

;
In[]:=
FullSimplify/@{P1[ϕ],P2[ϕ],P3[ϕ]}
Out[]=

4
Cos
ϕ
2

,
2
Sin[ϕ]
2
,
4
Sin
ϕ
2


In[]:=
L[ϕ_,m0_,m1_,m2_]:=
m0
(P1[ϕ])
*
m1
(P2[ϕ])
*
m2
(P3[ϕ])
In[]:=
FullSimplify@L[ϕ,m0,m1,m2]
Out[]=
-m1-2m2
2
m0
4
Cos
ϕ
2

m2
(
2
(-1+Cos[ϕ])
)
m1
(
2
Sin[ϕ]
)
In[]:=
DL[ϕ_,m0_,m1_,m2_]:=FullSimplify@D[FullSimplify@L[ϕ,m0,m1,m2],ϕ]
In[]:=
DL[ϕ,m0,m1,m2]
Out[]=
1-m1-2m2
2
m0
4
Cos
ϕ
2

m2
(
2
(-1+Cos[ϕ])
)
(-m0+m2+(m0+m1+m2)Cos[ϕ])Csc[ϕ]
m1
(
2
Sin[ϕ]
)
In[]:=
Reduce[DL[ϕ,m0,m1,m2]==0&&Element[{m0,m1,m2},PositiveIntegers]&&0<=ϕ<=2π,ϕ]
Out[]=
(m0|m1|m2)∈&&m0≥1&&m1≥1&&m2≥1&&(ϕ4ArcTanRootm1+2m2+(-8m0-6m1-4m2)
2
#1
+(m1+2m2)
4
#1
&,3||ϕ4ArcTanRootm1+2m2+(-8m0-6m1-4m2)
2
#1
+(m1+2m2)
4
#1
&,4||ϕ0||ϕ2π||ϕπ)
In[]:=
(*Definetheparameters*)​​m0=Symbol["m0"];​​m1=Symbol["m1"];​​m2=Symbol["m2"];​​​​(*Solvethepolynomialequationforx*)​​roots=Solve[m1+2m2+(-8m0-6m1-4m2)x^2+(m1+2m2)x^4==0,x];​​​​(*Extractthethirdandfourthrootsfromthesolutions*)​​thirdRoot=x/.roots[[3]];​​fourthRoot=x/.roots[[4]];​​​​(*Compute4*ArcTanofthethirdandfourthroots*)​​resultThirdRoot=4*ArcTan[thirdRoot];​​resultFourthRoot=4*ArcTan[fourthRoot];​​​​(*Displaytheresults*)​​{resultThirdRoot,resultFourthRoot}​​
Out[]=
-4ArcTan
4m0
m1+2m2
+
3m1
m1+2m2
+
2m2
m1+2m2
+
2
2
2
2
m0
+3m0m1+
2
m1
+2m0m2+m1m2
m1+2m2
,4ArcTan
4m0
m1+2m2
+
3m1
m1+2m2
+
2m2
m1+2m2
+
2
2
2
2
m0
+3m0m1+
2
m1
+2m0m2+m1m2
m1+2m2

In[]:=
Limit[DL[ϕ,m0,m1,m2],ϕ->#,Assumptions->Element[{m0,m1,m2},PositiveIntegers]]&/@{0,π,2π}
Out[]=
{0,0,0}