Why I am not able to do the integration??

%%%%%%%%% Integration w.r.t 'omega1'

%% Thank you in Advance!!

int((exp(-2*abs(omega1))*((4*2^(1/2)*(2^(1/2)*abs(omega1)^(3/2)*meijerG(1/4, [], [-5/4, 1/4, 5/4], [], -omega1^2/4)*(6352209995579977/36028797018963968 + 6352209995579977i/36028797018963968) + 2^(1/2)*abs(omega1)^(5/2)*meijerG(-1/4, [], [-7/4, -1/4, 7/4], [], -omega1^2/4)*(5081767996463981/144115188075855872 - 5081767996463981i/144115188075855872) + 2^(1/2)*abs(omega1)^(3/2)*conj(meijerG(1/4, [], [-5/4, 1/4, 5/4], [], -omega1^2/4))*(3176104997789989/18014398509481984 - 794026249447497i/4503599627370496) + 2^(1/2)*abs(omega1)^(5/2)*conj(meijerG(-1/4, [], [-7/4, -1/4, 7/4], [], -omega1^2/4))*(5081767996463981/144115188075855872 + 5081767996463983i/144115188075855872)))/abs(omega1) + (13*2^(1/2)*sign(omega1)*(2^(1/2)*abs(omega1)^(1/2)*meijerG(3/4, [], [-3/4, 3/4, 3/4], [], -omega1^2/4)*(1905662998673993/9007199254740992 + 1905662998673993i/9007199254740992) + 2^(1/2)*abs(omega1)^(3/2)*meijerG(1/4, [], [-5/4, 1/4, 5/4], [], -omega1^2/4)*(5081767996463981/72057594037927936 - 5081767996463981i/72057594037927936) - 2^(1/2)*abs(omega1)^(1/2)*conj(meijerG(3/4, [], [-3/4, 3/4, 3/4], [], -omega1^2/4))*(1905662998673993/9007199254740992 - 1905662998673993i/9007199254740992) - 2^(1/2)*abs(omega1)^(3/2)*conj(meijerG(1/4, [], [-5/4, 1/4, 5/4], [], -omega1^2/4))*(5081767996463981/72057594037927936 + 5081767996463981i/72057594037927936)))/10 + (39*2^(1/2)*sign(omega1)*(abs(omega1)^(5/2)*(2^(1/2)*meijerG(-1/4, [], [-7/4, -1/4, 7/4], [], -omega1^2/4)*(5081767996463981/72057594037927936 - 5081767996463981i/72057594037927936) + 2^(1/2)*conj(meijerG(-1/4, [], [-7/4, -1/4, 7/4], [], -omega1^2/4))*(5081767996463981/72057594037927936 + 5081767996463981i/72057594037927936))*1i - 2^(1/2)*abs(omega1)^(3/2)*meijerG(1/4, [], [-5/4, 1/4, 5/4], [], -omega1^2/4)*(6352209995579977/18014398509481984 - 6352209995579977i/18014398509481984) + 2^(1/2)*abs(omega1)^(3/2)*conj(meijerG(1/4, [], [-5/4, 1/4, 5/4], [], -omega1^2/4))*(794026249447497/2251799813685248 + 3176104997789989i/9007199254740992)))/40 - (13*2^(1/2)*sign(omega1)*(2^(1/2)*abs(omega1)^(5/2)*meijerG(-1/4, [], [-7/4, -1/4, 7/4], [], -omega1^2/4)*(2171165525833/35184372088832 + 2171165525833i/35184372088832) + 2^(1/2)*abs(omega1)^(7/2)*meijerG(-3/4, [], [-9/4, -3/4, 9/4], [], -omega1^2/4)*(5081767996463981/576460752303423488 - 5081767996463981i/576460752303423488) - 2^(1/2)*abs(omega1)^(5/2)*conj(meijerG(-1/4, [], [-7/4, -1/4, 7/4], [], -omega1^2/4))*(8893093993811969/144115188075855872 - 4446546996905983i/72057594037927936) - 2^(1/2)*abs(omega1)^(7/2)*conj(meijerG(-3/4, [], [-9/4, -3/4, 9/4], [], -omega1^2/4))*(1270441999115995/144115188075855872 + 2540883998231991i/288230376151711744)))/15 + (13*abs(omega1)^(7/2)*sign(omega1)*(conj(meijerG(1/4, [], [-5/4, 