I'm attempting to Minimize the following function with Minimize:
V0[Lambda_,m2_,k_,phi_] := Lambda + (1/2) m2 phi^2 + (1/4) k phi^4
and used the following code:
Minimize[V0[Lambda,m2,k,phi], phi]
I received the following output, which makes no sense to me since this function is easily minimized:
{Piecewise[{{Lambda, (m2 == 0 && k > 0) || (m2 > 0 && k >= 0) || (m2 == 0 && k == 0 && Lambda > 0) ||
2
4 k Lambda - m2
> (m2 == 0 && k == 0 && Lambda < 0)}, {----------------, m2 < 0 && k > 0},
4 k
> {-Infinity, (m2 == 0 && k < 0) || (m2 > 0 && k < 0) || (m2 < 0 && k <= 0)}}],
2 4
> {phi -> Piecewise[{{Root[2 m2 #1 + k #1 & , 1], m2 > 0 && k >= 0},
2
m2 2 4
> {Root[--- + 2 m2 #1 + k #1 & , 1], m2 < 0 && k > 0},
k
> {0, (m2 == 0 && k > 0) || (m2 == 0 && k == 0 && Lambda == 0) || (m2 == 0 && k == 0 && Lambda > 0) ||
> (m2 == 0 && k == 0 && Lambda < 0)}}, Indeterminate]}}
That being said, this function is readily minimized using:
Solve[D[V0[Lambda,m2,k,phi],phi] == 0, phi]
, which yields the correct solutions. However, I want to use the Minimize function as the "Solve" method fails for more complicated functions that I'm working with. For example:
f[x_] := (x^2/4) (Log[x] - (3/2));
fv[x_] := (x^2/4) (Log[x] - (5/6));
V1[m2_,e_,phi_,k_,xi_,xitilde_,Q_] := Module[{},
Z = e^2 phi^2;
G = m2 + k phi^2;
H = m2 + 3 k phi^2;
Zp = xitilde Z + 1/2 (G + Sqrt[G (G + 4 (xitilde - xi) Z)]);
Zm = xitilde Z + 1/2 (G - Sqrt[G (G + 4 (xitilde - xi) Z)]);
Zg = xitilde Z;
(1/(16 Pi^2)) (f[H] + 3 fv[Z] + f[Zp] + f[Zm] - f[Zg])
];
Any input is appreciated. Thanks!