Hi, I am trying to solve a complicated integral for my project and until now I couldn't solve it by Mathematica. Can you help me to find a way to solve it? I send the code that I have written.
Thank you for helping me solve this integral
k = 0.235;
q = 1;
Ki = 910980;
\[Theta]e = 90 Degree; (*\[Theta]e is a variable that takes different values*)
(*Step 2:Define \[Theta]1 and Kf*)
\[Theta]1 = ArcCos[(Ki - k Cos[\[Theta]e])/Kf];
Kf = Sqrt[Ki^2 + k^2 + k *Ki *Cos[\[Theta]e]];
(*Step 3:Define vectors Kf,Ki,and q*)
KfVec = {Kf Sin[\[Theta]1], 0, Kf Cos[\[Theta]1]};
KiVec = {0, 0, Ki};
qVec = KiVec - KfVec;(*Vector q*)
kVec = {k Sin[\[Theta]e], 0, k Cos[\[Theta]e]};
(*Step 4:Calculate dot products*)
KfDotKi = Dot[KfVec, KiVec];
KfDotq = Dot[KfVec, qVec];
qDotKi = Dot[qVec, KiVec];
kDotq = Dot[qVec, kVec];
\[Theta]2 = ArcCos[kDotq/k*q];
q' = k^2 + q^2 - 2*k*q*Cos[\[Theta]2];
q2 = Dot[KfVec, (kVec - qVec)];
(*Step 5:Define A,B,C,D*)
A = (q^2 + \[Lambda]^2) (k^2 + \[Beta]^2/
4) x^2 + (\[Beta] (q^2 + \[Lambda]^2) +
2 \[Lambda] (k^2 + \[Beta]^2/4)) x + (q' + (\[Lambda] + \[Beta]/
2)^2)
B = -2 (I \[Lambda] Kf - KfDotq) (k^2 + \[Beta]^2/4) x^2 -
2 (( \[Beta] (I \[Lambda] Kf - KfDotq) +
I (k^2 + \[Beta]^2/4) Kf)) x -
2 (q2 + I (\[Lambda] + \[Beta]/2) Kf)
cExpr = -2 (k^2 + I \[Beta]/2 k) (q^2 + \[Lambda]^2) x^2 -
2 ((2 \[Lambda] (k^2 + I \[Beta]/2 k) +
I (q^2 + \[Lambda]^2) k) ) x -
2 (k^2 + I (\[Lambda] + \[Beta]/2) k - kDotq)
dExpr = 4 I \[Lambda] (KfDotq) (k^2 + I \[Beta]/2 k) x^2 +
4 (I (k + \[Beta]/2) k Kf + I \[Lambda] KfDotq) x + (2 *
Dot[KfVec, kVec] - k Kf)
simplifiedA = Simplify[A];
simplifiedB = Simplify[B];
simplifiedcExpr = Simplify[cExpr];
simplifieddExpr = Simplify[dExpr];
(*Define the integrand*)
mT = 7*1873;(*amu*)
mp = 16*1873;(*amu*)
\[Mu]pT = (mT*mp)/(mT + mp)
zp = 6;
zT = 2.8683;
\[Alpha]pT = \!\(TraditionalForm\`\[Mu]pT*zp*zT/Kf\)
\[Alpha]T1 = 1*2.8683/0.235simplifiedA = Simplify[A];
simplifiedB = Simplify[B];
simplifiedcExpr = Simplify[cExpr];
simplifieddExpr = Simplify[dExpr];
integrand = (1/ A)*((A/(A + B))^(-I \[Alpha]pT))*((A/(A + cExpr))^(-I \[Alpha]T1))*
Hypergeometric2F1[-I \[Alpha]pT, -I \[Alpha]T1,
1, (B cExpr - A dExpr)/((A + B) (A + cExpr))];
integrandsimple = FullSimplify[integrand];
F[_\[Beta], \[Lambda]] =
Integrate[integrandsimple, {x, 0, \[Infinity]},
Assumptions -> {\[Lambda] > 0, \[Beta] > 0}]