1/4, 5/4], [], -omega1^2/4))*(5081767996463981/72057594037927936 + 5081767996463981i/72057594037927936) - meijerG(1/4, [], [-5/4, 1/4, 5/4], [], -omega1^2/4)*(5081767996463981/72057594037927936 - 5081767996463981i/72057594037927936)))/60 - (2^(1/2)*(2^(1/2)*abs(omega1)^(1/2)*meijerG(3/4, [], [-3/4, 3/4, 3/4], [], -omega1^2/4)*(1905662998673993/9007199254740992 - 1905662998673993i/9007199254740992) - 2^(1/2)*abs(omega1)^(3/2)*meijerG(1/4, [], [-5/4, 1/4, 5/4], [], -omega1^2/4)*(5081767996463981/72057594037927936 + 5081767996463981i/72057594037927936) + 2^(1/2)*abs(omega1)^(1/2)*conj(meijerG(3/4, [], [-3/4, 3/4, 3/4], [], -omega1^2/4))*(1905662998673993/9007199254740992 + 1905662998673993i/9007199254740992) - 2^(1/2)*abs(omega1)^(3/2)*conj(meijerG(1/4, [], [-5/4, 1/4, 5/4], [], -omega1^2/4))*(5081767996463981/72057594037927936 - 5081767996463981i/72057594037927936)))/(2*abs(omega1)) + (2^(1/2)*(2^(1/2)*abs(omega1)^(1/2)*meijerG(3/4, [], [-3/4, 3/4, 3/4], [], -omega1^2/4)*(1905662998673993/4503599627370496 - 1905662998673993i/4503599627370496) - 2^(1/2)*abs(omega1)^(3/2)*meijerG(1/4, [], [-5/4, 1/4, 5/4], [], -omega1^2/4)*(5081767996463981/36028797018963968 + 5081767996463981i/36028797018963968) + 2^(1/2)*abs(omega1)^(1/2)*conj(meijerG(3/4, [], [-3/4, 3/4, 3/4], [], -omega1^2/4))*(1905662998673993/4503599627370496 + 1905662998673993i/4503599627370496) - 2^(1/2)*abs(omega1)^(3/2)*conj(meijerG(1/4, [], [-5/4, 1/4, 5/4], [], -omega1^2/4))*(5081767996463981/36028797018963968 - 5081767996463981i/36028797018963968)))/(4*abs(omega1)) + (4*2^(1/2)*(2^(1/2)*abs(omega1)^(5/2)*meijerG(-1/4, [], [-7/4, -1/4, 7/4], [], -omega1^2/4)*(2171165525833/35184372088832 - 2171165525833i/35184372088832) - 2^(1/2)*abs(omega1)^(7/2)*meijerG(-3/4, [], [-9/4, -3/4, 9/4], [], -omega1^2/4)*(5081767996463981/576460752303423488 + 5081767996463981i/576460752303423488) + 2^(1/2)*abs(omega1)^(5/2)*conj(meijerG(-1/4, [], [-7/4, -1/4, 7/4], [], -omega1^2/4))*(4446546996905983/72057594037927936 + 8893093993811969i/144115188075855872) - 2^(1/2)*abs(omega1)^(7/2)*conj(meijerG(-3/4, [], [-9/4, -3/4, 9/4], [], -omega1^2/4))*(2540883998231991/288230376151711744 - 1270441999115995i/144115188075855872)))/abs(omega1) + (2^(1/2)*abs(omega1)^(3/2)*(2^(1/2)*meijerG(3/4, [], [-3/4, 3/4, 3/4], [], -omega1^2/4)*(5081767996463981/36028797018963968 - 5081767996463981i/36028797018963968) + 2^(1/2)*conj(meijerG(3/4, [], [-3/4, 3/4, 3/4], [], -omega1^2/4))*(5081767996463981/36028797018963968 + 5081767996463981i/36028797018963968)))/8 - (2^(1/2)*abs(omega1)^(3/2)*(2^(1/2)*meijerG(3/4, [], [-3/4, 3/4, 3/4], [], -omega1^2/4)*(5081767996463981/36028797018963968 + 5081767996463981i/36028797018963968) - 2^(1/2)*conj(meijerG(3/4, [], [-3/4, 3/4, 3/4], [], -omega1^2/4))*(5081767996463981/36028797018963968 - 