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I'm trying to calculate $\int_0^{\infty } \exp \left(t \left(-x^3+(1+i) x\right)\right) dx$ with mathematica, and different assumptions on t give different results:

f[x_, t_] := Exp[t ((1 + I) x - x^3)];

int[1][t_] = 
 Assuming[t > 0, Integrate[f[x, t], {x, 0, \[Infinity]}]] // 
   FunctionExpand // Simplify

int[2][t_] = 
 Assuming[Re[t] > 0 && Im[t] < 0, 
  Integrate[f[x, t], {x, 0, \[Infinity]}]]

int[3][t_] = 
 Assuming[Re[t] > 0 && Im[t] > Re[t], 
  Integrate[f[x, t], {x, 0, \[Infinity]}]]

int[4][t_] = 
 Assuming[Re[t] > 0 && Im[t] < Re[t], 
  Integrate[f[x, t], {x, 0, \[Infinity]}]]

enter image description here

When I tried to evaluate the integral at t=1-I, the result should be provided by int[2][t]. However, int[1][t] is actually the right result:

NIntegrate[f[x, 1 - I], {x, 0, \[Infinity]}]

Table[int[i][1 - I] // N, {i, 1, 4}]

enter image description here

So is this a bug? My version is"14.0.0 for Mac OS X ARM (64-bit) (December 1, 2023)"

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    $\begingroup$ Have you ever heard about branch cuts and related topics as Riemann surfaces? General expressions int[i] involve expressions like $t^{\frac{1}{3}},\; t^{\frac{1}{2}}$ and so you shouldn't assume these formule are valid in the whole complex plane. Documentation of special functions involve information about branch cuts and you should take care of them. $\endgroup$
    – Artes
    Commented Jun 13 at 13:01
  • $\begingroup$ @Artes: int[2][t_] = Integrate[f[x, t], {x, 0, \[Infinity]}, Assumptions -> Re[t] > 0 && Im[t] < 0, GenerateConditions -> True] // FunctionExpand // Simplify produces the same result in 14.0 on Windows 10. The result without FunctionExpand//Simplify is also incorrect. $\endgroup$
    – user64494
    Commented Jun 13 at 13:39
  • $\begingroup$ As I pointed out int[i] shouldn't be considered as well defined functions in the whole complex plane and your usage of these functions is incorrect since you jump between different branches of int[i]. $\endgroup$
    – Artes
    Commented Jun 13 at 15:03

1 Answer 1

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Confirm this bug in 14.0 on Windows 10. There is a workaround:

f[x_, t_] := Exp[t ((1 + I) x - x^3)];
Integrate[f[x, 1 - I], {x, 0, Infinity}] // FunctionExpand

((-1)^(1/12) \[Pi] AiryAi[(2^(5/6) E^((I \[Pi])/12))/3^(1/3)])/( 2^(1/6) 3^(5/6)) - ((-(1/2))^(1/6) E^(-((I \[Pi])/12)) \[Pi] AiryAi[(2^(5/6) E^((I \[Pi])/12))/3^( 1/3)])/3^( 5/6) + ((-1)^( 1/12) \[Pi] AiryBi[(2^(5/6) E^((I \[Pi])/12))/3^(1/3)])/( 3 2^(1/6) 3^(1/3)) + ((-(1/2))^(1/6) E^(-((I \[Pi])/12)) \[Pi] AiryBi[(2^(5/6) E^((I \[Pi])/12))/3^( 1/3)])/(3 3^( 1/3)) + (1/3 + I/3) HypergeometricPFQ[{1}, {4/3, 5/3}, 4/27 + (4 I)/27]

N[%]

1.8683 +1.28806 I

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