Why I am not able to do the integration??

Question

gourav pandey 2021-4-15

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https://ww2.mathworks.cn/matlabcentral/answers/802926-why-i-am-not-able-to-do-the-integration

评论： gourav pandey 2021-4-16

%%%%%%%%% 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)

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David Goodmanson 2021-4-15

编辑：David Goodmanson 2021-4-15

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 2021-4-16