5081767996463981i/36028797018963968))*5i)/8 - (2^(1/2)*abs(omega1)^(5/2)*(2^(1/2)*meijerG(1/4, [], [-5/4, 1/4, 5/4], [], -omega1^2/4)*(5081767996463981/72057594037927936 - 5081767996463981i/72057594037927936) - 2^(1/2)*conj(meijerG(1/4, [], [-5/4, 1/4, 5/4], [], -omega1^2/4))*(5081767996463981/72057594037927936 + 5081767996463981i/72057594037927936))*1i)/4 - (2^(1/2)*abs(omega1)^(5/2)*(2^(1/2)*meijerG(1/4, [], [-5/4, 1/4, 5/4], [], -omega1^2/4)*(5081767996463981/72057594037927936 + 5081767996463981i/72057594037927936) + 2^(1/2)*conj(meijerG(1/4, [], [-5/4, 1/4, 5/4], [], -omega1^2/4))*(5081767996463981/72057594037927936 - 5081767996463981i/72057594037927936)))/4 - (2^(1/2)*abs(omega1)^(7/2)*(2^(1/2)*meijerG(-1/4, [], [-7/4, -1/4, 7/4], [], -omega1^2/4)*(5081767996463981/288230376151711744 - 5081767996463981i/288230376151711744) + 2^(1/2)*conj(meijerG(-1/4, [], [-7/4, -1/4, 7/4], [], -omega1^2/4))*(5081767996463981/288230376151711744 + 5081767996463981i/288230376151711744)))/3 - (2^(1/2)*abs(omega1)^(7/2)*(2^(1/2)*meijerG(-1/4, [], [-7/4, -1/4, 7/4], [], -omega1^2/4)*(5081767996463981/288230376151711744 + 5081767996463981i/288230376151711744) - 2^(1/2)*conj(meijerG(-1/4, [], [-7/4, -1/4, 7/4], [], -omega1^2/4))*(5081767996463981/288230376151711744 - 5081767996463981i/288230376151711744))*1i)/3 + (39*2^(1/2)*sign(omega1)*(2^(1/2)*abs(omega1)^(3/2)*meijerG(1/4, [], [-5/4, 1/4, 5/4], [], -omega1^2/4)*(6352209995579977/36028797018963968 - 6352209995579977i/36028797018963968) - 2^(1/2)*abs(omega1)^(5/2)*meijerG(-1/4, [], [-7/4, -1/4, 7/4], [], -omega1^2/4)*(5081767996463981/144115188075855872 + 5081767996463981i/144115188075855872) - 2^(1/2)*abs(omega1)^(3/2)*conj(meijerG(1/4, [], [-5/4, 1/4, 5/4], [], -omega1^2/4))*(794026249447497/4503599627370496 + 3176104997789989i/18014398509481984) + 2^(1/2)*abs(omega1)^(5/2)*conj(meijerG(-1/4, [], [-7/4, -1/4, 7/4], [], -omega1^2/4))*(5081767996463983/144115188075855872 - 5081767996463981i/144115188075855872)))/20))/omega1^2, omega1, -2, 2)

[David Goodmanson]

Hello gourav,

Not every function has an analytic solution for its integral, and this one with all the Meijer G functions is so complicated that it probably does not. Sometimes we just have to take the bitter with the sweet. Anyway, there are problems for other reasons.

The integrand is complex and appears to have a singularity at the origin. As you approach the origin, the abs value goes like omega1^(-2) for awhile, and then for omega1 less than 1e-7 it goes as omega1^(-4). I don't know whether the latter behavior is due to numerical reasons, but let's say it is. However, the omega1^(-2) behavior by itself is enough to produce an infinite integral unless there is a rapidly oscillating phase factor to level out the singularity, and there is no sign of that happening.

[gourav pandey]

Thank you David.